A COUllSK OF
MODERN ANALYSIS -
AN INTRODUCTION TO THK GENERAL THEORY OF
INFINITE SERIES AND OF ANALYTIC FUNCTIONS;
WITH AN ACCOUNT OF THE PRINCIPAL
TRANSCENDENTAL FUNCTIONS
BY
E. T. WHITTAKEK, M.A.
FELLOW AND LECTURER OF TRINITY COLLEGE, CAMBRIDGE
O*" THE
^t^lVERsiTY )
CAMBRIDGE:
AT THE UNIVERSITY PRESS.
1902
^
SENERAL
Cambtilige :
PIlIXTEIl BT J. AND C. F. CLAY, AT THE UNIVERSITY PBESS.
PREFACE.
'I'hk first half of tlii.s book contains an account of those methods and lirocesses of higher mathematical analysis, which seem to be of greatest importance at the present time ; as will be seen by a glance at the table iif contents, it is chiefly concerned with the properties of infinite series and complex, integrals, and their applications to the analytical expression of functions. A discussion of inKnite determinants and of asymptotic expansions has been inciudoil, as it seemed to be called for by the value of these theories in connexion with linear differential equations and astroudniy.
In the second half of the book, the methods of the earlier part are apjiiied in order to furnish the theory of the principal functions of analysis — the Gamma, Legendre, Bes.sel, Hypergeometric, and Elliptic Functions. An account has also been given of those solutions of the partial differential equations of mathematical physics, which can be constructed by the help of these functions.
My grateful thanks are due to two members of Trinity College, Rev. E. M. Iladford, M.A. (now (if St John's School, Leatherhead), and Mr J. E. Wright, B.A., who with great kindness and care have read the proof-sheets ; and to Professor Forsyth, for many helpful consultations during the progress of the work. My great indebtedness to Dr Hobson's memoirs on Legendre functions must be specially mentioned here ; and I must thank the staff of the University Press for their excellent co- operation in the production of the volume.
E. T. WHITTAKER.
Cambridge,
1902 Auymt f)
CONTENTS.
PART I. THE PKUUESSES OF ANALYSIS.
CHAPTER I. COMPLEX NUMBERS.
SECTION PAGE
1. Renl numbers 3
2. l'<)iii])lex numbers 4
3. Tlie iiioiiuUi.s of a comjilex quuiitity ........ 5
4. 'I'lie geometrical interpretivtioii of complex Muiiil)ors ► 6
Mi.soEi.i..\XEors Ex.\5iri,i:s 7
\y
CHAPTER II.
THE THEORY OF ABSOLUTE CONVERGENCE.
5. The limit of a se<}ueiice of quautitie.s 8
6. The necessjiry and sufficient conditions for the existence of a limit . . 8
7. Convergence of an infinite serie.s ......... 10
8. Absolute convergence and .semi-convergence' 12
9. The geometric series, and the series 2rt~" 13
10. The comparison-tiieorem ........... 14
IL Discussion of a si^ecial series of importance ....... 16
12. A convergency-test which depends on the ratio of the successive terms
of a series ............. 17
13. A general theorem on those series for which Limit ( "*' ) is 1 . . . 18
„=. V a, )
14. Convergence of the hypergeometric series 20
15. Effect of changing the order of the terms in a serias ..... 21
16. The fundamentid property of absolutely convergent series .... 22
17. Kiemann's theorem on semi-convergent series ....... ii
18. ("auciiy's theorem on the multi[ilication of absolutely convergent scries . . i\
19. Merteiis' theorem on the niultiplic;ition of a .semi-convergent series by an
absolutely convergent series ......... 25
20. Aljel's result on the multiplication of series 26
21. Power-series 28
CONTENTS.
BKOTION
22. ( "oiivcrgpiicc of aeries derived from a ]iinver-sericH
23. InliniU- iirodiuts ..........
24. Some e.vjimplos of intinitc i)n>diut.s ......
25. Caiicliy's tlioorcm on products wliicli arc not absolutely convergent
26. Infinite determinants .........
27. Convergence of an infinite determinant
28. I'ersistence of convergence when the elements are changed .
Ml.SCELLANEOUS EXAMPLES
PAOK
30
:ii
32 34 35 36 37 37
CHAP'JER III.
THK FU.XDAMEXTAL I'KOl'KRTIES OF ANALYTIC FUNCTIONS; TAYLOR'S, LAURENT'S, AND LIOUVILLE'S THEOREMS.
The dependence of fine complex number on another
a c intei;
29.
30. Continuity
31. Definite integrals .......
32. Limit to the value of a definite integral
33. I'i'oiierty of the elementary' functions
34. ( )ccasional failure of the property ; singularities .
35. The analytic function ......
36. C'auchy's theorem on the integral of a function round
37. The value of a function at a point, expivs.sed as an
a contour enclo.sing the point ....
38. The higher derivates
39. Taylor's theorem
40. Forms of the remainder in Taylor's series .
41. The process of continuation .....
42. The identity of a function .....
43. Laurent's theorem .......
44. The nature of the singularities of a one-valued function
45. The jioint at infinity ....
46. Many-\alued functions ....
47. Liouville's theorem
48. Functions with no es.seutial singularities
MiSOELLANEODS EXAMPLES
ontour al t;
liken
40 41 42 44 44 45 45 47
50 51 54 56 57 59 60 63 64 66 69 69 70
•\ CHAPTER IV.
THE UNIFORM CONVERGENCE OF INFINITE SERIES.
49. Uniform convergence 73
50. Connexion of discontinuity with non->miform convergence .... 76
51. Distinction between absolute and uniform convergence .... 77
52. Condition for uniform convergence ........ 78
53. Integration of infinite series 78
54. Differentiation of infinite series 81
55. TTniform convergence of power-seiies 81
Miscellaneous Examples 82
CONTKNTS.
XI
CHAPTER V.
TIIK TIlKdliV OF KKSIIU'ES; Al'l'MCATIOX TO THK EVALUATION OK UKAl- DKKIXITK IXTE(iKAliS.
SKCTION VMIF.
Hosiiliio.s .............. H3
Evaluation i)f rwil definite integrals H4
Evaluation of the definite integral of a rational fuiution .... 91
Cauchy's integral 92
Tlie number of roots of an e^iuation contained within a contour . . 92
Connexion lietween the zero.s of a function and the zeros of its derivate . 93
Miscellaneous E.xamti.e.s 94
CHAPTER VI.
THE- EXPANSION OF FUNCTIONS IN INFINITE SERIES.
DarlH)u.\'s formula 96
The Bernoidlian numbers and the Hernoidlian ]inlynomials ... 97
The Maclaurin-Hernoullian exi)ansion ........ 99
Burmami'.s theorem ............ 100
Teixefra's extended form of Burmann's theorem 102
Evaluation of the coefficients .......... 103
Expansion of a function of a root of an equation, in terms of a parameter
occurring in the equation 105
Lagi'ange's theorem ............ 106
Kouclie's extension of Lagrange's theorem ....... 108
Teixeira's generalisation of Lagrange's theorem ...... 109
Liplace's extension of Lagrange's theorem 109
A further generalisation of Taylor's theorem . . . . . . 110
The expansion of a function .as a series of rational functions . . . Ill
Expansion of a function as an infinite product . . . . . . 114
Expansion of a periodic function iis a series of cotangents . . . . 116
Expansion in inver.se factorials 117
Miscellaneous Examples 119
CHAPTER VII.
FOURIER SERIES.
Definition of Fourier series ; nature of the region within which a Fourier
series converges ............ 127
Values of the coefficients in terms of the sum of a Fourier series, when the
.series converges at all points in a Ijelt of finite lireadth in the z-plane . . 130
Fourier's theorem 131
The representation of a function liy Fourier .series for ranges other than
0 to 2»r 137
The sine and cosine .series 138
Alternative proof of Fourier's theorem 140
Natui'e of the convergence of a Fourier series ...... 147
Determination of points of discontinuity 151
The uniqueness of the Fourier expansion 152
Miscellaneous Exami-les 157
CONTENTS.
CHAPTER VIII.
ASYMPTOTIC EXPANSIONS,
SECTION
87. Simple exauiplo nf :tii a,syiiii)tutiu expari.sioii
88. DotiiiitiDii of an a.symptotic expansion .
89. .\nutlior example of an a.syiniitotic expan.'^ion
90. Multiplication of asymptotic expansions
91. Integration of asymptotic expansions
92. Uniquene.ss of an asymptotic expansion Miscellaneous Examplfjs ....
PAQE
163
164 165 167 168 168 169
PART II. TRANSCJENDENTAL B^UNCTK^NS.
OHAPTER IX.
THE GAMMA-FUNaTION.
93. Definition of the (lamina-function, Euler'.s foiTii . . . . \ . 173
94. The Weierstrassian form for the Oainma-function . . . . . 174
95. The ilift'erence-equatioii .sati.sfied by the (iamiua-function . . . . 176
96. Evaluation of a general class of infinite products . . . . . 177
97. Connexion between the Gamma-function and the circular functions . 179
98. The multiplication-theorem of Gauss and Legendre 179
9S. Expansions for the logarithmic derivates of the Gamma-function . . 180
100. Heine's expre.ssion of r(s) as a contour-integral. ..... 181
101. Expression of r (z) as a definite integral, whose path of integration
is real 183
102. Extension of the definite-integral expression to the case in wliich the
argument of the Gamma- function is negative 184
103. Gauss' expression of the logarithmic derivate of the (lamuia-function as
a definite integral ........... 185
104. I'inet's expres.sion of logr(^) in terms of a definite integral . . . 186
105. The Euleri.an integral of the fir.st kind 189
106. Expression of the Eulerian integral of the first kind in terms of Gamma-
functions ............. 190
107. Evaluation of trigonometric integrals in terms of the Gamma-function . 191
108. Dirichjet's multiple integrals 191
109. The asymptotic expansion of the logarithm of the Gamma-function (Stirling's
.series) 193
110. Asymptotic expansion of the Gniouia-fuiiction ...... 194
Miscellaneous E.xamples 195
CONTENTS.
XUl
CHAPTER X.
LEGEND RE FUNCTIONS.
ION
Detiiiitioii of Legeiidre iwlynoniials
SihliiHi's integral for P„{s)
Koiliinuos' foiniul.i for the Legemire polynomials
Logi'nilro'.s ilitt'enMitial c<iuation ......
The integral-iiroiieitie.s of the Legendre polynomial.^ .
Logendro functions
'Pile recurrence-formulae
I'^valuation of the integral-expres.sion for /'„(-), as a power-ser
Laplace's integral-expression for /'„(;) ....
The Mehlcr-Diriclilet definite integral for I\{z).
Expansion of /'„(i) as a series of powers of 1/z . The Legiindre functions of the second kind
Expansion of <^„(z) as a power-series
The reeurreiK-e-fornudae for tlie TiCgeiidn' function of the second
Laplace's integral for the Legendre function of the .second kind Relation between Pn{z) and (^ni-)^ when n is an integer Expansion of (<-.i-)"' jis a series of Legendre iiolynomials Neumann's expansion of an arbitrary function as a series o polynonnals ..........
The a.s.sociated functions /'„'"(-) '"I'l Vh"'(2) The defniite integrals of the associated Legendre functions Expansion of /'„'"(;) as a definite integral of Laplace's type Alternative expression of /"„"' (z) as a definite integral of Laplace'
The function C/ (j)
Miscellaneous Ex^vmples
kinil
Legendr
s type
PAOB
204 20,5 206 206 207 208 210 213 215 218 220 221 222 224 225 226 228
230 231 232 233 234 235 236
CHAPTER XI.
HYPERGEOMETRIC FUNCTIONS.
The hyjxirgeometric series 240
Value of the series F (a, i, c, 1) 241
The differential equation satisfied by the hypergeometric series . . 242
The differential equation of the general hypergeometric function . . 242
The Legendre functions ius a particular ease of the hypergeometric function . 245
Transformations of the general hypergeometric finiction .... 246 The twenty-four pjirticular solutions of the hy[)crgeometric difi'erential
equation ............. 249
Relations between the particular .solutions of the hy|iergeoinetric differential
equation ............. 251
Solution of the general hypergeometric differential ecpiation by a definite
integral ............. 253
Determination of the integral which represents y"! 257
Evaluation of a doulile-contour integral 259
Relations between contiguous hypergeometric functions .... 260
Miscellaneous Examples 263
XI V CONTENTS.
CHAPTER XII.
BESSEL FUNCTIONS.
SKCTION PAGE
146. The Bes.sel coefficicuts 266
147. Bes.sers dift'erential equation 268
148. Bcssel'.-j equation as a case of the liypergeonietric- equation . . . 269
149. The general .solution of Bessel's equation by Be.s.sel functions whose order is
not necessarily an integer .......... 272
150. The recurrence-formulae for the Bessel functions 274
151. Relation between two Bessel functions whose orders differ by an integer . 275
152. The roots of Bessel function.s 277
153. E.xprcssion of the Bessel coefficients as trigonometric integrals . . . 277
154. E.xtcnsion of the integral-fornnila to the ca.se in which n is not an integer . 279
155. A second expression of ,/„ (;) as a definite integral whose path of integration
is real 282
156. Hankcl's definite-integral solution of Bessel's differential equatioti . . 28.3
157. E.xpression of J„ (z), for all values of n and z, by an integral of Hankel's type 284
158. Be.ssel functions as a limiting ca.se of Legendre functions. . . . 287
159. Bessel functions whose order is half an odd integer ..... 288
160. Expression of ,/"„(=:) in a form which furnishes an approximate value to J„ (z)
for large re;il positive values of z 289
161. The a.symptotic expansion of the Bessel functions ..... 292
162. The second solution of Bessel's equation when the ordi^r is an integer . . 294
163. Neumann's expansion ; determination of the coefficients .... 299
164. Pro<if of Neumann's expansi(jn .300
165. Schlomilch's expansion of an arbitrary function in terms of Bessel functions
of order zero ............ 302
166. Tabulation of the Bessel functions 304
Miscellaneous E.xamples . . . . 304
CHAPTER Xni.
APPLICATIONS TO THE EQUATIONS OF MATHEMATICAL PHYSICS.
167. Introduction : iilu.stration of the general method ..... 309
168. Laplace's equation ; the general .solution ; certain particular solutions . 311
169. The series-solution of Lai)lace's equation ....... 314
170. Determination of a solution of Laplace's equation which satisfies given
boundary-conditions 315
171. P.articular .solutions of Laplace's equation which depend on Bes.sel functions 317
gz y gz y
172. Solution of the equation ;^-;j -H . - . -I- 1'= 0 318
g2 y g2 y gz y
173. Solution of the equation ., „-|-.,^-[- ^-^ -I- !'=() 319
ox' cy' oz-
MlSCELLANEOnS EXAMPLES 321
CONTENTS.
XV
CHAPTER XIV.
THE KI.LII'TIC I'rXCTlON ^{i).
SECTION
174. IiitnxHiiction .........
175. Dotinitioii of ^(;)
176. Peri(xlicity, mid other properties, of ^ {:) .
177. The perioil-pjiralk'lograiiis ......
178. Kxpre.s.sioii of thv fuiiition ^ (z) by means of an integral
179. The lioniogoncity of the function ^ (z)
180. Tlie addition- theorem for the function ^f>(^)
181. Another form of the addition theorem ....
182. Tlie roots f,, e,' ''3
183. .Vildition of a half-periixl to the argument of ^(z)
184. Intfgration of (rt.c< + 46.!-^ + Cc.c- + 4o'.i- + <;)-4 .
185. .Vnother solution of the integration-problem
186. Uniformisation of curves of genus unity
.MlSCELL.\N'KOUS E.XAMl'LES
r.\uE 322 323 324 324 325 329 329 332 333 331 33.-) 33() 338 340
CHAPTER XV.
THE ELLIPTIC FUNCTIONS sn -n en.-, dm.
187. Construction of a doubly-periodic function with two simple poles in each
[leriod-parallelograra
188. Expr&ssion of the function f(z) by means of an integral .
189. The function sn i .
190. The fimctions en z and dn ;
191. E.xpression of cnz and dn* by means of integrals
192. The addition-theorem for the function dn 2 .
193. The addition-theorems for the functions sn z and cnz
194. The constant A'
195. The i>criodicity of the elliptic functions with respect to K
196. The con.stant A''
197. The jmritxlicity of the elliptic functions with i-espect to K + iK
198. Tlic ]>eri<Klicity of the elliptic functions with respect to /A"
199. The behaviour of the functions sn j, en ^, dn i, at the point ;
200. (ieneral de-scription of the functions sii ;, en i, dn ^ .
201. A geometrical illustration of tlie functions .sii ^, cnz, dn ^ .
202. ('omie.xion of the function sn z with the function ji>(j)
203. E.xpansion of sn z a.s a trigonometric series
.MiSCELL.VXEOUS E.XAMl'LES
lA"
342 343 34.-. 34(5 347 348 3.-.0 3.->l 3.-)l 352 3o3 3.-)3 3.')4
3.'ir) 3.-)(> 3.^7 3.^!»
XVI
CONTENTS.
CHAPTER XVI.
ELIJI-riC FUNCTIONS; GENERAL THEOREMS.
SKCTION
204. Holiition lictween the residues of an elliptic fiiiutioii
205. The onler of an elli|itic function
206. K.vpivs.sion of any elliptic function in terms of ^ (;} and ^' (;)
207. Relation between any two elliptic functions which admit the same 2>eriods
208. Relation between the zeros and poles of an ellii)tic function .
209. The function ^(z)
210. The qnasi-poriodicity of the function C (z) .
211. E.Kpres.sion of an ellii)tic function, when the principal part of its c.xpansioi
at each of its singularities is given ......
212. The function a- {z) .
213. The quaai-periodicity of the function o- (i)
214. The integration of an elliptic function .....
215. E.xpression of an elliptic function whose zeros and poles are known Miscellaneous E.xamples
PAOK
362 362 363 364 36.-) 366 367
367 368 369 372 372 374
Index 37
W. A.
PART I.
THE PROCESSES OF ANALYSIS.
CHAPTER I.
Complex Numbers.
1. Real Ntimbei's.
The idea of a set of numbers is derived in the first instance from the consideratiou of the set of positive integral numbers, or positive intecfers;
that is to say, the numbers 1, 2, 3, 4 Positive integers have many
properties, which will be found in treatises on the Theory of Integral Numbers : but at a very early stage in the development of mathematics it was found that they are inadequate to express all the quantities occurring in cjilculations ; aud so this primitive number system has come to be enlarged. In elementary Arithmetic, and in the arithmetical applications of Algebra, several new classes of numbers are defined, namely rational fractiwis such as A, negative numbers such as —3, and irrational numbers such as the number r-il4"2l.3..., which represents the square root of 2.
The object of the introduction of these extended types of number is that we may express the result of performing the operations of addition, subtraction, multiplication, division, involution, and evolution, on all integral numbers. Thus, the result of dividing the integer 1 by the integer 2 is inexpressible until we introduce the idea of fractional numbers: and the result of subtracting the integer 2 from the integer 1 is inexpressible until we introduce the idea of negative numbers.
The totality of the numbers introduced up to this point is called the aggregate of real numbers.
The extension of the idea of number, wliich ha.s ju.st been de.scriV>ed, was not efl'ected without .some opposition from the more conservative mathematicians. In tlie latter half of the 18th century, >[a.sere.s (1731—1824) and Frend (1757—1841) publi.shed works on Algebra, Trigonometry, etc., in which the use of negative quantities was disallowed, although Descartes had used them unrestrictedly more than a hundred years before.
1—2
4 THE PROCESSES OF ANALYSIS. [CHAP. I.
2. Complete Numbers*.
If we attempt to perform the operations already named — multiplication, etc. — on any of the real numbers thns recognised, we find that there is one case in which the result of the operation cannot be expressed without the introduction of yet another type of numbers. The case referred to is that in which the operation of evolution is applied to a negative number, e.g. to find the square root of — 2. To express the results of this and similar opera- tions, we make use of a new number, denoted by the letter i; this is defined as a quantity which satisfies the fundamental laws of algebra (i.e. can be combined with other numbers according to the a.ssociative, distributive, and commutative laws) and } Las for its square the negative number —1.
It is easily seen that all the quantities which can be formed by com- bining i with real numbers are of the form a + bi, where a and b are real numbers. A quantity a + bi of this nature is called (after Gauss) a complex number. Real numbers may be regarded as a particular case of complex numbers, corresponding to a zero value of the quantity b.
The complex quantity thus introduced may in the first instance be regarded as formed by the association of the pair of real numbers a and b ; as the quantities a, b, i are subject to the ordinary laws of algebra, we obtain for the addition and multiplication of two complex numbers a + bi and c+di the formulae
(a -I- bi) + {c + di) = (a -f c) +{b+ d) i, (a -I- bi) (c + di) = {ac — bd) + (ad + be) i.
But a complex number will usually be considered apart from its composition, as an irresoluble entity. Regarded in this light, it satisfies the fundamental laws of algebra ; so that if a, b, c are complex numbers, we have
a + b = b + a,
ab = ba,
(a + b) + c = a + (b + c),
ab . c = a . be,
a (b + c) = ab + ac.
It is found that the operations of multiplication, etc., when applied to complex numbers, do not lead to numbers of any fi-esh type ; the complex number will therefore for our purposes be taken as the most general .type of number.
The introduction of the complex number has led to many important developments in mathematics. Functions which, when real variables only
* For the general theory of complex numbers, see Hankel, Thcorie der complexen Zahlen- systeme (Leipzig, 1867), and Stolz, Vorlesimgen iiber allgcmeine Arithvietik II. (Leipzig, 1886).
2, 3] COMPLEX NUMHEKS. 5
are considered, appear as esscntiall}- distinct, are seen to be connected when complex variables are introduced : thus the circular fnnction.s are found to bo expressible in tenns of expunential functions of a complex argument, by the equations
cosi»= ^ (e'"^ + e~''*),
sin a' = ,7. (e'^ — e""'^).
Again, many of the njost important theorems of modern analysis are not true if the quantities concerned are restricted to be real ; thus, the theorem that every algebraic equation of degree n has ;? roots is true in general only when complex values of the roots are admitted.
Hiimilton's quaternions funiisli an exaniiile of a still further extension of the idea
of number. A quaternion
w+.i;i+>/j + zl-
is formed from four real numbcra w, .r, y, z, and four number-units 1, i, j, k, in the same way a.s the ordinary complex number .v + ii/ is formed from two real numbers x, y, and two number-units 1, i. Quaternions however do not obey the conmiutative law of multiplication.
3. The modulus of a complex quantiti/.
Let x-\-iy be a complex quantity; x and y being real numbers. Then the positive square root of or + y- is called the modulus of (x + yi), and is written
\x + yi \.
Let us consider the complex number which is the sum of two known complex numbers, x + iy and m + iv. We have
{x + iy) -f (m + iv) — {x -t- u) + i(y + v).
The modulus of the sum of the two numbers is therefore
[{x + uy + (y + vy}i, or {{a:r + y'') + (u^ + v") + 2(xu + yv)}^.
But
{\x + iy\ + \u + ivW'= !(■«' + >f)- + ("-' + v-y\-
= (x- + y-) + (u- +■(;-) -f 2 (x" + /)* (u- + v-)i
= (x- + y-) + {u- + V-) + 2 {(xu + yvT- + (xv - yu)-}i,
and this latter expression is greater than (or at least equal to) (x^ + /) + (m= + v')+2 (xu + yv). We have therefore
\x + iy\ + \u + iv\'^\{x + iy) + (u + iv) \ ,
or the 7nodulus of the sum of two complex numbers 'cannot be greater than the sum of their moduli; and in general it follows that the modidus of the sum
6 THE PROCESSES OF ANALYSIS. [CHAP. I.
of any number of complex quantities cannot be greater than tlie sum of their moduli.
Let us consider ne.xt the complex number which is the product of two known complex numbers x + iij and u + iv ; we have
{x + iy) (a + iv) = {xu — yv) + i (xv + yu),
and therefore
I {x + iy) (u + iv) I = [{xii, — yv)- + (xv + yu)-}^
= [ix^+f)iii^ + v^)]^
= \x + iy\ \u + iv\.
The modulus of the product of two complex quantities (and hence of any number of complex quantities) is therefore equal to the product of their moduli.
4. Tlie geometrical interpretation of complex numbers.
For many purposes it is useful to represent complex numbers by a geometrical diagram, which may be done in the following way.
Take rectangular axes Ox, Oy, in a plane. Then a point P whose coordinates referred to these axes are x, y, will be regarded as representing the complex number x+ iy. In this way, to every point of the plane there corresponds some complex number; and conversely, to every possible complex number there corresponds one and only one point of the plane.
The complex number *• + iy may be denoted by a single letter z. The point P is then called the representative point or affix of the value z ; we shall also speak of the number z as being the affix of the point P.
If we denote (a:^+ y'^)- by r and tan~' ("1 by 6, then r and 6 are clearly
*•/
the radius vector and vectorial angle of the point P, referred to the origin 0 and axis Ox.
The representation of complex quantities thus afforded is often called the Argand diagram*.
If P, and Po are the representative points corresponding to values z^ and Zn respectively of z, then the point which represents the value Zi + z., is clearly the terminus of a line drawn from Pj, equal and parallel to that which joins the origin to Pj-
To find the point which represents the complex number z^Zn, where z^ and z, are two given complex numbers, we notice that if
z^ = rj (cos ^1 + i sin 0,), 22 = r„ (cos 62 + i sin ^2),
* J. E. Argand published it in 1806 ; it had however previously been used by Gauss, and by Caspar Wessel, who discussed it in a memoir published in 1797 to the Danish Academy.
4] COMPLEX NUMUERS. 7
then by multiplication
z^z. = i\r., [cos (0, + 0.^ + i sin (^i + <?,)j.
The point which represents the value ZiZ., has therefore a radius vector niQasured by the product of the railii vectorcs of Pj and P„, and a vectorial angle equal to the sum of the vectorial angles of P, and P.^.
MiSCELLANEuUS EXAMPLES.
1. Shew that the represeut<ativo points of the complex numbers 1+4/, 2 + 7/, ;i + 10/, are colliuear.
2. Shew that a parabola can be drawn to pass through the representative points of the complex numbers
2 + /, 4 + 4/, fi + 9/, 8 + 16/, 10 + 2r)/.
3. Determine by aid of the Argand diagram the nth roots of unity ; and shew that the number of primitive roots (roots the powers of each of which give all the roots) is the number of integers including unity less than n and prime to it.
Prove that if 5,, d^, ^3, ... be the arguments of the primitive roots, 2cos/(5 = 0 when p is a, positive integer less than -r -. , where «, 6, c, ... k are the different constituent
primes of n; and that, when p= , — , , 2coS/o5=S — - — , , where u is the number of
'^ abc.k ^ abc.l:
the constituent primes.
(Cambridge Mathematical Tripos, Part I. 1895.)
CHAPTER 11. The Theory of Absolute Convergence.
5. The limit of a sequence of quantities.
Let 2,, ^^2, 2-3, ... be a sequence of quantities (real or complex), infinite in number. The sequence is said to tend towards a limiting value or limit I, provided that, corresponding to every positive quantity e, however small, a number n can be chosen, such that the inequality
\Zm-l\<e is true for all values of m gi-eater than w. If 2 is a variable quantity which takes in succession the values 2,, z„, z^, ... , then z is said to tend to the limit I.
Example. Consider the sequence of numbers ^, J, J,..., for which z„ = ^. This sequence tends to the limiting value 1 = 0; for if any positive quantity f be taken, and
\
if 7i denote the integer next greater than
is true for all values of m greater than n.
_loge
log 2
1 2S<^
, then the inequality
6. The necessary and sufficient condition for the existence of a limit.
We shall now shew that the necessary and sufficient condition for the existence of a limiting value of a sequence of finite numbers z^, z^, ^3. •■• is that corresponding to any given positive quantity e, however small, it shall he possible to find a number n such that the equation
is verified for all positive integral values of p. This may be expressed in words by the statement that a finite variable quantity has a limit if, and only if its oscillations have the limit zero ; it may be regarded as one of the fundamental theorems of analysis.
First, we have to shew that this condition is necessary, i.e. that it is satisfied whenever a limit exi.sts. Suppose then that A limit I exists ; then
'\
;■), ()] THK TIIKOUY OF ARSOI.UTE CONVERGENCE. 9
(§ 5) correspoiKiiug to ;iiiy positive iiuantity e, however small, a number n can be chosen such that
6
2'
\Zn-l^.<.
and I Zn+p — l\<7,, foi' ivll values of;) ;
therefore
I ^n+p - Zn W (^n+p - 0 " («n - 0 i ^ i Zn+p -l\ + \Zn-l\
e e
<2 + 2
which shews the necessity of the condition
and thus establishes the first half of the theorem, y
Secondly, we have to shew that this condition is sufficient, i.e. that if it is satisfied, then a limit exists. Suppose then that this condition is satis- fied. Let
Zr = Xr-¥iyr,
where a,v and iyr are the real and imaginary parts of z^. Then if
I Zn+p ~ -^ji I < 6,
we have | {x^+p -Xn) + i{yn+p- yn)\< e,
or (Xn+p - x„y + iyn+p- yn)- < e-,
and therefore a;„ — e < Xn+p < Xn + e,
and 2/n - € < y„+p < yn + f.
Now the number n is determined by the quantity e, which can be assigned
arbitrarily. Let %, n., n^, n^, ... be the numbers which correspond in this
way to the quantities -, j, -, r— Let u^ be the least of the quantities
Xn + e, x,,^ 4- ^ , x„^ + 2 , •■• Xni, + ofr ' ^'^ th^t the quantities u„, u^, u^, ... are a decreasing sequence ; and let v^ be the greatest of the quantities
so that the quantities v,,, f,, v„, ... are an increasing sequence ; and clearly
Then any of the numbers in the ^-sequence is greater than any of the numbers in the w-sequence, since we have
llr > I'r > Vg, if r > s,
and 11^ > u^ > V,, if ?• < s ;
10 THE PROCESSES OF ANALYSIS. [CHAP. II.
and the difiference % — tij. can be made as small as we please by increasing k. These two sequences ii and v therefore uniquely define a real number (rational or irrational) ^, such that f is less than any number in the w-sequence uiid greater than any number in the ?;-sequence, and the differences «(. — ^ and ^— «* can be made as small as we please by increasing k.
Then »* - ^ < iik - Vk < .^^.^ ,
so 1 a;„. - ? I < I X„, -U„\+\ Mjt - f I < ^ +^j < ^^.
Moreover, by hypothesis,
6
2* where p is an}' positive integer ; and so
e
2^
^m+p ^m i '^ on
^nt+p ? I <
Since ^^r.j can be made as small as we wish by increasing k, this inequality
shews that the sequence osi, a;^, n^, ... tends to the limit f. Similarly the sequence y,, y„, 1/3, ... tends to a limit 7;.
Thus if T be any small positive quantity, it is possible to choose a number 711 such that for all values of r greater than m we have
|a^r-|^!<^^, and | y^ - 7; j < J- ,
and therefore (a-v — f )- 4 (y,- — f )" < t-,
or \2r-l\<r,
where I = ^ + m;.
This inequality shews that the sequence of quantities z^, z^, z^, ... tends to
the limit I ; which establishes the required result, namely that the condition
expressed is sufficient to ensure the e.xistence of a limit.
7. Convergence of an infinite series.
Let «i, Mo, Mj, ... M„ be a series of numbers (real or complex). Let the sum
«i + M.J + . . . + Un
be denoted by S„.
Then the infinite series
III + «.> + Uj + Ui+ ...
is said to be convergent, or to converge to a sum S, if the sequence of numbers /Si, (S2, S3, ... tends to a definite limit S as n tends to infinity. In other cases, the infinite series is said to be divergent. When the series converges the quantity S — Sn, which is the sum of the series
7] THE THKOKY OF ABSOLUTE CONVERGENCE. 11
is called the remainder after n terms, and is frequently denoted by the symbol H,,-
The Sinn »„+j + "„+2 + ... + '(,i+p
will be denoted by S„^p.
It follows at once, by combining the above definition with the results of the last paragraph, that the necessary and sufficient condition for the convergence of an infinite series is that S„^p shall tend to the limit zero as n tends to infinity, whatever p is.
Since "n+i = (S^n, i. it follows as a particular case that m„+, must tend to zero as n tends to infinity, — in other words, the terms of a convergent series must ultimately become indefinitely small. But this last condition, though necessary, is not sufficient in itself to ensure the convergence of the series, as appears from a study of the series
.r , 1111
>f=l+, + 3 + 4+5 + ....
In this series,
"■"~n + l"^«. + 2 + n + 3+---''"2w'i_/v+-v. \ '
so '^n,n>;^^,
or 'Sn,„>2-
Therefore 5= 1 + ST,., + .S,.., + S,^, + S,,^ + -S,„.„ + ...
Ill *!'
> 1 + ,^ + 2 + 2+ •••' ^
which is clearly infinite ; the series is therefore divergent.
Infinite series were used by Lord Brouncker in P/dl. Trails. 1668, and the expressions convergent and divergent were introduced by Gregory in the same year. But the great mathematicians of the 18th century used infinite series freely without, for the most part, considering the question of their convergence. Thus Euler gave the sum of the series
111,,, , V
— + :5 + -^' +7 + 1 +-' + -+- + (")
as zero, on the ground that
: + c= + ^^ + ... = ^ (*)
and l + i + l + ...=_i,^ (c).
The error of course arises from the fact that the series (h) converges only when | ■: i < 1, and the series (e) converges only when \z >1, so the series (a) does not converge for any value of z.
The modern theory of convergence may be said to date from the publication of Gauss'
Disquiiritiotu;^ circa geriem infinitam 1 + -^+ ... in 1812, and Cauchy's Analyse Alg^briqii^
1 -y in 1821. See Reiff, Geschickte der unendlichen lieihen (Tubingen, 1889).
12 THE PROCESSES OF ANALYSIS. [CHAP. II.
8. Absolute convergence and seiai-converffence. In order that the series
)/, + U.,-\- i/,, + i*j+ ...
(which we shall frequently denote by tu,i), whose terms are supposed to be any comple.K quantities, may be convergent, it is sufficient, but not necessary, that the .series S | «„ | shall be convergent.
For wo have
I "n.p I = I ■**»+! + ""n+a + . . . + U^+p \
$ I «n+i I + 1 Mn+2 I + ••• + I Un+p \ ,
and this last expression is infinitely small, whatever p may be, when ?i. is infinitely great, provided the series S j m„ | is convergent.
Although this condition is sufficient to ensure the convergence of the series 2«„, it is not necessary, i.e. the series S«„ can converge even when the series S | «„ \ diverges. This may be seen by considering the series 11111 (-1)"+' ,
1 Z i 4 a ,j
This series is convergent ; for writing it in the form
(i-y-G-i)+a-5)--.
OI- 2 + r2 + 30+-'
we see that its sum is greater than ^ , and that the partial sum obtained by
truncating the series after its 2nth term increases as n increases ; on the other hand, by writing it in the form
'-G-j)-a-j)+--
we see that the sum is less than 1, and that the partial sum obtained by truncating the series after its (2n+l)th term decreases as n increases.
These partial sums must therefore tend to some limit between 5 and 1, and so the series converges. But the series of moduli is
which as already shewn is divergent. In this case therefore, the divergence of the series of moduli does not entail the divergence of the series itself.
Series whose convergence is due to the convergence of the series formed by the moduli of their terms possess special properties of great importance, and are called absolutely convergent series. Series which though convergent are not absolutely convergent (i.e. the series themselves converge, but the series of moduli diverge) are said to be semi-ccmvergent or conditionally convergent.
8, 9] THE THEORY OF ABSOLUTE CONVERGENCE. 13
■» 1
9. The qeometrical series, and the series S — .
n=lW'
The convergence of a particular series is in most cases investigated, not by the direct consideration of the sum S^ p. but (as will appear from the following articles) by a comparison of the given series with some other series which is known to be convergent or divergent. We shall now investigate the convergence of two of the series which are most frequently used as standards for comparison.
(1) The geometrical series.
The geometrical series is defined to be the series
1 + Z + Z- + Z'' + z*....
Considering the series of moduli
l + \z\ + \z\''+\z\'+ ..., we have for it Sn,p = \z\''+' + \z\"+- + ... +|«j"+p,
1 — l^ip
or S — z\ "+1 ' '
1 - !^ 1^ . Now if \z\< 1, then -^, , — v- is finite for all values of p, while I z 1"+' tends
' ' 1 — l'^ I
to zero as n tends to infinitj'. The series
1 +\z' + \z\-'+ ...
is therefore convergent so long as j 2 < 1, and therefore the geometric series is absolutely convergent so long as | ^ | < 1.
When |z I ^ 1, the terms of the geometric series do not tend to zero as n increases, and the series is therefore divergent.
c. m^ .11111
(2) TAesmes J^ + 2i + 3i + 4,+5, + ....
»: 1 Consider now the series 2 - , where s is any positive real quantity.
n = l 1
112 1 We have 2"' + 3, < 2» < 2«-i >
11114 1
4« "^ 5» "*" 6» ^ 7' ^ 4' "^ 4«~' ' and so on. Thus the sum of any number of terms of the series is less than the sum of the corresponding terms of the series
1 1 1_ J_
l»-i "*" 2»-' "^ 4»-' ^ 8»-' '
J_ J 1^ 1
14 THE PROCKSSES OF ANALY'SIS. [CHAP. II. '
and hence the convergi'iiee of this last series would involve that of the original series. But this last series is a geometrical series, and is therefore convergent if
that is, if s > 1.
"" 1 .
The series S — is therefore converr/ent if s > 1 ; and since its terms
are all real and positive, they are equal to their own moduli, and so the series of moduli of the terms is convergent ; that is, the convergence is absolute.
If s = 1, the series becomes
1111
I+2+3+4+-.
which we have already shewn to be divergent; and when s==l, it is d fortiori divei'gent, since the etfect of diminishing s is to increase the terms of the
", 1 .
series. The series ^ — is tlierefore divergent i-f s < 1 .
10. The Comparison-Theorem. We shall now shew that a series
will he absohiteli/ convergent, provided \ m„ | is always less than C \Vn\, where C is any finite number independent of n, and v^ is the nth term of another sei'ies which is knoivn to be absolutely convergent.
For we have under these conditions
I W„+, i + I Un+.2 I + ■•■ + I Mn+p I < C" I I V„+, 1 + I Vn+2 I + ••• + | V„+p |),
where n and p are any integers. But since the series %Vn is absolutely convergent, the series S j t'„ | is convergent, and so
kn+i 1 + I ^n+2 I + ••• +|«n+pl
tends to zero as n increases, whatever p may be. It follows therefore that
jw,i+i| + i «'n+2 I +••• + !"»+? I
tends to zero as n increases, whatever j) may be, i.e. the series S|t<,j| is convergent. The series Sif^ is therefore absolutely convergent.
Corollary. A series will be absolutely convergent if the ratio of its terms, to the corresponding terms of a series which is known to be abso- lutely convergent, is always finite.
Example 1. Shew that the serie.s 1
COS2+-„COS
is absolutely convergent for all real values of
cos 2 + -i cos 2z + — . cos32+ -i cos 42+ ... 22 32 42
10] THE THEOKY OK AUSOLUTK CONVERGENCE. 15
i COS nz\ ^ 1
For when .' is real, we have |cos«j, $ 1, and therefore : — j- <— , . The moduli of
the terms of the given series are therefore less than, or at most equal to, the corresponding terms of the series
1 1 i i
which by § i) is absolutely convergent. The given series is therefore absolutely convergent.
Example 2. Show that the series
1.1.1.1,
li{z-z,)^ 2^z-z^)^ 3Hz-z,)^ A^z-z,) -' where • z„ = (l+j\''', (« = 1, 2, 3, ...)
is convergent for all values of :, except the values z=Zi, z.,, £3, —
The geometric representation of complex numbers is helpful in discussing a question of this kind. Let values of the complex number z be represented on a plane : then the values 2,, J.,, C3, ... will form a series of points which for large values of n lie very near the circumference of the circle whose centre is the origin and whose radius is unity : so that in fact the whole circumference of this circle may be regarded as composed of points included in the values z„.
For these special values -„ of z, the given series is clearly divergent, since the term
-J-, , becomes infinite when z = z„. The series is therefore divergent at all points z
n'iz-z,,)
situated on the circumference of the circle of radius unity.
Suppose now that z has a value which is distinct from any of the values z„. Then
; is finite for all values of n, and less than some definite upiier limit c : so the moduli
of the terms of the given series are less than the corresponding terms of the series
c c c c
which is known to be absolutely convergent. The given series is therefore absolutely convergent for all values of z, except the values z„.
It is interesting to notice that the area in the 2-plane over which the series converges is divided into two parts, between which there is no intercommunication, by the circle 1^1 = 1.
Example 3. Shew that the series
2sin-+4sing + 8sin — +... + 2'»sin ^„ + ---
converges absolutely for all finite values of z. For when n is large, the quantity
I 2" sin |,
2"! 2 I 3» has a value nearly unity ; the given series is therefore absolutely convergent, since the
comparison series 2 ' is absolutely convergent.
16 THE I'UOCKSSES OF ANALYSIS. [CHAP. II.
11. Discussion of a special series of importance.
The theorem of § 10 enables us to establish the absolute convergence of a series which will be found to be of great importance in the theory of Elliptic Functions.
Let a>i and (o„ be any constants whose ratio is not purely real ; and consider the series
i + sl 1 1 _l
z' \{z- 2m<Bi - 2nto„y (2m(o, + -Inco,)"] ' where the summation extends over all positive and negative integral and zero values of m and n (the simultaneous zero values m = 0, n = 0 excepted). At each of the points z = 2»«<Ui + 2nw2 one term of the series is infinite, and the series therefore is not convergent. The absolute convergence of the series for all other values of z can be established as follows.
Let z have any value not included in this set of exceptional values.
The series may be written
1 ^ 1 I / z \-' _
z^ "^ (2m&)j + 2«(»o)- (V 2mmi+2na)J Now when | 2m&)i + 2na}., \ is large (and we can suppose the series arranged in order of magnitude of | 2 meoi + 2/10)2 1), we have
z
('
T . ., \ iincoj + 2nQ).J
Limit ^ " = 1.
1
2wQ)i + 2nG)2 The series is therefore absolutely convergent if the series
S 2.
(2ma)i + 2m&).,)' is absolutely convergent : that is, if the series
2 1
(2maj, + 2na)2y
is absolutely convergent. ' To discuss the convergence of the latter series, let ft), = ai + z/3i, 0)2 = 02 + 1/32, where a^, Hj, ySi, ySo, are real. Then the series of moduli of the terms of this series is
S 1 .
{(a,m + Ojn)^ + (ftm + ^m)-\^
This converges if the series
2 i (which we may denote by S)
(in- + 7i-y
11, 12] THK TIIKOKY OK AUSOMITK CONVERGENCE. 17
eoiivoi'fjos ; for the ([Udtient of corri'spiiiidiiig teiins is
where u = -
m
aud this is never zero or infinite.
We have thei-cfore only to stndy the convergence of the series S. Now
1
-»n=~-x (m=+n=)5
= 4 i :s ^
where in the summation the occurrence of the pair of valnes »i=0, ;; = 0 together is excluded.
Separating iS into the terms for wl)ich ;/( = n, in > ti, and ni < n, re- spectively, we have
^ ;/( - 1 1 n n-\
X 1 :r ;/( - 1 I
+ V . _1
= i(2»r)* i,t = \ ,1=0 (m-+n-)- " = i "i=ii (m- + n")-
But i 3 < 5 < — .
„=v(m-+ n-y- (ni-Y »«'
* 1 "^1 =' 1
Therefore J-,s'< S -f— + S — + S - .
"^ 1 °° 1
But the series S — , and S are known to be convergent. So the
series S is absolutely convergent. The original series is therefore absolutely convergent for all values of z except the specified excluded values.
Example. Prove tliat tliu .series
1
{n^^-\-'nu}+... + m,?f in which the summation extends over all positive and negative integral values and zero values of m,, nu, ...?»,., e.xtept the .set of simultaneous zero values, is absolutely convergent
if fi>-. (Ei.sen.stein, Crelle's Journal, xxxv.)
12. A convergency -test which depends on the ratiu of the successive terms of a series.
Wf shall now slu^w that a series
l(, + H.., + (/;; + "4+ ...
is ahsolidely convergent, provided that for all rallies of n greater than some w. A. 2
18 THK PROCESSES OF ANALYSIS. [CHAP. II,
fixed value r, the quantity j-^^j is less tlian K, where K is some positive quantity independent of n and less than unity.
For the terms of the series
\u,.+i\ + \Ur+^\+\Ur+,\ + ...
are respectively less than the terms of the series
which is a geometric series, and therefore absolutely convergent when K < 1.
Thus if
^^' tends as n increases to a limiting value which is less than unity, the series is absolutely convergent.
Example 1. If , f i< 1, shew that the series
converges absohitely for all values of z.
For the ratio of the (« 4- l)th term to the ?ith is
(,(n+l)3-nV,
or c2" + ie»,
and if I e |<1, this is ultimately indefinitely small.
Example 2. Shew that the series
a-b , , (a-6)(a-26)_,, (a-6) (a-26) (a-3&) ^ , '+ 27'""*" 31 '"^ 4] ^"^-
converges absolutely so long as | z | < -j- .
For the ratio of the ( « + 1 )th term to the rath is - —y z, or ultimately - bi : so the con- dition for absolute convergence is \bz <1, or |^|<-y- .
Example 3. Shew that tlie series 2 ——^-^-— converges absolutely so long as
:2i<i.
For when J2|<1, the terms of the series liear a finite ratio to those of the series 2m2""'; but this latter series is then absolutely convergent, since the ratio of the
(M-f l)th term to the Jith is ( 1 + - ) 4, which tends to a limit less than unity as n increases.
13. A general theorem on series for luhich Limit
n=ao
u„
= 1.
It is obvious that if, for all values of n greater than .some fixed value r,
l.S] THK THIOUKY OK AH.SOIA'TK CONVERGKNCE. 19
t "n+i I 'S greater than «„ I. then the tonus of the scries do not tend to zero as
u increases, and the series is therefore divergent. On the other hand, if "—
is always less than some quantity which is itself less than unity, we have shewn in t^ '2 that thi' scries is ab.solntely convergent. The limiting case
is that in which, as /; increases
a further investigation is necessary.
We shall now shew that a series
«j + U. + W3 + . . . ,
tends to the value unity. In this case
in which 1
tends to the limit unitij us n increases, will he absolutely con-
vergent if, for all values of n after some fixed value, we have
^1-
1 + c
where c is a positive quantity independent of n.
For compare the series 2 |«„ | with the convergent series 2i;„, where
Vn =
and .-1 is a constant ; we have
l+.T
Vn \n+l V «/
, ^+-2 . 1 1
= 1 — — — + terms m „ , -- n n- w
As n increases, ""'"' will therefore tend to the limit
Vn
1 + :
so that after some value of /( we shall have
«»
By a suitable choice of the constant A, we can therefore secure that for all values of n we shall have
i «n !<■"»■
As -Vn is convergent, S | m„ j is therefore convergent, aud so S«,i is abso- lutely convergent.
0 o
20
THE PROCESSES OF ANALYSIS.
Corollary. If
Wn+i
[chap. II.
can be expanded in descending powers of n in the
form
., .^1 jfXq A-t
1 + — +-^+ — +...,
where ^i, ^2, .^l;,, ... are independent of n, then the series is absolutely convergent if -i4, < — 1.
This is easily seen to follow from the fact that when n is large the terms
A.. A,
— ■+ 3+...
n- ?r become unimportant in comparison with A^.
14. Convergence of the hypergeometric series.
The theorems which have been given may be illustrated by a discussion of the convergence of the hypergeometric series,
a.b a{a+l)b{b+l) ., a (a + l)(a + 2)b {b + l)(b + 2) "^l.c"^ 1.2.c(c + l) ^ 1.2.3.c(c + l)(c+2) ^+--'
which is generally denoted by F (a, b, c, z).
If c is a negative integer, all the terms after the (1 — c)th will be infinite; and if either a or 6 is a negative integer the series will terminate at the (1 — tt)th or (1 — 6)th term as the case may be. We shall suppose these cases set aside, so that a, b, and c are assumed not to be negative integers.
The ratio of the (n + l)lh term to the ?;th is
Un+i _ {a + n —l){b + n—l)
Un n (c + I? — 1 )
Therefore
1+" |
-1 ^^6-1 |
||
Wji+I |
n n |
||
M„ |
1 + ^-^' n 1 |
As n tends to infinity, this tends to the limit \z\. We see therefore by § 12 that the series is absolutely convergent when 1 2: | < 1, and divergent when
|2|>1.
When 1 .2 I = 1, we have
= 1 + -
1 +
b-l
n
n n"
I, a+b-c-l , .111
= 1-1 + terms m — , — , , etc. .
1 n /('- ■«■■ 1
Now a, b, c are in the most general case supposed to be complex numbers.
14, l.->]
THK THEOKY OF AHSOLUTE CONVERGENCE.
21
Let thuiii be given in terms of their real and imaginary parts by the equations
a = a' + ia",
b = b'+ib",
c = c + ic".
Then (newlectiug tlie terms in ., , :, , etc.) we have
Wn+i
J a' + b'-c'-l + i{a" + b"-c")
= 1 +
a' + b' -c' -I
.11,
+ terms in -, -i, etc
n n- rV
By § 13, the condition for absolute convergence is
a' + b'-c'<0.
Hence when \s\ = 1, the condition for the absolute convergence of the hyper- geometric series is that the real part of a + b — c shall be negaiive.
15. Effect of changing the order of the terms in a series.
In an ordinary sum the order of the terms is of no importance, and can be varied without affectinef'the result of the addition. In an infinite series however this is no longer the case, as will appear from the following example.
Let
^^3 2^.5^7 4 + 9^11 (i^ ••
vj-1 ^4.1 1-^1 ^-t-
and let i„ and (S'„ denote the sums of their first n terms. These infinite series are formed of the same terms, but the order of the terms is different.
Then if /,■ be any positive integer.
—3* — >>ik —
1
1
2VA- + l+FT2^-+F^-) = ^^*^'^^-
But
1
1
1
Pk P'^-'-2k-l'^-2k k 2/,-l -Ik
Similarly ;,,_, -;.,., = ^j^^ - ,^. _ , •
A scries of equations like this can be formed, of which the last is
1
1
Adding these, we have
p,= \-],^\-\ + ...-^j_ = S.^.
22 THE PROCESSES OF ANALYSIS. [CHAP. II.
Thus Xk = S,, + l.%.
Making k indefinitely great, this gives
an equation which shews that the effect of deranging the order of the terms in )S' has been an alteration in the value of its sum.
Example. If in the series
1-Hi-H-
the order of the terms be altered, so that the ratio of the number of positive terms to the niuuber of negative terms in <S'„ is ultimately a", shew that tlie sum of the series will become log (2a).
(Manning.)
16. The fundamental property of absolutely convergent series.
We shall now shew that the sum of an absolutely convergent series is not affected by changing in any manner the order in luhich the terms occur.
For let S= Ui + u.^ + v^ + 1^+ ...
be an absolutely convergent series, and let jS" be a series formed by the same terms in a different order.
Suppose that in order to include the first n terms of S, it is necessary to take m terms of S'. So if k be any number greater than m, we have
^k = ''^'ji + terms of »S' whose suffix is greater than n.
Therefore
1 '^V — 'S I ^ I Sn — »S' I + the sum of the moduli of a number of terms of 8 whose suffix is greater than n
^\Sn-S\ + \ U„+, 1 + I Un+2 I + ! Un+3 \ + ■■■ ■
When n tends to infinit}', \ 8.^ — 8] tends to zero since the series 8 is con- vergent, and the sum
l'"n+l| + l«n+2| + l«,i+3!+---
tends to zero also, since the series is absolutely convergent.
Thus I Sk — 8 j tends to zero when /■ is indefinitely increased ; which establishes the required result.
17. Rieinanns theorem on semi-convergent series. We shall now shew that a semi-convergent series
"l + "-+ ":;+ ((4+ ...,
with real terms, may be made to converge to avy desired real value, by suitably disposing the order in u'hich tlie terms occur. This propeity stands in sharp contradiction to that proved in the last article ; an example of it was afforded by the result of § 1 5.
Hi, 17] THE THEOItY Ol' AliSOI.UTE CONVERGENCE. 23
To I'sUiblish the thooruiu, let thu positivo terms in the series be
"/'I • "ft ' ";'3 > • • • ' iind let the negative terms be
"«,. "«,, "„,,■■■■ Then the series
and - 1,,,^- a,,^- u,,^- ...
cannot be both convergent : for if they were, the original series would be absolutely convergent : one of them must therefore be divergent : and the other cannot be convergent, since in that case the original series would be divergent. It fallows that the series
lip, + ll,K + lip, + ■■■
and _ „^^ _ „^^ _ „^^^_ ...
are both divergent.
Now let S be any real number, and let it be desired to change the order of the terms in the original series, in such a way as to cause it to converge to the sum S. Suppose that a terms of the series
'ip, + iij>, + iip,+ ■•• have to be taken in order to obtain a .sum greater than S, so that
lip, + ",,.+ ■.. + I' p.. .,<S <Up^ + Up,, + . . . + «^, .
Take now a number b of the terms of the series
"h, + "», + "«, + ■■• , such as are required to make the sum
lip, + »p, + . . . + Up, + u„, + u,,,, + . . . + «„j
less than 6' : so that
"p. + "p, + ■■■ + lip.. + »„, + ... + u„^_, > ,S' > Uy^ + "/,, + ■ • • + lip.. + "«, + • • ■ + II, n-
Take ne.xt a number c of the terms of the series
Up, + lip., + . ■ • , such as are required to make the sum
Up, + Up., + . . . + "p.. + ""■ + "", + . . . + "„j + "■p,+, + . . . + Up,^,
greater than S ; and then take a number d of the terms of the series in such a way as to make the sum
!<;,, + . . . + Up, + W,„ + . . . + (/,„ + Mp„^, + . . . + ('j,,+. + «„.+, + . . . + U,,,^
less than i' again ; and so on.
Proceeding in this way, we obtain a series whose sum at any stage of
24 THE PROCESSES OF ANALYSIS. [CHAP. II.
the process, (liftVrs from »S' l)y le.ss than the last term iiichided. But the terms of the series
Vi + a., + U3 + ...
are ultihiately iiidofiuitely small, since the series is convergent ; we can therefore in this way obtain a series
//^,, + ... + «^,„ + "„, + ... whose sum differs from <S' by as little as we please ; and it consists of the terms of the original serie.s, disposed in a different order. This establishes the result above stated.
Corollary. If the terms of the original series are complex, they can be disposed in such an order as to give an arbitrarily assigned value to either the real or the imaginary part of the sum.
18. Caiichy's theorem on the multiplication of absolutely converc/eiit series. We shall now shew that if tivo series
IS = i(i + lu + 11.J+ ... and T= Vi + Vo + v, + ...
are absolutely convergent, then the series
P = «!», + u.x\ + UiV„ + . . . ,
funned by the products of their terms, ivritten in any order, is absolutely con- vergent, and has for sum ST.
Suppose that in order to include all the terms of the product
(h, + M., + U.J +.■..+ U„) {l\ + tl„ + . . . + Vn)
it is necessary to take tn terms of P ; and let k be any number greater than m.
Then
Pk = ("1 + "2 + . • • + Vn) {Vi + % + . . . + Vn) + terms «„tip in which either a or /3
is greater than n,
so \P,,-ST\^\SnT„-ST\ + tevvas \u^\\'v^\.
Let (Wf/J) be the greatest suffix contained in these suffixes a and /8. Then
\P,-ST\^\8nTn-8T\ + [\u„^,\ + ... + \un+j,\]{\v,\ + ...+\v,,+j,\]
+ [i//, j + ... +|M„|)(|'y„+i| + ... + ii'„+pl). Now when n tends to infinity,
"»i+i I + 1 Wn+2 1 + . . . + I Un+p i tends to zero, and j Vn+\ I + . . . + 1 Vn+p | tends to zero,
while their coefficients tend to finite limits.
Therefore \P^ — ST\ tends to zero, which proves the theorem.
IS, 19] THE THEORY OF AHSOLUTE COXVEKUENCE. 25
Extimplf I. Shew that tho series ubtiviiied by niultii>lying the two series
z z^ z' z^ l + 5+^ + ii + 25 + --'
■ ,111
niul l + : + .ii +^,+ -".
converges so long a.s the representative point of z Hes in the ring-shjiped region bounded by tlie circles :| = 1 and |:i = 2.
For the first series converges only when 1;'<2, and the second only when \z\>\, and both must converge if the product is to converge.
- + ....
Example 2. Prove by niiUtiplication of series that f cos Zz cos s>z 1 frr* 2 /cos 2^ cos Az W cos 3; cos 5z
For the coefficient of cos (2r+ 1) ^ in the product on the left-hand side of the equation is + iy- iklx (2X-)M(2^--2/--l)2^(2^ + 2r+l)2j '
2r-l 2k}'^\^k 2A + 2r + lj J '
or
9 (2,
9(2r
-^ 1 i // \
9(2r+l)3 3(2r+l)2tf, ^2^-2
^ 1 ' J 2 2 4 1 1
r+\y 3 {2.r-\-\f u^i \{^kf '*' (2/E:- 1)2 (2ifc - 2r - 1) (2)f-+2r + l)J ■*" 3 (2r-(- If '
n^ JV ^ (, 12 1 \ 2
9(2r+ 1)'^ 3 (2r+l)< 3 (2r-l-l)2 ^ "'"22 "^ IP + -P "'' ' ' 7 "'' 3 (2/-+ 1)* '
"" 1 ' - 9(2r + l)2 (2r+l)^ 1 |
2 |
n^ |
3(2r-fl)2 |
■ 6 ' |
(2;- + l)<' which gives the required rcsult.
19. Mertens tlieurem on the multiplication of a semi-convergent series hy an absolutely convergent series.
We shall now shew that )/" a series
S = «, + M., + Ug+ ...
is semi-convergent, and another srn-ies
r = I', + v., + ^3 + . . .
is absolutely convergent, then the series
P = P:-^p., + p,+ ...
w/'CT-e p„ = «,D„ + M„t)„_, + . . . + «„Vi,
w convergent, and its sum is ST.
For P„ = the sum of all terms u^v^ in which a + y8 ■^n + 1
= (". + "•+.••+ «n)(n + 1-2 + . . . + 1'„) - y.",, - Vs («„ + «„_,) - . . .
— w„(m„ + ((3+ ... + ;/„).
26 THE PROCESSES OK ANALYSIS. [CHAP. II.
Therefore
\P„-ST\^\S./r„-ST\ + \t,„\\v,\ + \v,\\u„ + Un-A + ...
+ \Vn\\u^ + Us+ ... +M„|.
Now let A' denote some number about half-way between 1 and n; let e be the greatest of the quantities
I Mm i , I W» + Un-i \, • • . | '«« + '"«-i + ■ ■ . + "„-* |,
and let 7 be the greatest of the quantities
I «„ + . . . + Un-k-i I , . . . ! '(„ + M,i_i + . . . + U2 I .
Then
\Pn-ST\^\SnTn-8T\ + e{\v,\+\v,\ + ... + \v,+,\]-i-y{\v,+,\+... + \v„\}.
As n tends to infinity, e and {| yt+s] + ... + ] m„1} are infinitesimal, while J Jlt^al + ... +ii'i+2ii and 7 are finite. So every term on the right-hand side ' of the last equation is infinitesimal, and therefore in the limit
P = ST, which establishes the theorem.
20. Abel's result on the multiplication of series.
We shall next prove a still more general theorem due to Abel*, which may be stated thus :
Let two series S »„ and 2 t)„ converge to the limits U and V respec-
« = 1 7* = 1
tively, and let the quantity
HiVn + >UV„-, + ... + UnVt
be denoted by w„. Then if the series
Wi + W2 + 'W.i + Wi+ ...
converges at all, it converges to the sum UV.
It will be noticed that none of the series considered need be absolutely convergent.
We shall follow a method of proof due to Cesarof.
Lemma I. If a set of quantities Sj, s,, S3, ••• tend to a limit s, then
. . 1 " Limit - S s; = s.
n = v. n i = \
For if e be any small positive number, we can find a number k such that the inequality
I s,. — s 1 < e '
is satisfied for all values of r greater than A*. We have therefore 1 jt 1 ^ 1 ' ~ " 1 "
- 2 Si = - 2 Sf 4- - 2 S -I- - S (Si - s).
n i=i n ,=1 n ;=*«., n ,=«:>,
• Crelle's Journal, i. (1827).
t Bulletin des Sciences math. (2) xiv. (1890).
20] TIIK Tlli;uKV Ot- AliSOLUTE CONVERCJENCE. 27
Thus j - i, s,- — s $ - S I s,- + - 2 Si — s I
1 A; , „-Ar-^l'
< -' 1 !s,-| + 6.
n ,-=1 w
1 * Now make /( infinitely great compared with /.• ; tlien - S is,] tends to zero,
" (=1
, /I - A: + 1 ,
and tends to unity,
1 ",
and so Limit - "^ S; - s < e ;
; n=x li 1 = 1
and as e can be made as small as we please, this establishes the Lemma.
Lemma II. I}, as n increases indefinitely, «„ and b„ tend respectively to the limits 'a and b, then
Limit - (a,bn + iij)„-i +... + f'„6i) = ab. « = 3c n
To prove this, let v be the greatest integer contained in ^n. Then if e be an}' small positive number, we can take n so great that the inequality
6, - 6 I < s
holds so long as i- > n — r. .
Hence | a, (b,, -b) + a., (6„_, — b)+... + ii, (6„_„+, - b)
< e{(0, ! + j flol + ... + 1 a^lj.
Hence Limit - ] «, (6„— b) + (t..{b„-, — b)+ ... + a„ (&,j_^^, - b) |
n = x ^
< e Limit - { j Oj [ + | aa | + . . . + i a^ | }
< e I a ! , by Lomma I.
The right-hand side of this inequality can be made as small as we please : hence
Limit - {ft, (bn -b) + a..(b„_, -b)+ ... +a^ (^„-^+i - b)] = 0, or Limit- (f/,t„+ «..6„_,+ ... +rt^?>„_^+,)
n=x n
= ^b X Limit- ((/i + f/.j+ ... + w^)
= ^aZ), by Lomma L Similarly
Limit - (tt„+i6„_^ + a^+..b,^,+, + ... + ('„6i) = ^ab.
28 THE PROCESSES OF ANALYSIS. [CHAP. II.
Adding the last two equations, we have
Limit (a,bn + n.f>n-\ + ■■■ + anb,) = ah,
which establishes Lemma IL
Now let W,t denote the sum of the ii first terms of the series
Wl + W, + if;, + . . . ,
considered in the above enunciation of Abel's result, we have
where f/„ and F„ are used to denote the sums of the first Ji terms of the series U and V. From this we have
W,+ W, + ...+ W„=U,V,,+ U,V„_, + ...+ UnV„ and so by Lemma IL it follows that
Limit -(W, + W,+ ... + W„) = UV.
But if the set of quantities TT,, W„, W^, ... tend to a limit W, we have b}- Lemma I.
Limit - ( IF, + W, + ... + Wn) = W.
Hence W=UV,
which establishes Abel's result.
Example 1. Shew that the .series
V2V3 v/4"^- is convergent, but that its square (formed by Abel's rule),
2 / 2 1
l-7. + (^, + 2)-t?4 + ,T6)+-
is divergent.
Example 2. If the convergent serie.'<
c , 1 1 1
'^=i-2'- + .r-4' + -
be multijilied by itself, the terms of the product being arranged as in Abel's result, shew that the resulting series is divergent if /• < l, but that it converges to the .sum <S''- when
(Cauchy and Cajori.)
21. Power-Series. A series of the type
in which the quantities cto, «i, a.,, a,., ... are independent of z, is called a series proceeding according to ascending powers of z, or briefly a poiver-series.
21] THE THEORY OK ABSOLUTE CONVEUOENCE. 29
We shall now shew that ;/' a power-series converges fur anij value s„ of z, it mil be absolutely convergent for all values of z whose representative points lire within a circle, which passes through z„ and ha.i its centre at the ongin.
X
For if : bo such a point, wo havo \z\ < | z,, j. Now since !£a„V' converges,
the quantity aoV must tend to zero as n increases indefinitely, and so we can write
where e„ tends to zero as n increases. Thus
\ Z ' \ z \^ ^ z
i Oo I + I «i I I ^ I + I a„ 1 1 2 I" + . . . = e„ + 6, — I + eo - + 63 i —
! *o I ^0 1 ^0
Now ultimately every term in the series on the right-hand side is less than the corresponding term in the convergent geometric series
2 -
n=0 I ■^o
the series is therefore convergent ; and so the power-series is absolutely convergent, as the series of moduli of its terms is a convergent series; which establishes the result stated.
It follows from this that the area in the ^-plane over which a power- series converges must always be a circle ; for if the series converges for any point outside the particular circle which has just been found, we can (by taking this point as the point z^) obtain a new and larger circle within Avhich the series will converge.
The circle in the ^-plaue which inolndo.s all tho values of z for which the power-series
Ua-\- a-^z + a.,z--irn^z'^+ ...
converges, is called the circle of convergence of the series. The radius of the circle is called the radius of convergence.
The radius of convergence of a power-serie.s may be infinitely great ; as happens for instance in the case of the series
z" ^ "-31 + 5!--'
which represents the function sinz\ in this case the series converges for all finite values of z real or complex, i.e. over the whole ^-plane.
On tho other hand, tho radius of convergence of a power-series may be infinitely small ; thus in the case of the scries
H-l! z + 2!^--f-:3!2=-t-'4! 2^-1- ...,
have
Wn+l
= n\z\
30 THE PROCESSES OF ANALYSIS. [cHAP. II.
which, for all values of m after some fixed value, is greater than unity when z has any value different from zero. The series converges therefore only at the point 2=0, and its circle of convergence is infinitely small.
A power-series may or may not converge for p(jints which are actually on the circumference of the circle ; thus the series
~ , z z' :? z*
1-1 1 1 1 ^-
•* ~ \» 2* 3* 4"
whose radius of convergence is unity, convei'ges or diverges at the point z=-\ according as s is greater or not greater than unity, as was seen in § 9.
22. Convergency of series derived from a poiver-series.
Let a„ + «! 5- + UnZ^ + a.jZ'-' + UtZ* + ...
be a power-series, and consider the series
ffij + 2a.,z -f 'ia^z- -f- A^a^z^ -f ... ,
which is obtained by differentiating the power-series term by term. We shall now shew that the derived series has the same circle of convergence as the original series.
For let 2 be a point within the circle of convergence of the power-series ; and choose a positive quantity r, intermediate in value between | z \ and the
radius of convergence. Then, since the series S(/,„j-" converges absolutely, its
n = 0
terms must decrease indefinitely as n increases ; and it must therefore be possible to find a positive quantity M, independent of 7i, such that the inequality
I «« ! < ,T.
is true for all values of ».
Then the terms of the series
in|a„iU|"-' are less than the corresponding terms of the series
M
n z
In— 1
V
", *,n— 1
But in this series we have
n r \ n) r
which, for all values of n greater than some fixed value, is constantly less than unity ; this comparison-series therefore converges, and so the series
22, 23] TllK THKOKY OK ABSOLUTE CONVERGENCE. 31
converges; tliat is, tlic series S na„z"~^ converges absolutely for all ])()ints z
H-I
00
sitiiiitcJ within the vm-\v of convergence of the original series 2 «««", and the
11=0
two series have the same circle of convergence.
Similarly it can be shewn that the series 2 — - , which is obtained by
»=o H + 1 . "^
integrating the original power-series term by term, has the same circle of
oc
convergence as 2 «„^".
23. Infinite Products.
We proceed now to the consideration of another class of analytical ex- pressions, known as infinite products.
Let 1 -|-(/i, 1 + a.,, 1 +0;,, ... be an infinite set of quantities. If as 11 increases indefinitely, the product
(1 +0l)(l + «'=)(l+«3)---(l +«,.)
(which we may denote by n„) tends to a definite limit other than zero, this is called the value of the infinite product
n=(l+o,)(l + <(.,)(l+a,) .... and the product is said to be convergent.
The product is often written H (1 + «„).
« = i
If the value of the product is independent of the order in which the factoi-s occur, the convergence of the product is said to be absoiute.
The condition for absolute convergence is given by the following theorem : in order that the infinite product
(1 +ch){l+a„){l+a,)... may be absolutely convergent, it is necessary and sufficient that the series
Ui + a., + a3+ ... should be absolutely convergent.
For n„= elOK(I+o,)+log(l + a,)+...+IoK(l+o„)^
so that n is absolutely convergent or not according as the series
log (1 + «,) + log (1 + aa) + log (1 + a-,) + ... is absolutely convergent or not. But since log(l + «,) is nearly equal to a^ when Ur is small, the terms of this series always bear finite ratios to the corresponding terms of the series
«! + u., + a3+ ..., and so the absolute convergence of one series entails that of the other ; which establishes the result*.
* A discussion of the convergence of infinite products, in which the results are derived without makiug use of the logarithmic function, is given by Priugsheim, Math. Ann. .\xxni. pp. 119—154.
32 THE pkoces.sp:s oi- analysis. [char II.
Example. Shew that tlio iiitiiiit(' product
sin: .sin i: sin |: sin J:
is absolutely convergent fur all values of z.
. z sin -
For when /( is large, _ is of the form 1 — " , where X„ is finite ; and the series
* X ■ ' \
2 -" is absolutely convergent, as is seen on comparing it with 2 j- "^^^ infinite pro-
tl = l ^i H-l ^
duct is therefore absolutely convergent.
24. Some examples of infinite products. Con.«iider the infinite product
(-S)(i-,4.)('-.iS.)-.
which represents the function .
In order to find whether it i.s ab-soliitely convergent, we must consider the
<» 2" Z'- '^ \
series 2 -„— ;, or — S — ; thi.s series is absohitely convergent, and so the product is absolutely convergent for all finite values of z. But now let this product be written in the form
The absolute convergence of this product depends on that of the series
z z z z
TT TT ZTT ItT
But this series is only semi-convergent, since its series of moduli
\z\ \z\ \z\ \z\ TT IT Sir lir
is divergent. In this form therefore the infinite product is not absolutely convei'gent, i.e. if the order of the factors ( 1 + is deranged there is i
a risk of altering the value of the product.
Lastly, let the same product be written in the form
[('-i)-H(>n)-}l(-fJ-}|('-s)^-*i--
in which each of the expressions
1 + — e «'" imrj
24] THK THEOIIY OF AUSOI.UTK CONVEIUiENCE. 33
is counti'ti as a singk' tcnii of tho iuHiiito product. The absolute convergence of this product depends on that of the series
,.g(l_f) + i|,i,og(l + f)-f).{,o,(l-^^J + .^^-,....
(-^+-) + ri+-) + (-2J^=
+ -) + {-J^+-
and tlie alisolute convergence of tliis series follows frmn that of t hr series
The infinite product in this last form is therefore again absolutely
±-^— convergent, the adjunction of the factors e "' having changed the con- vergence from conditional to absolute.
Example 1. Prove that n \{\ ) «"f is absolutely convergent for all values of
z, if c is a constant other than a negative integer.
For the iuRuite product is absolutely convergent provided the series
i [(\-~^\e^-l\ is, „=i (V e + nj J
. -, '^ izc-isz- ^ . 1 1 "I .
I.e. if 2 -I ^ — t- terms ni — , —, etc. V is,
„=i I n^ n' ' M* J '
and on comparison with the convergent .series 2 -r, , this is seen to be the case.
11=1 '^"
Example i. Sliew that n -jl-(l — ) -^""f converges for all points z situated
outside a circle whose centre is the origin and radius unity.
For the iiitiiiitc product is absolutely convergent provided the scries
. / \\-'' 2 1-- -"
n=2\ «/
is absolutely convergent. But a.s n increases, (1 — I tends to the finite limit c, so the ratio of the « + l)th term of the .series to the nth term is ultimately - : there is therefore absolute convergence when - < 1 , or ; | > 1 .
Example 3. Shew that
1.2.3...(«-I) ,
z(j+l)(2+2)...(i+n-l)"
tends to a finite limit as n increases indefinitely, unless « is a negative integer.
W. A. 3
34 THE PROCESSES OF ANALYSIS. [CHAP. II.
For the expression can be regarded <is a product of whicli the «th term is
This product is therefore absolutely convergent, [ii'ovidod the series
z{z-\)
+ terms in —
1 1
,^,,etc.^
is absolutely convergent ; and a comparison with the convergent series 2 — ^ shews that
n = l 'i
this is the case. When ^ is a negative integer the expression clearly becomes infinite owing to the vanishing of one of the factors in the denominator.
Example 4. Prove that ,
•('-.-)('-i)('*-:)(-l;)(-/.)('*4)-«
For the given product
= Limit
rV 2 3 4 2 2t-l 2it^t/
= Limit e'i {'^'l + l'-'^WrrYk) z(l--\e''{\ + ^--\e'l(\-^ e^' (l +
27r
since the product whose factors are
(>-i)'
is absolutely convergent and so the order of its factors can be altered.
Since log2 = l -i + |- J + i- ... ,
this shews that the given product is equal to
«-„'"« = sin 2.
25. Cauchy's theorem on jnvducts which are not absolutely convergent. We shall now shew that if
«! + a., + a~i + ai+ ...
is a semi-convergent series of real tet'nis, then the infinite product ■ (l+a,)(l+a.)(l + a,)...
25, 2()] TllK THKOUY OF ABSOLUTE CON VKKUEXCE. 35
conveiyes (tliouijli nut itbsoluteli/) or diverges (tu the value zero), according as the series
a,-+ ((,,'-+ «/+ ...
■IS convergent or divergent.
For the infinite product in (lucstion converges (though not absolutely) or diverges (to the value zero) according as the series
log(l +a,) + log(l +a„)+...
is semi-convergent or diverges to the value — oo .
n= w
Now since the series 2 <»„ i.s convergent, the quantities a„ ultimately 11 = 1
diminish indefinitely, and tht'refore we can write
log (1 + «„) = a„ - ^' ( 1 + en), where len| tends to zero as ii tends to infinity.
If the series S a„'- diverges, it is clear therefore that the series 2 log(l +«„)
n = l
must diverge to the value — x : if on the other hand the series 2 a„= con-
71 = 1
n = x
verges, the series S log(l + a„) is convergent. From this the results relating
n = l
to the infinite product follow at once. 26. Infinite Determinants.
Infinite series ami infinite products are not by any means the only known cases of infinite processes which can lead to convergent results. The re- searches of Mr G. W. Hill in the Lunar Theory* brought into notice the possibilities of infinite determinants.
The Jictuiil inve.stigation of the convergence i.s due not to Hill but to Poincaiv, Bull, de la Sui: Matli. de Franci; si v. (1886), p. 87. ^^'e shall follow the exposition given by H. von Koch, .{<.ta Matli. .\vi. (1892), p. 217.
Let J.,-;t (i. A' = — 3c , . . . 4- X ) be a doubly-infinite set of given numbers, and denote by
Dm — L-^mJi',/l= -111, ... + m>
the determinant formed of the quantities ^1,^ {i, k = — m — . . . + ni) : then if, for indefinitely increasing values of m. the quantity J),n ha-s a determinate limit D, we shall say that the infinite determinant
is convergent and has a value D. In the case in which the limit D does not exist, the determinant in question will be said to be divergent.
* Beprioted in Acta Mathematica, viii. pp. 1 — 30 (188G).
3—2
36
THE PROCESSES OF ANAI.VSIS.
[CUAP. II.
The elements ^,-,(t = — oo ... + oo ) are .laid to form the princifal diuyonal of the (leterniinaiit D ; the elements A;if{k = oo ... + x ) are said to form the line i ; and the elements A n^ii = — x> . . . + cc ) are said to form the column k. Any element Ai^ is called a diagonal or a non-diagonal element, according as i = k or i "^ k. The element j4o.o is called the origin of the determinant.
27. Convergence of an infinite determinant.
We shall now shew that an infijiite determinant converges, provided the product of the diagonal elements converges absolutely and the sum of the non- diagonal elements converges absolutely.
For let the diagonal elements of an infinite determinant D be denoted by 1 + a,t(i = — 00 ... + cc ), and let the non-diagonal elements be denoted
^y Uik ( '^ >^> /, _ „ " , . ) ' so that the determinant is
oo
-l-CO
..!+»_,_, |
a-io |
«_„ ... |
«0-l |
1 +aoo |
«01 ••• |
Ol-l |
f'lu |
l + a„ ... |
Then since the series
i,k= -00
is convergent, the product
p= n (1+ S \aa
IS convergent.
Now form the pi'oducts P.
n 1
- aik
m /
. = .n (
1= -7)1 \
1+ S |cu
k= - m
then if, in the expansion of P,„, certain terms are replaced by zero and certain other terms have their sighs changed, we shall obtain D,„ ; thus, to each term in the expansion of D,,, there corresponds in the expansion of P^ a term of equal or greater modulus. Now D,n+p — Dm represents the sum of those terms in the determinant i)„,+p which vanish when the quantities O'ik {i< A- = + (m + 1) ... + (m + p)] are replaced by zero ; and to each of these terms there corresponds a term of equal or greater modulus in P,„+j; — Pm-
Hence
-^m+p -^m I ^ ■^771+p -^n
As the quantities P^, Pm+i, ••■ tend to a fixed limit, the quantities Dm, D,„+i, ■■■ will therefore tend to a fixed limit. This establishes the proposition.
^7, 28] TUK TlIKOltY ol- AUSOMTTK CONVERnENCE. .'?7
28. We shall now sliew that a determinant, of the convergent form alreadij considered, remains convergent when the elements of any line are replaced by any set of quantities whose inoduii are all less than some fixed positive number.
Roplaco, for example, the elements
• • • .^rt, — III > • •• -^'^O • • • ■'^o. '" ■ ' •
of the line 0 by the quantities
• • • P'—m > • • ■ Po "' Mm • * •
which satisfy the inequality
\Pr\< M.
where p is a positive number; and let the new values of Z),„ and D be denoted by Z),„' and D'. Moreover, denote by P,,/ and P' the products obtained in suppressing in P„, and F the factor corresponding to the index zero ; we see that no term of Z),„' can have a greater modulus than the cor- pesponding term in the expansion of pP,,,' ; and consequently, reasoning as in tlie last article, we have
I D'm+p - Dm 1 ^ pP'm+p - pP.n,
which establishes the result stated.
Example. Shew that the necessary and sufficient condition for the absohite conver- gence of the infinite determinant
1 a, 0 0 ... /3, 1 n,, 0 ... 0 02 1 ".I •••
is that the series
shall be absolutely convergent.
a|,i[-|-u.232 + (i3,ii3+ .
(von Koch.)
MiSCELLANEOU.S EXAMPLES.
1. Find the range of values of ^ for which the series
2sin-i — 4siu-'3 + 8sia«2-...-|-(-l)'' + i2''sin-"i + ... is convergent.
2. Shew that the series
z z + l"^ Z+-2 i + 3 "*"■■■ is semi-convergent, except for certain exceptional values of : ; but that the series 111 1 1
1 1
z z+\ z+p-1 z+p z+p+l
+
,+...,
z+2p+q-\ z + 2p+q in which {p + q) negative terms always follow p positive terms, is divergent. (Simon.)
38
TIIK PROCKSSES OF ANALYSIS.
[chap. II.
3. Shew that tlie scries
IS convergent.
4. Shew that the series
i.s convergent.
5. Shew that the series
1111
r 2^ 3° 4^
a + ^2 + a^ + /i< + ...
•-{(-;,)"-
(1<«0)
(Cesaro.)
(0<o</S<l)
(Cesaro.)
-(."- !){..-- (l+-j I converges absolutely for all values of z, except the values
(a = 0, 1; /• = 0, 1, ...m-1 ; »i = l, 2, ... x).
6. If «„ denote the sum of the first n terms of a convergent series whose sum is s, shew that
T . .. „ f a cfi «■> 1
i™ "' r "*"''' r"'"'''-2l''"*\3! + -/
■r
7. In the series whose general term is
■■(••+1)
tt„ = j"->'^ 2
(0<y<l<.r)
where v denotes the number of figures in the expression of n in the ordinary decimal scale of notation, shew that
Limit ?<„" = ^,
71 = 00
and that the series is convergent, although the quantity -^^' is infinitely great when n is infinitely great and of the form 1 + 10"-^. (Lerch.)
8. Shew that the series
4
where q„ = q^'^«, {0<(ji<i)
is convergent, although the ratio of the (M+l)th term to the /tth is gi-eater than unity when « is not a triangular number. (Cesaro.)
9. Shew that the series
„=o («> + «)''
where ?o is real, and where («' + »)« is undei'stood to mean c«'og("' + »), the logarithm teing
taken in its arithmetic sense, is convergent for all values of s, when the imaginary part of
X is positive, and is convergent for values of s whose real part is positive, when x is real.
« (_l)» + i 10. Shew that the yth jjower of the convergent .series 2 -- ^ is convergent when
/( = 1 '*■
<r, and divergent when >r.
q ?
(Cajori.)
MISC. KXS.j
THE IHKOKY OF AUSOLUTE UONVEKGENCE.
39
1 1. If tlie two senii-convergeut scries
2 ^- '- aui\ v^ iL_- , „ 1 II'' ii'
wliorc /■ iiml s lie between U ami 1, be multiplieil together, aud tlic product arranged as in Aliel's result, shew that the necessary and sufficient condition for the convergence of the resulting series is ;- + s>l. . (C'ajori.)
12. Shew that if the series
1-J + i-H-
be multiplied by itself any number of times, the terms of the product being an-anged as in AViel's result, the resulting series converges. (Cajori.)
13. Sliew that the lyth power of the series
((,sin ^ + «2sin -26+ ...+a„»ui n6-\-...
is convergent whenever <;•, ;• being the ma.ximum number satisfying the relation
n' for all values of n.
14. Shew that if 6 is not equal to 0 or a multiple of iJTr, and if the quantities U(|, M,, ».j, ... are all of the same sign and continually diminish in such a way that the limit of u, is zero when n is infinite, then the series 2«„ cos («5 + a) is convergent.
Shew also that, if the limit of »„ is not zero, Imt all the other conditions above are
B 8
satisfied, the sum of the series is oscillatory if - is commensurable, but that, if is in-
commensurab'e, the sum may have any value between certain limits whose ditferenoe is a coseci^, where a is tlie limit of u„, when n is infinite.
(Cambridge Mathematical Tripos, 1896, Part I.)
15. Prove that
where k is any positive integer, converges aV)solutely foi- all finite complex values of;.
X
16. Let 2 d^ l>e an absolutely convergent series. Shew that the infinite determinant
n=l
A(C) =
(c- 4)^-^0 4='-5„
2^-60
■6,
i^-e.
-62
4' -6c
4- |
-^0 |
- |
6^ |
22 |
-6, |
C2 |
-6, |
0=! |
-60 |
- |
6, |
2- |
-00 |
- |
6, |
42 |
-6„ |
-6,
42 |
-6, |
- |
62 |
22 |
-6, |
- |
6, |
02 |
-60 |
; + 2)2_e„ |
|
22 |
-60 |
- |
6, |
4»- |
-0, |
42-^0
22 -d„
02-^0 '
22-^0
(c + if-6, 42 -d„
converges : and shew that the equation is equivalent to the equation
A(f) = 0 8in2 Vc = A (0) sin2 ^6^.
(Hill.)
CHAPTER III.
The FiTNDAMENTAi. Properties of Analytic Functions ; Taylor's, Laurent's, and Liouville's Theorems.
29. The (hpevdmice of one complex miuiher on another.
The problems with which Analysis is mainly occupied relate to the dependence of one complex number on another. If z and f are two complex numbers, so connected that the value of one of them is determined by the value of the other, e.g. if f is the square of z, then the two numbers are said to depend on each other.
This dependence must not be confused with the most important case of it, which will be explained later under the title of analytic functionality .
If f is a real function of a real variable s, then the relation between f and r, which may be written
can be- visuali.sed by a curve in a plane, namely the locu.s of a point whose coordinates referred to rectangular axes in the plane are {z, f). No such' .simple and convenient geometrical figure can be found fof the purpose of visualising adequation
considered as defining tlie dependence of one complex nuiuber f=^ + J>/ on another complex number z = x + iy. A representation strictly analogous to the one already given for real variables would require four-dimensional space, since the number of quantities ^, ij, X, y, is now four.
One suggestion (made by Lie and Weierstrass) i.s to use a doubly-manifold .system of lines in the quadruply-manifold totality of lines in three-dimensional space.
Another suggestion is to represent ^ and ?; separately by means of surfaces
?= ^ (■<••. .y), n=i(.^,y)-
A third suggestion, due to Hetfter*, is to write
then draw the surface r = r{x,y) — which may be called the modidar-surface of the function — and on it to express the values of 6 by surface-markings. It might be possible to modify this suggestion in various ways by reijresenting 6 by cur\'es drawn on the surface r=r{x, y).
* Zeitfchrift fiir Math. ii. Phijs. xLiv. (181)!)), \>. 235.
20, ;W] TIIK KUNDAMENTAI, I'ltdl'KKTI KS (IK AN'Ar.TTIC l-TXCTIOXS. 41
30. I'untinuity.
Let f{z) be a quantity which, tor all values of s lying within given limits, depends on 2.
Let z, bo a point situated within those limits. Then /(z) is said to be continuous at the point z,, if, corresponding to any given positive ([uantity e, however small, a finito positive ijuantity tj can be found, siieh that the inoi|uality
1/(2) -/(^.) I <e'
is satisfied so long as | z — ^, I is less than ■»;.
If /(z) is continuous at z = Zi, a:id if its real and imaginary parts be denoted by a and v, then a and v depend continuously on z..
For it' f{z) = u + iv, we have
; («. — ((,) + i{v — V,) I < e,
and so ((( — »i)'- + (y — r,)'- < e",
which gives {u — Ui)'- < e- and {v — t;,)- < e-,
and so \u — Ui\ < e and [v — Vi\ < e.
The popular idea of continuity, so far as it relates to a real variable f depending on another real variable z, is somewhat different to that just considered, and may perhaps best he expressed by the definition " The (juantity f is said to depend continuously on z if, as ; passes through the scries of all values intermediate between any two adjacent vahies ^^ and z.,, f passes through the scries of all values intermediate between the corresponding values f, jiind f^."
The question thus arises, how far this popular definitipn is equivalent to the analytical definition given above.
C'auchy shewed that if a real variable f, depending on a real quantity z, satisfies the analytical definition, then it also satisfies what we have called the popular definition. But the converse of this is not true, as was shewn by Darboux. This fact may be illus- trated by the following example*.
Let E{x) denote the integer next less than .v ; and let
/(.r) = . [l -E lj^^n^^E{^^} sin -—^
At .v=0, we have/(x) = 0.
Between x= -I and i-= + 1 (except at .i" = 0), we have
/(.r) = 8in£.
From this it is easily seen that/' (a;) depends continuously on .i' near x = 0, in the sen.se of the popular definition, but is not continuous in the sense of the analytical definition.
* Due to Mansion, Matliesi^, ix. (1899).
42 THE PHOCESSES OF ANALYSIS. [CHAP. III.
31. Definite integrals.
Let Zo and Z be any two values of z ; and let their representative point.s A and B in the .z-plane be connected by an arc (straight or curved) AB; and let z^, z^, z,, ... Zn be a number of poiuts taken on the Hue AB in any manner.
Let f{z) be a quantity' which, for variations of z along the arc AB, depends continuously on z.
Let ^o' be any point situated in the interval z^z-^ of the curve : let z^' be any point .situated in the interval z-^z.,: and so on: and consider the .sum
S =f(Zo')(z, - 2„) +f{z/){z, -Z,)+... +f{Zn')iZ-Zn).
We shall shew that if the number n increases indefinitely, in such a way that each of the quantities \Zr — Zr-i\ tends to zero, then this sum will tend to a fixed limit, independently of the way in which the points
^I » -^2 > • ■ • ^n , ^0 , Zi , . . . Zn ,
are chosen.
For let e be a given small positive quantity. Since f{z) is continuous, for each point ^ = a of the arc AB we can find a quantity rja such that
l/(^)-/(a)|<e,
so long as \z — a\< rja.
Let T] be the least value of ■>;„ corresponding to points a on the arc AB. We shall suppose the subdivision of the arc has been carried so far that each quantity 1 Zr — ^r-i 1 is less than 77, and shall first find the effect of putting in further subdivisions.
Suppose then that the interval z^z^ is subdivided at points z„, ^02. ••• ■^or^; that the interval z^z^ is subdivided at the points ^„, z^^, ... z^r, ', and so on : so that the sum s becomes
«' =f{^<,"){^oi - ■^o) +f(Zoi')(Zm - ^m) + ■■■ + f{z"){Zn - Z,) +f{Zn'}{Zi2 - Zn) + ■■■
+ ... ,
where zj' is any point in the interval z„Zt,i, z^i' is any point in the interval ZoiZaa, and so on.
Then
S -S= {f(Zo") -fizj)} (2„, - 2o) + l/(-Sol') -./'(2«')l (^02 - ^oi) + ... + !/(^/') -/(^.')1 (-.. - 2,) + !/(2.>') -fiO] {^^. - Zu) + ■■■
+ ... .
SI] THE IT NOA MENTAL IMtOl'EUTIES OF ANALYTIC FUNCTIONS. 43
'I'herefore
.S-' - ,s I < f { I r„, - «„ I + 1 2„3 - ^0, 1 + . . . ]
< e X thp length of the broken line connecting the points z^,, z^i^, z^, ■■■
<€l.
where / is the length of the :irc AB.
Now by making <■ indefinitely small, we can make the right-hand side of this equation as small as we please ; and therefore the sum .s tends to a definite limit when the number of subdivisions is indefinitely increased, provided that at each change in the subdivisions the old points of division are retained.
The restriction contained in the last phrase has still to be removed. To do this, suppose that two different methods of division, in each of which the quantities \Zr — Zr-i\ are less than t), furnish sums s, and s.,. Now combine the two methods of division, so that every point of division in eithej- of the original schemes becomes a point of division in the new scheme. Let the sum corresponding to this new method of division be 5,2. Then since ty the above
I «1 — «!; I < e' ■I'lfJ Us — *12 i < ^l,
we have ! s, — So j < 2e?,
which shews that s, and s., tend to the same limit. The theorem is thus established.
The limit thus shewn to exist is called the definite integral of /(z), taken along the arc AB: it is denoted by
L
/:
.f(^)dz;
AB
in cases where there is no ambiguity as to path, it may be denoted by
/{z)dz.
A
A.s an example* of the evaluation of a detinite integral directly from the definition, suppose it is required to find the definite integral of the continuously dependent quantity (l—z^)~', taken- along the straight line (part of the real axis) joining the origin (j = 0) to a point 2 = Z, where Z is real. Denote the definite integral by /. Then by definition,
/= Limit 2 fLiiJlil,
n = » r=0 (1 — 3/2^4
and the mode of choosing the points r, and r/ "s arbitrary, within the limits already explained ; we shall take
2, = sinrS,
where d = sin - ■ JST.
re+1
* Netto, Xeitschrift fiir Mulli. xi,. (1895).
44 THE PROCESSES OF ANALYSIS. [CHAP. III.
Thu.s /= Limit 2 ^inCr+il* -«■"':»
« = « r=0 COS()- + 4)8
" 8
= Limit 2 -2 sin =
= Limit 2 (« + l) sin -
= sin ' iT Limit —-^
2
= sin-iZ The value of ttie definite integral is therefore sin"' Z.
32. Limit to the value of a definite integral.
Let M be the greatest value of j f{z) | at points oa the arc of inte- gration AB.
Then j/(V) (^1 - Zo) +f{z^) {z, -z,)+ ... +/( V) {Z - z,,) \
^ |/(V) i I ^1 - ^0 1 + \f{z;) \\z.,_-z,\ + ...+\ f{z,:) \\Z--z,,\
^M[\z^-z^\ + \z.,- z^\-\- ...^-'Z-Zn'^]
^Ml,
where / is the length of the arc of integration AB.
We see therefore, on proceeding to the limit, that
/(z)dz
' AB
cannot be greater than the quantity Ml.
33. Property of the elementary f auctions.
The reader will be already familiar with the word function, as used (in te.xt-books on Algebra, Trigonometry, and the Differential Calculu.s) to denote analytical expressions depending on a variable z ; such for example as
z', e^, log z, sin~' z-.
These quantities, formed by combinations of the elementary functions of analysis, have in common a remarkable propertj-, which will now be investi- gated.
Take as an e.xample the function e^ Write e^ =/(«)•
Then if z' be a point near the point z, we have
f(z') -f\z) _ e^' - e^ _ , e^J2"' ' 1 z — z z — z z — z
,1 z-z (z- z'f ]
o'2 — 85] THE Kl'NDAMENTAI, I'Kdl'lCUriES OK ANALYTIC KITNCTIONS. 45
;m(l lieiuf, if tho puiiit z ti'iuls to uoiiiindf witli z, tliu liiiiiliiig value of the i|Uiptii'iit
f(z')-f{z)
t
z — z IS (F.
This shews that the limiiinr/ value of
/(fl_-/(^) z'-z
is in this case independent of the direction of the short path by lultich the point z' moves towards coincidence with z, i.e. it is independent of the direction in which z lies as viewed from z.
It will be found that this property is ^shared by all the well-known elementary functions; namely, that if f(^z) be one of these functions and /( be any small complex quantity, the limiting value of
l|/(^ + A)_/(^)|
is independent of the mode in which h tends to zero.
34. Occasional failure of the property.
For each of the elementary functions, however, there will be certain points z at which this property will cease to hold good. Thus it does not
hold for the function at the point z = a, since the limiting value of
1 f 1 1
h [z — a — h z — a] is not finite when z = a. Similarly it does not hold for the functions \ogz and 2^ al the point ^ = 0.
These exceptional points are called singular points or singularities of the function f{z) under consideration ; at other points the function is said to be reguhn:
35. The analytical function.
The property noted in § 33 will be taken as the basis of our definition of an analytic function, which may be stated as follows.
Let an area in the 2-plane be given ; and let u be a quantity which has a definite finite value corresponding to every point z in that area. Let z, z + Bz be values of the variable z at two neighbouring points, and u, u + Su the corre.sponding values of n. Then if at every point z within the area
^ tends to a finite limiting value when Bz tends to zero, independently of
46 TIIK PROCESSES OF ANALYSIS. [CHAP. III.
the way in which Sz tends to zero, u is said to be an analytic function of z, regular within the area.
We .shall generally use the word " t'miction " alone to denote an analytic function, as the functions studied in this work will be almost exclusively analytic functions.
In the foregoing definition, the function u has been defined only within a certain area in the ^-plane. As will be seen subsequently, however, the function a will generally exist for other values of z not deluded in this area; and (as in the case of the elementary functions already discussed) may have singularities, for which the fundamental property no longer holds, at certain points outside the limits of the area.
The definition of functionality must now be translated into analytical language.
If /(^) be a function of z, regular in the neighbourhood of a particular value z, then, by the definition, the quantity
fW)-f{^)
z — z
tends to a definite limit, depending only on z, when z tends to z. Let this limit be denoted by the symbol/' (2).
Then (by the definition of a limit) for every positive quantity e, however small, it is possible to find a quantity 77, such that
z —z
is less than e, so long as \z' — z\ is less than ??. If therefore we write
/(/) =f{z) + {z' - z)f' {z) + e {z' - z),
we see that | e' 1 is less than e, so long as \ z' — z \ is less than 7/ ; that is, the function _/' (2) must be such that the quantity e', defined by the equation
/(/) =f(z) + {z' - z)f' {z) + /(/ -z\
tends to the limit zero as z! tends to z.
The necessity for a strict definition of the term "function" may be seen from the following consideration.
Let ij denote tlie temperature at a certain place at time t. As t varies, y will vary, and y may loosely be called a "function" of t. But y cannot be expressed in terms of t j by a Maclaurin's infinite series '
for if it could, the knowledge of the temperature for a single day would enable us to determine the quantities
»-.. (i),., (If),., -■•
3()J IHK KIJ.N'DAMKXTAI, PUOI'KlfTIES OK ANALYTIC Kl'NCTIONS. 47
iiid tlieti from the Maolaurin's expiiiision it wimUi l>o [xjs.siblo to piviiict the temponiture for the future !
Miiclfturin's series is in fact, as will appear subsequently, applicable only to analytic functions, in the sense in which analytic functions have been defineii above.
36. Cauchifs theorem on the integral of a function round a contour.
A .-iiinplo closed curve in the plauc of the variable z is often called a contour: if A, B, C, D be points taken in order along the arc of the contour, and \if{z) be a quantity depending on z and continuous at all points on the arc, then the integral
J A BCD A
taken round the contour, starting from the point A and returning to A again, is called the integral of the quantity f{z) taken round the contour. Clearly the value of the integral taken round the contour is unaltered if some point in the contour other than A i.s taken a.s the starting-point.
We shall ihiw prove a result due to Cauchy, which may be stated as follows. If f{z) is an analytic function, regular at all points in the interior of a contour, then
jf(z)dz = Q,
where the integration is taken round the contour.
For let A, B, C, D be poiuts in order on the contour. Join A to C hy an arc A EC, which will divide the region contained within the contour into two distinct portions. Then the integral taken round the contour ABCDA is equal to the sum of the integrals taken round the two contours ABCEA and A EC DA ■ for
f f{z)dz+\ f{z)dz
J ABCEA -'ABCDA
= \ f{z)dz+j f{z)dz+j f{z)dz+i f{z)dz
J ABC J CEA J ABC J CD A
= 1 f(z)dz,
J ABCDA
since the integrals along CEA and A EC neutralise each other.
Now join any point E on the arc AEC to D by an arc EFD, and join E to B hy an arc EGB ; then in the same way we see that the integral round ABCEA is equal to the sura of the integrals round ABGEA and EGBCE, and the integral round xiECDA is equal to the sum of the integrals round A EFD A and DFEGD.
Thus the original contour-integral is equal to the sum of the integrals
48 THK I'ROCEHSKS OF ANALYSIS. [cilAP. III.
round the four coutours ABGEA, ICdBCE, AKFDA, DFEGD, into which it has been divided bv drawintf the cmss-linos.
Proceeding in this way by drawing more cross-lines, we see that the original contour-integral can be decomposed into the sum of any number of integrals round smaller contours, which constitute a network filling uj) the original contour.
Now suppose that each of these small contours has linear dimensions of the same order of magnitude as a small quantity /. Let z^ be a point within one of them. Then on this small contour we have
f{z) =f{z„) + (z- z„)f' (z„) +(z- Z,) 6,
where e is infinitely small when I is infinitely small.
Thus jf(z) dz = jf{z,) dz + jiz - z,)f{z„) dz + jiz - z„) edz,
where all the integrals are taken round the small contour.
Now j / (^„) dz =f{Zo) j dz
=/{Zii) X the increase in value of z after once describing the small contour
= 0. Similarly jf(z^i) (z - z„) dz = ^f{Zo) \d\(z- ^o)'-) = 0,
wdien the integral is taken round the small contour.
Thus, if 7) be the greatest value of | e | for points on the small contour, we have
jf(z)dz ^7] l\z~z„\\dz\,
where the integrals are taken round the small contour.
Now the right-hand side of this equation is clearly of the order tjl- of small
quantities. The value of \f{z)dz, taken round the small contour, is there- fore a small quantity of order 7)1".
Now the number of such small contours in a given area is of the order t;;. If rj' be the maximum value of rj for all the small contours in the area, we see therefore that the total sum of the integrals for all the small contours in the area is at most of the order tj'I'-x y,, or tj' ; and t)' can be
made indefinitely small by decreasing I.
It follows, therefore, that the sum of the integrals round all the small
36] THE FUNDAMENTAL PROPERTIES OF ANALYTIC FUNCTIONS. 49
contoui-s is zero; that is, the integral round the original contour is zero, which establishes Ca\icliy's result.
Coivllor;/ 1. If there are two paths 2„AZ and 2„BZ from z„ to Z, and if /{:) is a regular function of z at all points in the area enclosed by these two
paths, then | f{z)dz has the same value whether the path of integration is
z„AZ or z^BZ. This follows from the fact that Zf,AZBzo is a simple contour, and so the integral taken round it (which is the difference of the integrals along z^AZ and z^BZ) is zero. Thus, if/(s) be an analytic function oi z, the
value of I f{z) dz is to a certain extent independent of the choice of the
. AB
are AB, and depends only on the terminal points ^1 and jB. It must be borne in mmd that this is onhj the case when f(z) is an analytical function in the sense of § 35.
Corollary 2. Suppose that two simple closed curves C„ and Cj are given, such that Co completely encloses C^, as e.g. would be the case if C„ and C^ were concentric circles or confocal ellipses.
Suppose moreover that f{z) is an analytic function, which is regular at all points in the ring-shaped space contained between C'„ and Cj. Then by drawing a network of intersecting lines in this ring-shaped space, we can shew exactly as in the theorem just proved that the integral
jf{z)dz
is zero, cohere the integration is taken round the ivhole boundary of the ring- shaped space; this boundary consisting of tivo curves Co and C\, the one described in a positive (counter-clockivise) direction and the other described in a negative (clockwise) direction.
Corollary 3. And in general if any connected region be given in the .2-plane, bounded by any number of curves Co, Ci, Cn, ..., and it f(z) be any function of z which is regular everywhere in this region, then
jf(z)dz
is zero, where the integral is taken round the whole boundary of the region; this boundary consisting of the curves Co, Cj, ..., each described in such a sense that the region is kept either always on the right or always on the left of a person walking in the sense in question round the boundary.
An extension of Cauchy's tlieorem j f{z)d: = 0, to curves lying on a cone whose vertex is at the origin, has been made by Raout (Xouv. Annates de Math. (3) xvi. (1897), w .\. •!■
50 THE PROCESSES OF ANALYSIS. [CHAP. III.
pp. 365-7). Osgood (Bull. A mei: Mut/i. 'SoCjISW^) has shewn tliat tlie property jf(:)dz=0
tnay be taken as tlio defining-property of an analytic function, the other properties being deducible from it.
Example. A ring-.9haped region is bounded by the two circles \z\ = l and U| = 2 in the 2-plane. Verify that the value of I -^ , where the integral is taken round the boundary of this region, is zero.
For the boundary consists of the circumference |j| = l, described in the clockwi.se direction, together with the circumference U| = 2, described in the coiuiter-clockwise direction. Thus if for points on the first circumference we write z = e'', and for points on the second circumference we write 2 = 2e'*, then 6 and (f> are real, and the integral becomes
j„ e>« "^jo " 2e'* "'
or -27rj + 2jr(', i.e. zero.
37. The value of a function at a point, expressed as an integral taken round a contour enclosing the point.
Let C be a contour within which f{z) is a regular function of z.
Then if a be any point within the contour, the expression
/(£l z— a
represents a function of z, which is regular at all points within the contour C except the point z = a, where it has a singularity.
Now with the point z = a as centre, describe a circle <y of very small radius. Then in the ring-shaped space between 7 and G, the function
z — a
is regular, and so by Corollary 2 of the preceding article we have
r f{z)dz [fi^)dz^^ J c z-a Jy z -a
where I and I denote integrals taken in the positive or counter-clockwise sense round the curves C and 7 respectively. . ,
But (§35) fmA^^ffia) + (^-a)f'(a) + e(z-a)^^^
J y ^ (-t J y Z Oj
37. 38] THE FUNDAMENTAL PROPERTIES OF ANALYTIC FUNCTIONS. 51
where e is a quantity which tends to zero whi^ii the nulius of the circle 7 is indefinitely diminished. Thus
JC Z-(l 'JyZ-a Jy Jy
Now if at points on the circumference 7 wc write
z — a= /-e", where r i.s the radius of the circle 7, we have
JyS-a Jo re'" Jo
and i (^^=1 'ire'" (10 = 0;
J y .'0
also I edz | < 1? . i-rrr,
where -q is the greatest value of |6J for points ^ on 7; and therefore in the limit when /• is made indefinitely small we have
edz = 0. Thus
"/( |
2)d: |
■ = 27r7/(a), |
-C 2 |
— (I |
|
' ./•(") = |
1 27ri |
f f{z) dz ] Jc z-a |
or
This remarkable result expresses the value of a function/(2) at any point ■ a within a contour C, in terms of an integral which depends only on the value of f(z) at points on the contour itself.
Corollarji, If f{z) is a regular function of ^ in a ring-shaped region bounded by two curves C and C, and a i.s a point in the region, then
•^ zin J cz — a 277-1 J c- z — a
where C is the outer of the curves and the integrals are taken in the positive or counter-clockwise sense.
38. The Higher Derivates.
The quantity/' (2), which represents the limiting value of
f{^ + h)-f{z) h
when h tends to zero, is called the derivate of / {z). We shall now shew that /' {z) is itself an analytic Junction of z, and consequently itself possesses a derivate.
4—2
52 THE PROCESSES OF ANALYSIS. [CHAl*. III.
For if 0 be a contour surrounding the point z, and situated entirely within the region in which f{z) is regular, we have
/'(«) = Limit -^^^^Z^^)
//=()
h
= Limit \ If /(^^-f/(^^' 7t=o 2Trih\j c 2 — a - It Jcz — a
= Limit - — . / ^-^^-^ p;
/,=(, 27ri J c {z - a) (z - u - li-)
Now
^_l_r /(f)iL^+ Limit -^f /<^>^^
2Tri j c {z - a)- /,=() '2-jTi ] c {z - uf {z — a — h)'
j c (2 — a)" (« — «—/
is a finite quantity, since the integrand
(^^ — a/ (^ — a — /i)
is finite at all points of the contour C, and the path of integration is of finite
length. Hence
T- ■ /' r f{z)dz
Limit ^— . ^ --^ r- =0,
A=o 2-jriJc{z-aY{z-a-h)
w
and consequently , f (a) = -r — . ^ r„ ,
\ 2inJciz-ay
a formula which expresses the value of the derivate of a function at a point as an integral taken round a contour enclosing the point.
From this formula we have, if li be any small quantity,
f'(a + h)-f'(a)^ 1 ff(z)dz\ 1 1_
h 27riJ c h \{z — a-h)- {z — a)'
2m j c^ ^^' (z-a- h)"- (z - af
2-7ri] c {z — ay
where J. is a quantity which is easily seen to remain finite as h tends to zero.
Therefore as h tends to zero, the expression
/'{a + h)-f'(a) h
tends to a limiting value, namely
2 /• f{z) dz V 2m}c{z- ay
3.S] TlIK KINDAMKNTAI. I'Ki IPKUTIKS OK ANALYTIC FUNCTIONS. 53
Tlu' (|ii;uit.ity f (a) is tln'iet'ore an analytical function of «; its derivate, which is represented by the expression just given, is denoted by /'" (a), and is called the second derivate of /(a).
Similarly it can be shewn that f" (a) is an analytical function of a, possessing a derivate equal to
2.3 /• /(£) d^ K 2Tri}c{s-ay'
this is denoted by /'" (a), and is called the thifd derivate of f{a). And in general an Hth derivate ofy'(a) exists, expressible by the integral
n! r / (z) dz
2^ijc{2-ar+"
and having a derivate of the form
! r fiz)dz .
this can be proved by induction in the following way.
27ri
Let
Then
/'"'(a + A)-/"'i(a) ^ _n| f y(z)rfg f 1 1^ \
h 2in]c h |(^-a-A)»+' (^-a)"+']
^ ji^ |- /(g) rf^ {( _ h \-"- _ :27n'Jc(^-a)»+'/i (V ^ - a/
(n + l)!r /(^)d2
2771' }c{z- «)"+' + terms which vanish when h tends to zero.
Thus /(«+» (a) = ^"^^>' f iMll.
•^ ^ ^ 2'7ri Jc(^-a)''+='
c(^-a)" which establishes the required result.
A function ^vhich possesses a first derivate at all points of a region in the g-plane therefore possesses derivates of all orders.
Example '[. Verify the theorem
f{z)dz
/'"'(")=£i/.,
(z-aY by use of Taylor's Theorem.
By Taylor's Theorem we have
(j-a)»
dz.
n![ f(z)dz n'. f .f'a) + (^-a)f'(aH ... + ^'-f fKa) + ..
54 THE PROCESSES OF ANALYSIS. [CHAP. III.
But when k is an integer other than unit\-, I ; -,, ia zero, since , n— r resumes
its original vahie after describing tlie contour. So the only .surviving part of the right-
hand side is 5-^ ./(■') (a) j -^, or/W (a).
Example 2. Verify the same theorem by means of integration by parts.
We have
«j_ r f{z)dz f {n-\)\ f(z) \ {n-\)\C f'{z)dz
27iijc(!-u)"*'- o\ 27r/ (j-a)"/"*" 2jn' J c (z-a)"
and the first term is zero, since ^-^ resumes its original value when z makes the circuit
[z—a)
of the contour C. Proceeding in this way, we have
^ r f{z)dz _ 1 f f^"H^)dz_
2ffijr(2-a)"-'' Znijc z-a ^ ^'■
39. Taylors Theorem.
Consider now a function f{z), which is regular in the neighbourhood of a point z = a. Let C be the circle of largest radius which can be drawn with a as centre in the ^-plane, so as not to include any singular point of the function f{z)\ so that/(2) is a regular function at all points of G. Let z = a + hhe any point within the circle C. Then by §37, we have
<,jcz — a — h
Ztti Jcz — a — h
_ l_f , \ 1 h A"
SttJc-^^ ■ [z-a ■*" {z-ay'^ ' ' " "^ (z - a)"+^ "^ (^ _ a)»+i (^ - a - A)|
,-/ \ 7/'/ X /*"/■/// X A" JM„, / X 1 /" f(z)dz.h
=/(a) + V («)+2!/ («>+-"+n!-^'"'(«>+-2^Je(^i»-(^
But at points z on the circle 0, the modulus of -^ 1- will not exceed
^ z—a—h
some finite quantity ilf. Therefore
R
I 27rt J c (2 - a)""^' (z-a-h)
where R is the radius of the cii-cle C, so that 2TrR is the length of the path jof integration in the last integral, and R = \z — a\ for points z on the cir- cumference of C.
The right-hand, side of the last inequalit}' tends to zero as n increases indefinitely. We have therefore
/ (a + h) =f{a) + hj ' (a) + fj" („)+...+ J-"/ <») («) + ...,
I
89] THK ITN'DAMEN'TAI, I'UOPKUTIES OF AXALYTIC FUNCTIONS. 55
which wo can write
J\z) =/(a) + iz- a)/' (a) + ll^/"(„) + ... +<i^7(«. („)+....
This result is known iis Tmjlors Tlieurem; the proof we have given is due to Cauchy, and shews exactly for what range of values of z the theorem holds true, namely for all points s which are nearer to a than the nearest singularity o{J {z). It follows that the radius of convergence of a power-series is alw(ii/s such as just to exclude from the circle of convergence the nearest singularity of the function represented by the series.
At this stage we may introduce some terms which will be frequently used.
If/(a) = 0, the function /(2) is said to have a zero at the point z= a. If at such a point/" («) is different from zero, the zero of /(«) is said to be simple; if, on the other hand, the quantities /'(a), /"(a), .../'"~" (a) are all zero, so that the Taylor's expansion of f(z) at z = a begins with a term iu {z — a)", then the function f{z) is said to have a zei'O of the nth order at the point z = a.
Example 1. Find a function f{z), which is regular within the circle C of centre at the origin and radius unity, and has the value
a - cos 6 . sin 6
i
a^-2acos6-\-l a'' — 2acos0+l (where « > 1 and 6 is the vectorial angle) at points on the circumference of C. We have
J ^' 2nijc 2"-''
"' ['^ -nifl ja «-cosfl + t.sin5
= =—. I e "'* . idO . —r, — ~- , putting 2 = e'*
2m jo a'--2acose+\ '
= 2irlo ^=^ = 2;;..?„a-^jo^'""'"^^
ml = — -r-,, since the only non-zero term is that from k = n.
Therefore by Maclaurin's Theorem*,
or/(z)= - - for all points within the circle.
This example raises the interesting question, What is/(i) for points outside the circle?
Is it still 1 This will be discus.sed in 5§ 41, 42.
a-z J3 '
Example 2. Prove that the arithmetic mean of all values of j-" 2 a^z", for points z
on the circumference of the circle 1^1 = 1, is n„, itSa^s'' is regular at all points within the circle.
• Theresult /W=/(0) + 2/"'(0) + ^/" (0)+ ,..,
which is obtaioed by pnttiug a = 0 in Taylor's Theorem, is usually called Maclaurin's Theorem.
56 THE I'ROCESSES OF ANALYSIS. [CHAI'. III.
Let 2 ai,2''=/(2), so that a„=-^ -^ . Tlion the required mean is
„ , , whore z^e'".
In .
-r — . I - 4r,- ) where C is the circle, 2jrJ J c ^
/'"'(O) n! '
Example 3. Prove that if A is a given constant, and (1 -2zh + h-) ' is expanded in the form
l + hP,{z) + /,-^P,{z) + h^l\{z) + (A),
where P„ (z) is easily seen to be a polynomial of degree n in z, then this series convei-ges so long as 2 is in the interior of an ellipse whose foci are the points 2=1 and s= -1, and
whose semi-major axis is ^ (/i + j ) .
Let the series be first regarded as a function of h. It is a power-series in h, and therefore converges so long as the point h lies within a circle on the /j-plane. The centre of this circle is the point A = 0, and its circumference will be such as to pass through that singularity of (l—2zh + h^)~- which is nearest to h=0.
But l-2zh + h-' = {h-z+ V^^^) (A - ; - Vs- - 1 ),
so the singularities of {l-2zh+k-)~^ are the points k = z-{z--l)- and /i=z + {z^--l)-, at which it is infinite.
Thus the series (A) converges so long as \h\ is less than either U--(.-2-l)i|or:2-|-(22-l)i|.
Now draw an ellipse in the 2-plane passing through the point z and having its foci at the points 1 and - 1. Let a be its semi-major axis, and 6 the eccentric angle of z on it.
Then z = acose + i{a:'-l)- aind,
which gives 2 ± (s^ - 1 )* = {a ± (a^ - 1 )*} (cos d + isind),
so l2±(22-l)i| = a + (a2_l)J.
Thus the series (A) converges so long as h is less than the least of the quantities o-f(a2-l)' and a-(a^ — lY, i.e. so long as h is less than a— (a^- 1)*. But
A = « — («- — !)' when a = -[h + -.j.
Therefore the series (A) converges so long as z is within an ellipse whose foci are 1 and - 1, and whose semi-major axis in- (h + jj.
40. Forms of the remainder in Taylor's Series.
The form found in the last article for the remainder after n terms in
Taylor's series is
^J_ /• f{z)h-dz
" ^■n-iJc{z-ciT{z-a-hy
40. 41] THK KUNDAMENTAI. ritdPEllTIES OK ANAIA'IIC FUNCTIONS. 57
It is not (litticult to derive from this cxjiicssion the forms of the remaimler usually given in treatises on the Diti'erential and Integral Calculus. For
r'' (h-ty-'dt (z-a- n"+'
on integrating by parts the (juantity // | — — - — ^„^, , we have
\"-dt
[>■ (h-t)"-'dt h" , , /•* {h-t)"dt
by successive repetition of this process,
So
{z -a)»(z-a-h)'
R =-■ r-, / <"' (a + eh) {h - 0"-' dt,
( /( — 1 ) : Jo
or R„ = . \., f (h - t)"-' f "" (<f + t) dt,
{n — 1)1 Jo
which is a newjormfor the remainder.
Xow suppose that all the quantities concerned are real. Then along the line of integration, {h -^ 0"~' has a fixed sign, so
where H lies between the greatest and least values of/'"'(a + 0 between t = 0 and t = /(. We can therefore write H =/"" (a + dh), where 0 < ^ < 1, and then
1
('"
or \ R = -J'^">(a + eh),
n ;
which is Lagrange's form /or the remainder.
Darboux gave in 1876 {Journal dn }[ath.{Z) il. p. 291) a form for the remainder in Taylor's Series, which is applicable to complex variables and resembles the above form given by Lagi-ange for the case of real variables.
41. The Process of Continuation.
Near every point P {z^) at which a function f{z) is regular, we have seen that there is an expansion for the function as a series of ascending positive integral powers of {z — z^), the coefficients in which are the suc- cessive derivatfis of the function at z^.
Now let A be the singularity of /(^) which is nearest to P. Then the circle within which this expansion i.s valid has P for centre and PA for radius.
58 THE PROCESSES OF ANALYSIS. [cHAP. III.
Suppose that wo arc given the values of the function at all points of the circumference of this circle, or more strictly speaking, of a circle slightly smaller than this and concentric with it: then the preceding theorems enable us to find its value at all points within the circle. The question arises, How can the values of the function at points outside the circle be found ?
In other words, c/iven a pvwer-series tvhich converi/es and rej)reseiits a function only at points within a circle, to derive from it the values of the function at points outside the circle.
For this purpose choose any point P, within the circle, not on the line PA. We know the value of the function and all its derivates at P,, from the series, and so we can form the Taylor series with P, as origin, which will represent the function for all points within some circle of centre Pj. Now this circle will extend as far as the singularity which is nearest to Pj, which may or not be A ; but in either case, this new circle will generally* lie partly outside the old circle of convergence, and fo7' points in the region which is included in the new circle but not in the old circle, the neiu series will firnish the values of the function, although the old series failed to do so.
Similarly we can take any other point Pn, in the region for which the values of the function are now known, and form the Taylor series with Pj as origin, which will in general furnish the values of the function for other points at which its values were not previously known ; and so on.
This method is called continuation-^. By means of it, starting from a representation of a function by any one power-series we can find any number of other power-series, which between them furnish the value of the function at all points where it exists; and the aggregate of all the power-series thus obtained constitutes the analytical expression of the function.
+ r. + ...
Example. The series |
|
a + a^ + a^"*" |
|
represents the function |
|
/»%■.. |
only for points z within the circle \z\ = a.
But any number of other power-series exist, of the type 1 , 2-6 , {z-hf (z-bY
a - b^ {a-bf^ {a-bf^ (a-b)* ' which represent the function for points outside this circle.
* The word "generally" must be taken as referring to the cases which are likely to come under the student's notice before he reads the more advanced parts of the subject, t In German, Fortsetzimg.
42] THE KUN'OAMEXTAL I'HOPEKTIES OF ANALYTIC FUNCTIONS. 59
On functions to which the continuation-process cannot be applied.
It is not always jiossible to carry out the process of continuation. Take as an example the function f{z) defined by the jxiwer-series
which clejvly conveivos in the interior of a circle whose radius is unity and whose centre is at the origin.
Now as c approaches tlie value +1 bv roil values, the value of /(;) obviously tends towaitls + X ; the point + 1 is therefore a singularity oi f{z).
But f{z) = z^-\-f{z'\
so if ; is such that ^-=1, and therefore f{z-) is infinite, then f{z) is also infinite, and so « is a singularity of/(j) : the point ;= - 1 is therefore a singularity of /(z).
Similarly since
f{z)=z^ + ^+f{z^\
we see that if z is such that ;^ = 1, then z is a singularity of/(-) ; and in general, any root of any of the equations
z-=\ z*=\ '■*=! 'i''=l
is a singularity o{ f{z). But these points all lie on the circle \z\ = \ ; and in any arc of this circle, however small, there are an infinite number of them. The attempt to carry out the proce.ss of continuation will therefore be frustrated by the existence of this uubroken front of singularities, beyond which it is impossible to pass.
In such a case the function f{z) does not exist at all for points z situated outside the circle |z| = l ; the circle is .said to be a limiting circle for the function.
42. The identity of a /unction.
The two series
1+Z+ z- + z''+...
and -i+(z-2)-{z--2)- + (z-2f-(2-'2Y + ...
are not simultaneously convergent for any value of z, and are distinct expansions. Nevertheless, we generally say that they represent the same function, on the strength of the fact that they can both be represented by the
same rational expression .. — .
This raises the question of the identity of a function. Under what circumstances shall we say that two different expansions represent the same function ?
We shall define a function, by means of the last article, as consisting of one power-series together with all the other power-series which can be derived from it by the process of continuation. Two dififerent analytical expre.ssions will therefore be regarded as defining the same function if they represent power-series which can be derived from each other by continuation.
It is important to observe that a single analytical expression can represent different functions in different parts of the plane. This can be seen from the following example.
60 THE PROCESSES OF ANALYSIS. [CHAP. III.
Cousidor the series
Ihl) \i {' - J) iih^ - 1 +'."-) •
The sum of the first n terms of this series is
2 J 1+2"
h^'
The series therefore converges for all finite values of z. But since when n is infinitely great, z"' is infinitely small or infinitely great according as \z\ is less or greater than unity, we see that the sum to infinity of the series is
s when \z\<l, and - when [2l>l. Tins series therefore represents one
z
Junction at points in the interior of the circle \z\ = 1, and an entirely different function at points outside the same circle.
Example. Shew that the series
Z^ ^ ZT 2 + 32 + 52+^2+...
and MJi_,_2.1.p. V + ?.i.l.fJ^Y_ I
2ll-£2 3 .3 Vl-^V 3 5 5 Vl-2V J
represent the same function in the common part of their domain of convergence.
43. Laurent's Theorem.
A very important extension of Taylor's Theorem was published in 1843 by Laurent ; it relates t,o the expansion of functions under circumstances in which Taylor's Theorem cannot be applied.
Let C and C be two concentric circles of centre a, of which C is the inner; and let f{z) be a function which is regular at all points in the ring-shaped space between G and C. Let a + h be any point in this ring-shaped space. Then we have (§ 37, Corollary)
/ (a + h) = i- ( ~l^l~ dz-^( Mil- dz •' liri ] cz — a — h 1-ni Jc z — a- h '
where the integrals are supposed taken in the positive or counter-clockwise direction round the circles.
This can be written
If (1 h A" /t"+' 1
1 f a z-a {z - g)" {z - a)"+' \
'^ 2iTi]c'-'^^'\h^ h''^-^ h^^ ^h^^'l'z-a-h)\
dz.
. 4.'{] THK FUNDAMENTAI, I'KOI'ERTIES OK ANAI.VTIC FUNCTIONS. Gl
\Vc find, as in tlio jiioof of Taylor's 'J'lu'oriin, tliat
tend to zero as /; increases indefinitely ; and thus we have
J\a + h) = «„ + (;,/( + aJr + . . . + ^^' + ^|^ + . . . ,
This result is Laurent's Theorem; changing the notation, it can be expressed in the following torni : //' z be any point in the rinr/shaped space tvithin which /(z) is regidar, and which is bounded by the two concentnc circles C and C" of centre a, then J (z) can be expanded at the point z in the form
y(.) = «o + .,(.-«) + «.(.-ay+... + -A-^^ + ^-^, + ...,
where . «« = ^^J^ (^i)"-« ^"' '" = slL^^- ''^"-'/('^'^^-
An important case of Laurent's Theorem arises when there is only one singularity within the inner circle C", namely at the centre a. In this case the circle C can be taken to be infinitely small, and so Laurent's expansion is valid for all points in the interior of the circle G, except the centre a.
Example 1. Prove that
JH)
/„ {.V) + z J, {X) + .'2 Jo (,x) + ...+ z" J-„ (.r) + . . .
1 Pt
where •^n(^)=:s~ I cos{n6—xiiin6)dd.
2ir J 0
For Laurent's Theorem gives
-(z--) b h
"^=Ljf''^^-^'"-Lh.
where «„=., ,/e^^ ' ::STi and 6„ = 5--. e"^ '^ s"-'dz,
and where C and €' are any circles with the origin as centre. Taking C to be the circle of radius unity, and writing z=e' , we have
girslnS _ 0-"'' l.dd
1 fi'
1 1^" = — / cos {n6- X sin 6) dd,
62 THE I'ltOCESSKS OF ANALYSIS. [cHAP. 111.
since the iiarts of I sin (h^) -. ''siii 5)fW which arise frciiii 6 iuid 27r - 5 will destroy each other. Thus
Now 6„ = (-l)"a„, since the function expanded is unaltered if - be written fori.
Thus
6„ = (-!)■'./„ (.I'), which completes the proof.
Examjjle 2. Shew that, in the ainiulus defined by
!a|<|2 <\H,
( hz li
the expression ■!; z-r, u can be exi)anded in the form
^ |(s-«)(6-2)J
where ^,= ^2^ W--^TVUT^)\ \h) '
For by Laurent's Theorem if C denote the circle \z\ = r, where |a|<?-<|6|, then the coefficient of i" in the required expansion is
_1_ /" cfo_ f hz 1 i
27ri jc^"* 'l(z- a) (6- -')/ '
Putting 3 = «'"*, this becomes
hi:- -<«('-r")''('-"-"")"-
The only terms which give integrals diti'erent from zero are those arising from k = l + n. So the coefficient of 2" i.s
1 r^"- ...^ 1.3...(2Z-1) 1.3...(2; + 2m-1) a' — ' do £, —
-■i\
6"
■ Similarly it can be shewn that the coelficient of — is S^a'^.
Example 3. Shew that
U2+- ., b, 6,
1 /■■^'^ where «n= 5~ I «''"■'""' '"^ * cos {(tt - -y) sin d - nd] dO,
1 r^ir
and &n = s- I (i'"+'-''™'**cos{(t»-w)sin5-n^}t;5.
^n J 0
44] THE FUNDAMENTAL PROPERTIES OF ANALYTIC FUNCTIONS. G3
44. The nature of the singularities of a one-valued function.
Consider now a function f{z) which is regular at all puiuts of a certain region in the 2-plane, except a point 2 = «; so that the point a i.s a singularity of the function f{z).
Surround the point a by a small circle 7, with a as centre. Then in the ring-shaped space between 7 and some larger concentric circle C, the function f {z) can by Laurent's Theorem be e.xpauiled in the form
A, + A,{z- a) + A.,{z-ay + A.,{z-af+...
■B, Ba B-i
+ — ~ +f — .„ + , ., + ■■■■ z — a {z — a)- {z — af
The terms in the last line are called the Pnncipal Part of the expansion of the function at the singularity a ; if they were non-existent, the function would clearly be regular at the point ; so they may be regarded as consti- tuting the analytical expression of the singularity.
Now these terms of the Principal Part may be unlimited iu number, i.e. the series
B-, Bn Bn
^ H 1- — h...
z — a (z — a)- (z— af
may be an infinite series; in this case the point a is said to be an essential singulariti/* of the function / (z). Or on the other hand, they may be limited in number, i.e. the series just written down may be a terminating series ; so that the expansion can be written in the form
In this case the function is said to have a pole of order n at thelpoint a. When /( is unity, so that the expansion is of the form
■^ ■\-A„+A,{z-a)->rA.,{z-a)--^...,
the singularity is said to be a simple pole. Example 1. Find the singularities of the function
c
£
Near * = 0, the function can be expanded in the form
a''"2!o2"'"3!a3"'"'"
* The name e»»ential ningularitij is also applied to any singularity of a one-valued function \rhich is not a pole, i.e. to singularities for wliich no Laurent expansion at all can be found.
64 THE PHOCESSES OF ANAIA-.SIS. [CHAP. III. I
c
e'a .a -t/c , IN , .,. .
or e a ( - + „ +|«i.sitivo i)cj\vt'i'.s ot ^.
z \a 2/ ' '
There is thereforo a simple pole at z = 0. Similarly there is a simple pole at each of the points 2jr«ia (re= ±1, ±2, ±3,...).
Near z = a, the function can be expanded in the form
c
e.e " -1
c . c2
^ + 7^ + 2!(- --•'+-
/ : — II
e 1 + : +.
V a
which gives an expansion involving all positive and negative powers of (i-a). So there is an essential singularity at r = a.
There is also an essential singularity at r = QO, as will be seen after the explanations of the next article.
Example 2. Shew that the function defined by the series
has a simple pole at each of the points
s = (l+-\e'" {l: = 0,\,2,...n-l; ?»=1, 2, ...oo ).
(Cambridge Mathematical Tripos, Part II., 1899.)
45. The point at infinity.
The behaviour of a function /(z) for infinite values of the variable z can be brought into consideration in the same way as its behaviour for finite values of z.
For write ^ = - , so that the infinite values of z are represented by the
point z'=0 in the z'-plsine. Let f{z) = (f>{z'). Then the function 4){z'} may have a zero of order m at the point z' —0; in this case the Taj'lor expansion of (f> (z) will be of the fonn
Az''" + Bz'"'+^ + Cz"""*- +...,
and so the expansion oi J (z) valid near z = cc will be of the form
A ^ C
J \^'~ ^m "^ ^m+i gm+2 + . . . .
In this case, / {z) is said to have a zero of order 7n at z = <x> .
45] Tin: l-UNDAMENTAL PROPKUTIES OF ANALYTIC FUNCTIONS. 05
Again, the function (f> (z) may luive a pole of order m at the point z' = 0; ill this case,
and so for large values of z,f{3) can be expanded in the form
N P J {z) = Az"' + Bs"'-' + Gz-^-- +...+ Z^ + j¥ +-+-„+... .
Z Z'
In this case, « = x is said to be & pole of order m for the function f(z).
Similarly J (z) is said to have an essential singulariti/ at z = x> , \{ <}) (z) has an essential singularity at the point z' = 0. Thus the function e* has an
essential singularity at ^ = ao , since the function e^' or
l + ? + 2i^ + 3?3 + - has an essential singidarity at z' = 0.
Example. Discuss the function represented by the series
The function represented by this series has singularities at « = — and z= -,
(n=l, 2, 3, ...), since at each of these points the denominator of one of the terms in the series is zero. These sing\ilarities are on the imaginary axis, and are infinitely numerous near the origin c = 0 : so no Taylor or Laurent expansion can bo formed for the function valid in the region immediately surrounding the origin.
For values of z other than these singxdarities, the series converges absolutely, since the
ratio of the (7i + l)th term to the nth is ultimately . i-, ■> i which is very small when n
is large. The function is an even function of z (i.e. is unchanged if the sign of : be changed), is zero for all infinite values of z, and is regular at all points outside a circle C of radius unity and centre at the origin. So for points outside this circle it can be expanded in the form
j2 + ^+i«+-'
«> 1 n~2n
where, by Laurent's Theorem,
" ^nl J c
and the coefficient of - on the right-hand side of this equation is ^ — j .
w. A. 5
66 THE PHUCKSSES OF ANALYSIS. [CUAP. III.
Therefore, since only terms in - can furnisli non-zero integnU.s, we have
2m n=o J a n]
" (-1)*-'
,,=0 n!a"™
= (-l)*-'e''".
Therefore for large values of z (and indeed for all points z outside the circle of radius unity) the function can be expanded in the form
J- _L -L
The function has a zero of the second order at i = oo, since the expansion begins with
■ 1 a term in ^ .
46. Many-valued functions.
In all our previous work we have supposed the function f(z) to have one definite value corresponding to each value of z.
But functions exist which have more than one value corresponding to each value of z. Thus the function z- has two values (viz. + 'Jz and — vz) corresponding to each value of z, and the function tan~' z has an infinite number of values, expressed by the formula tan~' z ± kw, where k is any integer.
We may however for many purposes consider + V^ and — */z as if they were two distinct functions, and apply to either of them separately the theorems which have been investigated in this chapter. When we in this way select some one determination of a many-valued function for considera- tion, it is called a branch of the many-valued function. Thus the values log z, log z + 2iri, log z + 4771, . . . , would be said to belong to different branches of the function log z.
There will be certain points for which the values of the function given by different branches coincide : these points are called branch -poi7its of the function, and must be included among its singularities. Thus the function z^ has a branch-point at ^ = 0, since either branch there gives the same value, zero, for the function.
It must not however be supposed that the branches of a many-valued function really are distinct functions. The following example shews how the different branches of a many-valued function change into each other.
Let f{2) = 2K
•iU] THE FUNDAMENTAL I'UOl'EKTIE.S OK ANALYTIC FUNCTIONS. 67
Write 2= ('(cos 0 + 1 sin ^), wheiv 0 < 0 < 2Tr. Then the two values of f (z) are
,- e . . o\ , ,- 1 e . . e
+ v»- 1 COS ^ + I sin ^1 and - vr ( cos ^ + i sm .^
Lot us take the tornier of those vahies, and consider its changes as the point z iloscribos a circle round the origin (z = 0). As the point travels, r is unchanged, but d constantly increases, and when the point roaches again the starting-point after completing the circuit, 0 has increased by 27r. The function has therefore become
r : d + 2-ir . . 0+2-rr- + vr (cos — 2 1" * si'i — 2~
r ( 6 ■ ■ 0
"1 — V )• I cos ^ + I sin -
In other words, the branch of the Junction with which we star-ted has passed over into the other branch.
In following the succession of values oi f(z) along a given path, the final value is deduced without ambiguity from the initial value ; and all con- ceivable paths lead to one of two final values, viz. V^r and — V^. But it appears from the above that it is not possible to keep these branches per- manently apart as distinct functions, because paths lead from one value to the other.
The idea of tlic different branches of a function helps us to understand many of the "paradoxes" of matliematics, such as the following.
Consider the function
for which -r- =2'(l+log2).
When £ is negative and real, -; is not real. Now if 2 is a negative quantity of the form Q-^ (where p and q are positive or negative integers), u is real.
If therefore we draw the real curve
ti = z',
we have for negative values of z a series of conjugate points, arranged at infinitely small intervals of z : and thus we may think of proceeding to form the tangent as the limit of
the chord, just as if the curve were continuous ; and thus -=- , when derived from the
ctz
inclination of the tangent to the axis of x, would appear to be real. The question thus arises. Why does the ordinary process of differentiation give a non-real value for -y ? The
68 THE PROCESSES OF ANALYSIS. [CHAP. III.
explanation i.s, that these conjugate points do not all arise from the same branch of the function m = 2'. We have in fact
and log z has an arbitrary additive part 2kni, where k is any integer. To each value of k corresponds one branch of the function «. Now in order to get a real value of u when z is negative, we have to choose a suitable value for k : and this value of k varies as we go from one conjugate point to an adjacent one. So the conjugate points do not represent values of u arising from the same branch of the function 'ii,=z', and consequently we cannot expect
the value of -^ to be given by the tangent of the inclination to the axis of x of the
tangent-line to the series of conjugate points.
Example 1. If log 2 be defined by the equation
I log z = Limit n (j" - 1 ),
shew that logs is a many-valued function, which increases by 27r« when 2 describes a closed path round the origin.
For put z = r (cos 6->ri sin 9).
Then one of the values of log 2, on this definition, is
\r^ cos - + I sm - —IV, I V " 'V J
Limit n \r" ( cos - + i
71 = 00
1
where r" is the positive nth root of r. This can be written
Limit n {r"- \} + id.
When 2 describes a closed path round the origin, the first term in this expression remains unaltered, while the second increases by 2^1 ; hence the result.
Example 2. Find the points at which the following functions are not regular.
Answer, s=od . Answer, 2 = 0, ±n, ±2jr, iSir, ....
Answer, 2 = 2, 3.
Answer, 2 = 0. Answer, 2=0, 1, 00 .
Example 3. Prove that if the diSerent values of a', con'esponding to. a given value of z, are represented on an Argand diagram, the representative points will be the vertices of an equiangular polygon iu.scribed in an equiangular spiral, the angle of the spiral being independent of a.
(Cambridge Mathematical Tripos, Part L, 1899.)
{a) |
2". |
(b) |
cosec 2. |
(«) |
2-1 |
22- 52-1-6 ■ |
|
(d) |
1 e". |
(e) |
{(2-l)2}l |
47, 4.S] THE FUNDAMENTAL PKOPERTIES OF ANALYTIC FUNCTIONS. 69
47. Liouville's Theorem.
Wo know by § 38 that '\i J\z) bo any function of z which is regular at all points of the z-planc within a circle G, of centre a and radius r, then
Now let M bo the greatest value of \f{z)\ at points on the circle G. Then this equation gives (§ 32)
i/'"'(«)k2^,-ii-'--"''
n\M
From this inequality an important consequence can be deduced. Suppose that / {z) is, if possible, a regular function of z over the whole z-plane, including infinity, i.e. that it has no singularities at all.
Then in the above equation M is finite when r is infinite, whatever n is ; and therefore /*"' (a) is zero for all values of n and a, i.e. /(a) is a constant independent of a. We thus arrive at Liouville's theorem, that tfie only function which is regular everyiuhere is a constant.
As will be seen in the next article, and again frequently in the latter half of this volume, Liouville's theorem furnishes short and convenient proofs for some of the most import;int results in Analysis.
48. Functions with no essential singularities.
Wo shall now shew that the only one-valued functions which have no singularities in either the finite or infinite part of the plane, except poles, are rational function,').
For let/(^) be such a function; let its singularities in the finite part of the plane be at the points Cj, c.,, ... Cj,- : and let the principal part (§44) of its e.xpansion at the pole Cr be
rtr 1 ar o ar tir
Z — Cr {Z — Cr)' '" {z — C^"-'
Let the principal part of its expansion at the pole ^ = x be
-^^ ^ a,3 + a-fi" + . . . + «„2" ;
if 2 = X is not a polo, but a regular point for the function, then the coefficients in this e.xpansion will be zero.
Now the function
'^ \ r=Az-c,'^ {z-cry---^{z-crrA
- a,z — a^'- — ... — «„«"
l^
70 THE PROCESSES OF ANALYSIS. [CHAP. III.
has clearly no singularities at the points c,, Cj, ... Cj, x ; it has therefore no singularities at all, and so by Liouville's theorem is a constant ; that is,
f{z) = constant + a^z + a^z^ + . . . + «»«"
f{z) is therefore a rational function, and the theorem is established.
Miscellaneous Examples.
1. Obtain the expan.sion
2. Obtain the expan.sion
111 I
+§rsfJ.I/"<»)-/"<'»
+2r4rI^^-^'*'(»)--^'''(^)J
^^0)]
+ ,... (Corey.)
3. Obtain the expansion
+ ... (Corey.)
2 fz-a\^\ + 3-.'
4. In order that vahies U-\- Vi, whicli are given as continuous functions of the arc of a circle, .should be the boundary values of an analytic function, shew that it is necessary and sufficient :
{a) That — ^ — -^-^-j — ^^ ^ at the place i|/' = 0 should be uniformly integrable for
all values of a ; (6) That the \-alues of V shall be given by
r(a) = ^ {" {U{a-^)-U{a + yir)}coth^d^ir. (Tftuber.)
MISC. KXS.] TlIK KUNDAMKNTAI, I'KOI'EKTIES OF ANALYTIC FUNCTIONS. 71 b. .^liow that for the series
2 ^-
n =0 •• T '
the region of convergence consists of two distinct areas, namely outside and inside a circle of riulius unity, and that in each of these the series represents one function and represents it ciimjiletely.
(Weierstrass.) 6. Shew that
1.^1^3.4^ ^1.3...(2«-1)
(1-.^) J=l+ .= + -4 + ...+
.2n
2 '2.4 2.4...2«
(Jacobi & Scheibner.)
7. Shew that
(I :)-n_-i I ^. I m{m + \)_, ^ ^ _ ^ TO(TO + l)...(TO+ra-l),„ 1 2 ! ■■■ « !
7n(m + l)...(m + M) ,, , /'^ „ , , ,
(Jacobi & Scheibner.)
8. Shew that
, n_.2^-J (w+2)(m+4)...(m+2n) /■%„ + 2„m _,o._i . + ^' -■' («i + l)(TO + 3)...(m + 2«-i)j„' ^' '^ '*'•
(Jacobi & Scheibner.)
9. If, in the expansion of (a + a^z + a.^:-)'" by the multinomial theorem, the remainder after n terms be denoted by R, so that
shew that
Ji-(a+a,.+a^) jo («+«,< + «./)-*■ '•
(Jacobi & Scheibner.)
10. If (aj + aii + a./-)-"*-' I (a^ + aj^ + «./-')"' (i<
.'0
be expanded in ascending powera of z in the form
A,: + A.a'' + ..., shew that the remainder after («— 1) terms is
{aa + a^z + a.yz'^)-'^-^ I* (a„ + a,< + «./)"' {?ia„J„- (2nj + ;i + 1) a.,A„_it} t"~i di.
(Jacobi & Scheibner.)
11. Shew that the series
where X„(.)= -1+^-^", + |^ - ... + |^, .
72 THE PROCESSES OF ANALYSIS. [CHAP. III.
and where cj){z) is a regular function of z near z = 0, is convergent in the neighbourhood of the point ^ = 0 ; and shew that if the sum of the series be denoted by/(z), then f{z) satisfies the differential equation
/ ' (z) =f{z) - 4> (z). (Pincherle.)
12. Shew that the arithmetic mean of the .squares of the moduli of all the values of the series 2 a^z* on a circle |2| = ?', situated within its circle of convergence, is equal
0
to the sum of the squares of the moduli of the .separate terms.
(Gutzmer.)
13. Shew that the series
2 g-2(am)*2m-l m=I
converges when |c| < 1 ; and that the function which it represents can also be represented when |z| < 1 by the integral
v4 /■" .-z d.v
©7:
and that it has no singularities except at the point 2 = 1. (Lerch.)
14. Shew that the series
- (2 + 2- ) + - 2 |(i_ 2„_2y'2i)(2v + 2i''2i)2"'"(l-2i/-2v'2-ij)(2i' + 2i''z-it7j ' in which the summation extends over all integral values of u, v\ except the combination (i' = 0, v =Q), converges absolutely for all values of z except pm'ely imaginary values ; and that its sum is + 1 or - 1, according as the real part of z is positive or negative.
(Weierstrass.)
15. Shew that sin I" ( 2 + 7 ) f can be expanded in a series of the type
ao + ai2 + «22-+... + -J + -H.-.,
z z
in which the coefficient of either 2" or 2~" is
1 C^^ ^— I sin (2m cos 6) cos iiB ad.
16. If
hew that/(2) is finite and continuous for all real values of 2, but cannot be expanded as Maclaurin's series in ascending powers of 2 ; and explain this apparent anomaly.
CHAPTER IV. The Uniform Convergence of Infinite Series.
49. Uniform Convergence.
We have seen* that the sum of <a convergent series of analytic functions of a variable z can have discontinuities as z varies. It was found by Stokes"!* and SeidelJ in 1848 that this can never happen except in association with another phenomenon, that of non-uniform convergence, which will now be investigated.
Consider the series
,, 2^ , Z+Z^Z-l)
(I +2z){l +2z + z') {1 + 2z + z"-){l +3z + z')^ '"
z + 2"{z-l)
"^(1 +nz+ z"){l+(n+ l)^ + 2:»+'}'^ "■■■
We shall first shew that this series is convergent for all values of z except certain isolated points.
For, except for the roots of 1 + nz + z'^ = 0, the nth term can be put in the form
1 1
l+nz + z"^ 1 +(n + l)2 + 2"+i'
so the sura of the first n terms is
1 1
Sn =
l + 2z l + (n + l)z + z"+''
which, as n becomes infinitely great, tends to the value = ^ for all points
except z = 0 : and for z = 0, we have S ~ 0.
Thus (except at the roots of the equations 1 + nz + 2" = 0) the series converges ; and it represents a regular function, except at ^ = 0, where it has a discontinuity.
• In § 42.
t Collected Papers, Vol. i. p. 236.
i Miinch. Abh.
V
74 THE PROCESSES OF ANALYSIS. [CHAP. IV.
What lies at the root of the discontinuity ? The remainder after n tevms is
R = 1
For ordinary values of z, say z=l, this remainder decreases rapidly as
71 increases. Thus if ?i = 10, z = 3, the remainder = „ A — on' ^ negligible
quantity. But now let z approach near to its discontinuity 0 : say
•^"innnnnn' '^^^'^ with this value of z, the remainder after 1000 terms is
nearly 1, and the remainder after lOOOOOO terms is still nearly ^. This
shews that, as z approaches the discontinuity at z = 0, the terms which contrilmte sensibly to the sum tend to recede to the infinitely distant part oj the series, so the first 1000 terms do not furnish a good approodmation at all.
We can express this analytically as follows : — The number of terms n, which we have to take in order to make \Rn\ less than a given small positive quantity e, tends to oo as we approach the point of discontinuity.
This circumstance is the basis of the following definition : —
Let iti {z) + Ma {z) + «3 {z) 4- v-i («) + ...
be a series of functions of z, which is convergent at all points z within a given area in the ^-plane. Let R^ be the remainder after n terms. Then since the series converges, if we take a small finite quantity e we can find at any point on the area a number r (varying from point to point) such that \Rn\ < e so long as n > r. If the numbers r corresponding to the aggregate of points in the vicinity of a given point z are all less than some definite finite number, the series is said to he, uniformly convergent at the point z; but if near any point z the number r tends to infinity, so that no definite upper limit can be assigned to it, the convergence of the series is said to be non-uniform* in the neighbourhood of the point z. s/
Example 1. Shew that the .series
which converges al)sohitely for all real values of z, is discontinuous at 2 = 0 and is non- uniformly convergent in the neighbourhood of 2 = 0.
The sum of the first n terms is easily seen to be 1+^-— t^ jt-^i- So when z is not
(1+2^)" '
zero the sum is 1 +2^, and when z is zero the sum is zero.
* An interesting geometrical treatment of uniform convergence is given by Osgood in Vol. iii. of the Bull, of the Amcr. Math. Soc. p. 59 (1896).
49] THE UNIFORM CONVEROENCE OF INFINITE SEUIES. 76
The remainder after >t toruis is rx j; — ■ • This can bo luiido smaller than any
(1+2^)"-' ^
log-
assicncd small tinite positive (uiantitv < bv choosing » so that »-!>] ]!ut as
' ' • • " log(l+i-)
t tends to zero, , , , .,, tends to inlinitv, so n must tend to infinity, i.e. we have to
include an infinite number of terms in order to get the remainder less than t. This series is therefore non-uniformly convergent in the neighbourhood of 3 = 0.
Example 2. Shew that at 2 = 0 the sum of the series
+ /. , 1X/0., 1S + -" + ./„ ,x.. ■...- . .. + -
\{z+\y {z+\){iz+\y-^ {{n-\)z+\]{m-{-l is discontinuous and the series is non-uniformly convergent.
The sum of the first n terms is easily seen to be 1 , : so when z is zero the
sum is 0.
The remainder after n terms of the series is : so when z is nearly zero, say
t = one-hundred-millionth, the remainder after a million terms is ^^— or 1 - , so
Too+1
the first million terms of the series do not contribute one per cent, of the sum. And in general if c be small, it is necessary to take n large compared with the large quantity
- in order to make the remainder after 7i terms small. There is therefore non-uniform z
convergence in the neighbourhood of 2 = 0.
Example 3. Discuss the series
- zl7i(n + l)z'-l}
„=, {l + ?lV}{l-f(n-^-l)2 2i!}• The »!th term can lie wi'itten r k-„— ,— ^ rrrro. so the sum to infinity is ,— — ,, ,
l-f»V l+(,j+l)2i2' •' 1-1-2^'
and the remainder after n terms is ,— ^ ~-« •
l+{n + iyz^
. However great ?t may be, if we take ; equal to , this remainder will have a finite
value, namely i ; the series is therefore non-uniformly convergent at 2 = 0.
Note. In this example the sum of the series is not discontinuous at 2 = 0.
Cayley* regards non-uniform convergence as consisting essentially in the occurrence of a discontinuity in tlie sum of a series. The condition for a discontinuity in a series
«1 (2) + «2(-')+«.1 (')+•••
at the point z=a is that the series
* Mr(a)-M,.(2) r=l a-z
shall have an indefinitely large sum when (a—z) is indefinitely small.
• "Note on Uniform Convergenoe," Proc. Roy. Soc. Edinb. xix. (1891-2), pp. 203-8.
76 THE PROCESSES OF ANALYSIS. [CHAP. IV.
Thus in the series
which is non-uniformly convergent and discontinuous at 3=1, we have
M. (a) - u. (z) ,
a-z '
'*' U (dii "" U ( Zl 1
so the sum of the scries 2 " "^--^ is :; — , which is infinite for z = l.
„=i a-z \-z'
50. Connexion of discontinuity with non-uniform convergence.
We shall now shew that the sum of a series of continuous functions of z, if it is a uniformly converr/ent series for values of z witliin certain limits, cannot he discontinuous for values of z within those limits.
For let the series be f{z) = Mj {z) + u„ (z)+ ...+ «« (2) + . . . = <S„ {z) + i?„ {z), where J?„ is the remainder after n terms.
Since the series is uniformly convergent, we can to any small positive number e find a corresponding integer n independent of z, such that
\Rn{z)\ < -^ for all values of z within the area.
o
Now n and e being thus fixed, we can, on account of the continuity of 8n {z), find a positive number rj such that, when \z — z'\ <r), the inequality
\S,Az)-S„{z')\<^ is satisfied.
We have then
l/(^) -/(^') I = ! {Sn (^) - <S„ {z')] + E,, (z) - R„ (z) I
< I ,S'„ (Z) - Sn (/) \ + \Rn(2)\+\ Rn i^') I
which establishes the result.
Example 1. Shew that at 2 = 0 the series
1 1
where u-^ (2) = 2, m„ (2) = 2^" - ' - 2^" " \
and real values of 2 are concerned, is discontinuous and non-uniformly convergent.
1 The sum of the first n terms is 2'^""^ ; as n tends to infinity, this quantity tends to 1, 0, or - 1, according as 2 is positive, zero, or negative. The series is therefore absolutely convergent for all values of 2, and has a discontinuity at 2 = 0.
The remainder after n terms, when 2 is small and jjositi ve, is 1 - 2-" ~ ■ ; however great n may be, by taking 2 = e~(2"-i) we can cause this remainder to take the value 1 — , which is different from zero. The scries is therefore non-uniformly convergent at z=0.
50, .')!] THE UNIFOKM CONVERGENCE OF INFINITE SERIES. 77
Example 2. Shew that at 2 = 0 the series
- 2? (1 +?)»-'
S
„I,{1+(1+J)-'}{1 +(!+-')"} is discontinuous and non-uniformly convergent.
The Hth term can be written
1 -(1 +z)" so the sum of the first >i terms is ■ — — — ~. Thus considering real v;ihies of z greater
than - 1, it is seen that the sum to infinity is 1, 0, or — 1, according as 2 is negative and greater than -2, zero, or positive. There is thus a discontinuity at 2=0. This discon- tinuity is explained by the fact that the series is non-uniformly convergent at 2 = 0 ; for the remainder after n terms in the series when z is positive is
-2
l + (l-l-2)»'
and however great n may be, by taking z= this can be made to take the value
— 2
z — , which is different fi-oni zero. The series is therefore non-uniformly convergent
at 2 = 0.
51. Distinction between absolute and uniform convergence.
The uniform convergence of a series does not necessitate its absolute
convergence, nor conversely. Thus the series (§ 49, Ex. 1) "S,— — con-
(1 + z )"
verges absolutely, but (at z = Q) not uniformly : while if we take the series
i |
(- 1)"-' |
z^ + n |
|
00 |
1 |
its series of moduli is
rZl\n+2'-\'
which is divergent, so the series is only semi-convergent ; but for all real values of z, the terms of the series are alternately positive and negative and numerically decreasing, so the sum of the series lies between the sum of its first n terms and of its first {n + 1) terms, and so the remainder after n terms is less than the »ith term. Thus we only need take a finite number of terms in order to ensure that for all real values of z the remainder is less than any assigned quantity, i.e. the series is uniformly convergent.
Absolutely convergent series behave like series with a finite number of terms in that we can multiply them together and transpose their terms.
Uniformly convergent series behave like series with a finite number of terms in that they are continuous and (as we shall see) can be integrated term by term.
78 THE PROCESSES OF ANALYSIS. [CHAP. IV.
52. Conditifm fur unifur III convergence.
A sufficient tliougli not necessary condition for the uniform convergence of a series may be enunciated as follows : —
If for all values of z within a certain region the moduli ol the terms of a
series
S=iu{z)+ ii.{z)+ u.,{z)+ ...
are respectively less than the corresponding terms in a convergent series of positive constants
then the series S is uniformly convergent in this region. This follows from the fact that, the series T being convergent, it is always possible to choose n so that the remainder after the first n terms of T, and therefore of S, is less than an assigned positive (quantity e ; and since the value of n thus found is independent of z, the series 8 is uniformly convergent.
Corollary. The theorem is still true if the moduli of the terms of S, instead of being less than the terms of T, are to them in a variable but finite ratio.
Example. The series
cos 2 + iT, COS^ 2 + 55 C0s3 3 +...
is uniformly convergent for all real values of z, because the moduli of its terms are not greater than the corresponding terms of the convergent series
, 1 1
whose terms are po.sitive constants.
53. Integration of infinite series.
We shall now shew that if S {z) = iii (z) + Mj (2:) + . . . is a uniformly con- vero-ent series of continuous functions of 2, for values of z contained within
o
some domain, then the series
I ?i, (z) dz + I Uo (z) dz
+
where all the integrals are taken along with some path C in the domain, is convergent, and has for sum jS(z)dz.
For let n be some definite finite number, and write
S (z) = w, {z) + It, {z)+ ... + u„ (s) + En (z),
so
1 8 (z) dz = iu,{z)dz+ ...-^ jun (z) dz + j Rn {z) dz.
52, 53] THE UNIFORM CONVERGENCE OF INFINITE SERIES. 79
Now siuce the scries is uniformly convergent, to every positive number e there corresponds a number ?• independent of 0, such that when n%r we have I R„ (z) I < e, for all values of z in the area considered.
Therefore if I be the length of the path of integration, we have (§ 32)
\l'
\R„{z)dz\<el.
Therefore the modulus of the difference between I S (z) dz and the sum
of the n first terms of the series 2 1 »„ {z) dz is less than any positive number provided n is large enough. This proves both that the series 2 I Un{z)dz is convergent, and that its sum is I S{z)dz.
Example 1. As an example of the necessity of this theorem, consider the series
„=i {1 +»i-sin2 22j{l + {n + \f sin2i2} *
The nth term is
%zn cos 2- 2z (n + 1 ) cos z^
\+i\?sixfiz^ l + (n + l)2sin202'
and the sum of n temis is therefore
2s cos i- 2j(»+1)cosz- 1+sin-*'- l+(?t + l)-sin222'
The series is therefore absolutelj' convergent for all real values of z : but the remainder after n terms is
iz{n + \) cos 2* ■ iT(/i+l)2sin2^2'
and if n be any number however infinitely great, by taking z - —— this has the finite value 2. The series is therefore non-uniformly convergent at 2 = 0.
» 2z cos 2"
Now the sum to infinity of the series is ; — . ., ., , and so the intestral from 0 to 2 of •' 1 + sm- 2- ' "
the sum of the series is tan"' (sin z^). On the other hand, the sum of the integrals from
0 to z of the first n. terms of the series is
tan " ' (sin z-) - tan " ' (;i + 1 sin 2-), and for n = 00 this tends to
tan " ' (sin 2^) — - .
Therefore the integral of the sum of the series differs from the sum of the integrals of the terms by - .
Example 2. Discuss the series
I 2e''2{l-?t(e-l) + e''-"'2^} „_,n(n+l)(l+e"22)(l+e» + i2!)
for real values of c.
80 THE PROCESSKS OF ANALYSIS. [CHAP. IV.
Tlie /ith term of the .series may bo written
2e"z . 2e"*h
M(l+eV) (?H-l)(l+«'' + >«2)' The sum of the first n terms i.s
1+622 (n+l){l+e''-'h^)'
2ez
The .sorie.s therefore couveree.s to the vahie = ;, ; and .since the terms are real and
1 + ez-
ultiniately of the same sign, the convergence is absolute. The integral from 0 to j of the
sum of the serie.s is
log (1+632).
IS
The .sum of the fir.st m term.s of the .series formed by integrating the terms of the series log(l + e.-2)-_l^log(l+6" + i.2),
which for » = oo tends to
log(l+ez2)-l
Tlii.s discrepancy is accounted for by the non-uniform convergence of the series at 3 = 0 ; for the remainder after n terms in the original series is
2e" + ij 2
and however great n may be, on taking z= this takes the value unity ; so the series
is non-uniformly convergent at 2 = 0.
Example 3. Discuss the series
M,+M.., + %+...,
where
for real values of z.
The .sum of the first n terms is nze~^'-'^, so the sum to infinity is 0 for all real values of z. Since the terms «„ are real and idtimately all of the same sign, the convergence is absolute.
In the series
I ^^^dz + \ u.,dz+ \ ')i^dz-\-..
the sum of n terms is J (1 — e-"^), and this tends to the limit ^ as n tends to infinity ; this is not equal to the integral from 0 to a of the sum of the series 2 ?«„.
The explanation of this discrepancy is to be found in the non-uniformity of the convergence near 0=0, for the remainder after n terms in the series M1 + M2 + ... is -nze-'^;
and however great n may be, by taking 3=- we can cause this to tend to the limit —1,
which is different from zero: the series is therefore non-uniformly convergent near z = 0.
54, ").'>] THK rNMFdUM CONVERGKNCE OK INFINITE SERIES. 81
54. Differentiation of infinite sei'ies.
The converse of the hist theorem may be thus stated :
// S (z) = », (s) + ii„ (;)+... is ti convergent series of analytic functions of z, which are regular when the variation of z is restricted to be within a certain
domain, and if the series 'S.{z) = -r- '<i (z) + j- u„{z) -i- ... is %iniformly convergent
titithin this domain, then this latter series represents -, S (z).
For by the preceding result, if a and z arc two points within the domain, we have
f ' X{t)dt = j' u,' (0 dt + f' u.; (t) dt + ...
J a J a -a
= it, (2) - H, (a) + ... + Un{z)-Un{a) + ....
Since ^
Ui{z)+u.{z)+ ... and »i («) + (/, (a) + ...
are each of them convergent series, we can write this
('l{t)dt = {u,{z) + u,iz} + ...}- I U, {u) + «., («) + . . . ] J a
= S{z)-S{a), and hence
We may note that a derived series may Ue iinn-uniformly convergent even when the original series is uniformly convergent : for instance the series
sin z— ^ sin 22+ J sin 3^ + ...
is non-uniformly convergent at z = ir; although the series from which it can be derived, namely
— C0.S 2 + .^ cos 22 - .jr, cos 32 + . . . , is uniformly convergent for all real values of z.
55. Uniform convergence of Power-Series.
We shall now shew that a poiuer-series is uniformly convergent at all points within its circle of convergence.
For let i? be a region, forming part of the area of the circle, and let r be a quantity greater than the modulus of every point of R, but less than the radius of convergence. Then if z be any point of R, the moduli of the terms of the series
Oo + «i2 + (t»s- + . . . W. A. 6
82 THE PROCESSES OF ANALYSIS. [CHAP. IV.
are less than the moduli of the corresponding terms of the convergent series
«„+«,)• + a.r-+ ....
But the latter series does not involve z, and so (§ .52) the power-series is uniformly convergent within the region jR ; as 7? is arbitrary, the series there- fore converges uniformly at all points within the circle of convergence.
It must be observed that nothing is proved regarding points on tlie circumference ; we do not even know that the series is convergent there at all.
Co7-oUary. A power-scries is continuous within its circle of convergence : and the series obtained by differentiating and integrating it term by term are equal to the derivate and integral of the function respectively.
Example. As an example of this, consider the series
which is convergent at all points within a circle of radius 1. We can integrate it term by term, so long as the path of integration lies in this circle ; the result is
z' .3 5
Now I 2 clearly represents that value of ta,n~'^z which lies between - - and +- .
So the series represents this value of tan ~ ' z and no other.
Miscellaneous Examples.
1. Shew that the series
„=i(l-0»)(l-0" + i)
represents tj when \z\<l and represents ^ when |2|>1.
Is this fact connected with the theory of uniform convergence ?
2. Shew that the series
2 sin- +4sin - + ... + 2" sin — - + ...
dZ VZ tj".i
converges absolutely for all values of z, but does not converge uniformly near 2 = 0.
3. If a series ^r (2)= 2 {c^- c^ + i)s\n{2v+l)zn (in which Cf, is zero) converges uni- formly in an interval, shew that ff (2) -. ; is the derivate of the series
/(2)= 2 ^sin2i-27r. (Lerch.)
CHAPTER V.
The Theory of Residues : Application to the Evaluation of Real Definite Integrals.
56. Residues.
If a point z = a is a pole of order m for a function f{z), we know by Laurent's theorem that the expansion of the function near ^ = a is of the form
(7^^ -^{z- «r-> + • • • + ^ - a + -^ (^>'
where <f) (z) is regular in the vicinity of ^ = a.
The coefficient a_, in this expansion is called the residue of the function f{z) relative to the pole a.
Consider now the value of the integral \f{z)dz, where the integration is
taken round a circle 7, whose centre is the point a and whose radius is a small quantity p.
r r=\ f dz C
We have lf(z)dz=l a_, — + ^ (2) dz.
J y r=m J y \^ ~ ^''J J y
Now I <h {z) dz = 0, since <f) (z) is a regular function in the interior of the ■' y circle 7 : and (putting z — a = pe'*) we have
jy{z-ay .',, p'e"^ ^ j„
-" r e""^"* 1 = «-'■+' , when ?• 4= 1
= 0, when ?-4= 1- But when »• = 1 we have
f J±^ = ^\de = 27rt'. JyZ — a Jo
Hence finally f{z)dz = 2Tria-,.
J y
Now let C be any contour, containing in the region interior to it a number
6—2
8* THK PROCESSES OF ANALYSIS. [CHAP. V.
of poles a, b, c, ... of a function f(z), with residues a_,, 6_i, c_i, ... respec- tively : and suppose that the function f{z) is regular at all points in the interior of G, except these poles.
Surround the points «, b, c, ... by snmll circles ct, /3, 7, ... : then since the function fiz) is regular in the region bounded by C, a, /3, 7, ..., its integral taken round the boundary of this region is zero. But this boundary consists of the contour C, described in the positive sense, and the contours a, /3, 7, ... described in the negative sense.
Hence 0 = f f{z) dz - i f(z) dz - j f{z) dz...,
J C J a. J ^
or 0 = I f(z) dz — 27n'o_, — 2Tri6_i
J c
Thus we have the theorem of residues, namely
f f{z)dz = 27ritR, J c
where SJ2 denotes the sum of the residues of the function y (2) relative to
those of its poles which are situated within the contour C.
This is an extension of the theorem of Chapter III. | 36.
57. Evaluation of real definite integrals.
A large number of real definite integrals can be evaluated by the use of contour-integrals and the theorem of residues. The following examples will serve to illustrate the various ways in which these aids to the evaluation may be applied.
Example 1. To find the value.s of
i^''e'^^^co^{ne-iime)d6 and i^" e"""^ sm{ne-s\nd) d6. Denoting these integrals respectively by / and J, we have
Jo
Jo
Write e'^ = z, and let C be a circle of radius unity round the origin in the z-plane. Then as 6 assumes the sequence of real values from 0 to 2;r, z describes the circle C.
I-iJ= [%cos6+,si„9-;,.9^
= r't^'^e-'nedd.
Hence l-iJ=-. \ e' z " ^ dz
4/.
= 2n- X the residue of — r-. at j = 0
,n + l
2,7
57] THE THEOKY OF UESIDUES. 85
Thei-efore l=—r,
71.1
J=0.
Example 2. The method used in Example 1 can bo very generally applied to trigonometrical integrals taken between the limits 0 and in. As another example, consider the integral
f= I , ^ {a>b).
Jo a + b cose ^
AVrite e'*=3; and let C be the circle on the c-plane whose centre is at the origin and whose radius is unity.
Then /=[ '■ ,„ ^f ^ „
Jo iz(2a+bz+bz-^)
2 /■ . dz
i Jcbz' +
%iz+b
= 4jr X sum of residues of -;-= — r i at i)oles contained within C.
bz^ + '2az + b '
„ 1 1 r
Now .— = — i ,= —
-1 1
62«+2cK+6 2Va^-6^- a y/a^-b^ a -Ja^-b^
a \/<^ - 6^ , a \^a^ - b^
1--
, and the residue b b
^ at the former (which is the only one within C) is
Hence /= -
Therefore the two poles are at s= — ; ; and z= - r-\ — ■
0 0 b b
J. _ 2\/^^'
Example 3. Shew that
Jo (a + 6cos?)2~(a2-6*P'
Example 4. Find the value of
/"" X sin mx , I — S-, — ^ ax.
J _oo x' + a-
Let C be a contour formed by the real axis together with a semicircle y, consisting of that half of a circle, whose centre is at the origin and whose radius is very large, which lies above the real axis.
Then ^ 2 's a function of z which has only one pole in the interior of C, namely at
., — ., rfc = 2ni X residue of -^ -, at its pole ai. But writing
cz' + a- z^ + a-
z=ai+(, we have
ze"^ _ (ai+C)e~'^*"'*i z^Ta^~ 2aif+f^
v"""'?0^«9^'+""^^-K'^2y'
= -a). + positive powers of f.
86 THE PROCESSES OF ANALYSIS. [CHAP. V.
2gmsi ^ — ma
Therefore -»— -5 — a/ -n + l"^«itive powers of {z - ai).
Thus the residue of -r. r, at ai is - e"'"".
2- + a^ 2
/■ -glim c z-" /■ \ jg""!
Therefore 7rie-""'= I ! .,dz=\\ + \ \ ., —^dz.
1
Since „ — „ is infinitesimal comiiared with ,--, at iioints on y, the integi-al round v is
f dz\ infinitesimal compared with I — or in, and is therefore zero.
Therefore ■nie~'^= \ „ —„dz.
J -^z^+a^
Equating imaginary parts, we have
/"" X sin 7n.v ,
Example 5. To find the value of
I e" "<>' '»' sin (a sin b.v) -^ r, .
Jo \v^ + r^
We have / e" ™' ''•'^ sm (a sui 6.r) „ — =^ = I —.e^ „ , „.
^0 x^ + r^ J-x2i x'+r^
Take a contour C compo.sed as in Example 4 of an infinite semicircle y and the real
axis.
Then j — .e"' ^' „ „ dz=27ri x residue of ^ e"" *' „ „ at its poles inside C. J c 2i z^+r^ 2i z^ + r'
But -.e"^ '^ — — has only one pole in the interior of C, namely at the point z=i-i.
Now if 5=n + f, we have
1 tizi Z 1 -br-bii ri+C 1 „ -br ... , ^
—.ef" -= „ = —.€"' „ ... -z^ = -rrie'" + ijositive powers or C-
1 _^ Therefore the residue is —.e^. %
At *
Thus ^^e-^r^l l,,b.i Z r 1 ,,W^ ^^_
But at points on y, e''-<^' = 0, so ea« '^' = 1, and so
/" 1 >i -^^-^ _ 1 [ fil'^l
jy2i .!^ + ?-'^ 2VJy .!• 2'
j__ T ^-— f T* n t*
Therefore - e"'" = s + ^""^"^ '"' s'" (" ^in 6.i-) -o-r-; ,
or / e'":osiu- sij, (oj,sin6.x-) '^ -% =^ (e'»«~'"' - 1).
jo 'x^ + r- 2'
We may note that in the above / stands for the limit of I where k is infinitely great, and is not equal to the limit of I where k- and I are different.
F.XS.] TIIK THKOKV OF KESIDUES. 87
Example 6. Prove by inU'grating
ju^ + z'- 1-ouiid the contour used in Exiiiuples 4 and o, tliiit
/■" cos a; _ »r _^j /■* sin.r _
Example 7." Find the vahie of
/,
sni mx , -.ax.
Consider a contour C, formed of 1° a semicircle r whose centre is at the origin and whose radius is very hvrge, 2° a semicircle y whose centre is also at the origin and whose radius is very small, and 3" the portions of the real axis intercepted between these circles. The semi- circles are to be drawn in the upper half of the c-plane, i.e. the half above the real axis.
Then | — ri; rT7, = 2)r!'x the residue of -^.^ — ,, at the singularity z = ai.
But if we write 3 = at + f, where f is small, we have
i(l+»uf + ...)(^l + i-.) (1+^. + .
j(2* + a2)2 "" (^-l-f)(2aif+f2)2 " 2^^2 (1 +»"f + •••) [^ + --j [^^^i
g — nio / 2\-
Thus the residue at ai is — -^- 1 m + - ) .
Therefore - ^-^(^,„ + -j = j^ ^^^-^,^^ = {J_^ +j^ _ jj___.
Now r-= — iT^ is infinitely small at points oil r, so the integi-al taken round r vanishes. (2»-|-a')^ ^ I '
[ e'»"dz fdz II . ] \ ( dz m
Also I ,-, , ,,,= 1 — -^^ + powers ot s^ = -. I — = -t.
jyz{z- + a')- J z [a* ^ I C' J y z a*
Therefore j_^ ^-^^__^^ = _ _ ^^(^„, + ^J.
In this, I means \ + \ ^ where the two t's arc the same : but in the final result
we can j)ut f =0, since the final integrand is finite at the origin.
Equating the imaginary parts on both sides of this equation, we obtain
r* sinnixdx _ir jre-"™" / 2'\ j _. x(3?+a^Y ~a*~ ""2a3~ v" "^ a) '
, f" ain mxdx w ne~"^ / 2\
*"^ '"'' 1 0 IJ^T^^ = 2^ - -4^r + a) ■
Example 8. Find the value of
/:
cos 2ax - cos 26.r , — ax.
x^ Take the contour C formed a.s in Examjile 7 by an itifinito semicircle r, a small semi-
88 THE PROCESSES OF ANALYSIS. [CHAP. V.
circle y round the origin, and tlie parts of the real axis intercei)tcd between them. Within this contour the function --^ has no singularities.
Therefore 0 = I —^ dz = I — .^- dz - I —^ dz + | —5- dz.
] c z J ^ ^ J y ^ J -•» z
In tliis equation | must be regarded as an abbreviation for I + I where e is the
J — <» J e J _ao
radius of y.
g2ai« ^ J
Now at points on r, —^ is zero compared with - , so the integral round r is zero.
/g2aw g2atz
—5- ^2 = one-half of 2^1 x the residue of ., at the origin y z^ z^
. H-2aiz+... = n-i X the residue 01 3
= - ina. Therefore I —^dz=—2na,
J —00 z r " giaie _ g2&w
and so I ^ '^^ = 2n(b- a).
J —GO Z
Taking the real part of this we have
/"" cos 2aj; - cos 26.r , ^ ,, , I 2 (ij; = 27r (6-a),
cos 2aic — cos 2bx .^.. , „ , , j. ■ ^ f" ^
and smce -^ is fimte when x = 0, we need no longer restrict ( to mean
Example 9. Find the value of
/;.»-^sinff-6.)^, («>0).
We have J^ .x-^ sin (^-6x)^,
( - .i-)"-' e*^}
rdx
x^-'rr^
1 /-".i^-i r ?i!r.to -?|^+6;x-i rdx
= 2J„-T- i^' -'' ■' >
1 /■"
2 7 0
2j_a>^ ' .(.•'i + ?-''
Consider a contour C, formed as in Examples 7 and 8 by an infinite semicircle r, a small semicircle y romid the origin, and the parts of the real axis intercepted between them.
1 /* Tdz 1 T t •
Then - | (-2i)<'->e«"'^ — r = 27ri'xthe residue of - (-20«->e*»' ^-r-,at its singulanty 2yc j^-l-r^ 2^ 2''+r^
EXS.] THE THEORY OF HESIUUES. 89
Putting t = ri+( and neglecting jxiwei'S of f, we see that the expansion of
1
2^ "■' " s' + r'
^(_,,-)«-l«<6. •
t i'
begins with u term
■1 f
so the required residue is a ■ - •
Therefore ;; ,-<-i <;-'•'•= ^ | (_,-,•)«-!«»• J" ^
2 .' (' -""r
rdz
.2 •
At points on r tlie integrand is infinitesimal compared with -, and so the integral
round r is zero.
•2r
At points on y the integrand is approximately ^ — ^^ — z""', and so if a >0 the integral
, ., rdx TT „ . 1.
, dx
round y is zero.
Therefore I .<■""' sin (-:^-6.>;) .7—^; = ;:: I { — -vi) } I, V2 /.),- + r- 2_/_x
/■» ,
£a:am/>ie 10. Find the value of 1 e"™siiisin(asin6.i,-)
.' 0
We have {" e'">^'"&m(asmb.v)— =~. T e^^—,
where in the latter integral I must be regarded as an abbreviation for | + / where
J -« y « J -»
f is a small quantity.
Take a contoiu- C, consisting as in Examples 7, 8, 9, of an infinite semicircle r, a small semicircle y of radius t round the origin, and the parts of the real axis intercepted between them.
Then 0= { e"^'^ "^ = ( e--"" - - { e'"''"' ^' + /' " e^'-^'' ^ .
Jc •!; jr -^ }y ^ .'-» ^
At points on r, we have «*■'' = 0, e<^<^"' = l, and so
J r .V J r .V
At ix)ints on y, e*"= 1, so
] y X J y X
Therefore f e«'*"^"=7ri(««- 1),
and so / e° costi gin (a sin hx) "' = | (e« - 1 ).
Jo X z
/e^dz ~^- round the same contoiu- as that used in Examples
7, ^, 9, 10, shew that ['^^dx=l. } Q X 2
90 THE PROCESSES OF ANALYSIS. [CHAP. V.
Example 12. To find the value of
/ - -- dx, a.ud I , -dx (0<a<l).
7: rf.r-, and ff= / f— <;ia;.
0 1+^ yo 1-^
As will be seen from the working below, the integral A" has a meaning only when I is
/•« / l-r'
understood to mean j +1 , where / is a small positive quantity.
Consider a contour C formed of (a) that half r of a cii'cle, whose centre is at the origin and whose radius is a large quantity R, which is above the real axis, (6) that half y of a circle whose centre is at the origin and whose radius is a small quantity r, which is above the real axis, (c) that half y' of a circle, whose centre is at the point (-1) and whose radius is a small quantity /, which is above the real axis, {d) the parts of the real axis intercepted between the.se semicircles.
-j , where the many-valued fvuiction is supposed to have that one of
its determinations which is real and positive when z is real and positive. The integrand is regular in the interior of the contour C, and so
Now on y the integrand is sensibly equal to 2""', and so the integral to — , which is infinitesimal, since a > 0.
On y, the integrand is sensiblv equal to ^^-— i — ; putting I + z = /«'*, the integral
1 +z
along y is / idd, or iw{-iy~\ J »
On r, the integrand is sensibly equal to .,^^ , the modulus of which is infinitesimal
compared with j — : ; so the integral along r is zero.
Ml
Therefore (-l)'-i7ri= ["■!!Zi?L^'+ //■-'-'' + [" \ ■^^•!; = /+(_ i)a-iA'. Jo l+x \J -^ J -1+// 1+-^-
Thus 7rt = (- 1)'~°/+A'= -/(cos an- -I sin a7r) + A'.
Therefore equating real and imaginary parts, we have
sm an K= TT cot arr.
Example 13. By using the result
r°°.r°-'rf.y_ n
In 1 + X siu an '
1:
shew that t-^=— . Limit 2 -, . (Kronecker.)
58] THE THKOUY OK RESIDUES. Dl
68. Evaluation of the ilf/inite intei/ral of a rational function.
The principle.s which have bt'un apj)liu(l in the preceding paragraph can iilso be used to evahiate an integral of the form
r /{■r)dx,
J - X.
where /(j-) is a rational function of .c, in the cases when this integral has a meaning.
For suppo.se that f{x) is brought to the form of a quotient ^ , where
g {x) and h (x) .are polynomials in x. In order that the integral may have a meaning unconditionally, it is necessary that the degree of g {x) should be at least two units lower than that of li{x), and that the equation h{x) = 0 should have no real roots.
Consider now a contour C, forme(i of the real axis together with a semicircle T of large radius, whose centre is at the origin, and which lies in the iipper half of the 2-plane.
We have | f {z) dz = iwi x sum of residues of /(2) at the poles of f(z) contained within G.
Now 1=1 + I • '^"'1 since /(z) has a zero of at least order 2 at
;8 = 00 , it follows that I is zero.
Hence I f {x) dx = 'I-n-i x sum of residues of /(./•) at those of its poles
which are contained in the upper half of the ^-plane.
If the degree "of g (x) is lower than that of /( (.r) by only one degree, or if /; {x) has real non- repeated roots, the integral will .still have a meaning provided we make certain restrictions,
Le. that I shall be undei-stood to mean the limit, when /• tends to oo and c to zero, of
+ I , where c is a typical root of the equation A {x) = 0. -t J c+.
Example 1. The function T^irr ,■,3 ha.s a single polo in the upper half of the i-plane,
namely at z = i, and the residue there >s - tt. ; we have therefore
/
dr
_.(a:2+l)3 8 x*dji;
Example 2. Shew that | •; 1 »ti= -i-^
92 THK PROCESSES OF ANALYSIS. [CHAP. V.
59. Cauchy's integral.
We shall next discuss a class of contour-integials which are very fre- quently found useful in analytical investigations.
Let C be a contour in the ^-plane, and let f{z) be a function regular everywhere in the interior of (7. Let ^{z) bo another function, which in the interior of G has no singularities except poles ; let the zeros of 0 {z) in the interior of C be a,, a„, ..., and let their degrees of multiplicity be 7*1, r.,, ...; and let its poles in the interior of 0 be 6i, 6.>, ..., and let their degrees of multiplicity be s, , s.,,
Then by the fundamental theorem on residues, we have „ ~. f(z) %-^{ dz = sum of residues of •' f-J^ , in the interior of C.
Now / , N can have singularities only at the poles and zeros of (j> (z). At one of the zeros, say a, , we have
<}>{z) = A{z- a,)'' +B{z- a,)'''+' + . . . .
Therefore 4>' (z) = Ar,(z- UiY'-' + B (?•, + 1 ) (^ - a^Y' + . . . , and f(z) =/(«,) + {z- a,)/' (a,) + ....
Therefore "^ . ; } = -^ — - + a constant + positive powers of (z — a,). (}){Z) z — cti ^
Thus the residue of ,T} . at the point z = ai, is r,/(aj).
Similarly the residue at ^ = 6; is — Sif{bi) ; for near z = bi, we have
<t>(z) = Ciz- b,)-" + D(z- 60-*'+' + . . . ,
and f(z) =/{b,) + (z - 60/' (6,) + .-.,
f{z)(j>'(z} -s,/(M .,,-.- f I
so . ; . = =^-r — + a constant + positive powers ot z — Oi.
<})(z) z-b,
the summations being extended over all the zeros and poles of 0 (z).
60. The number of roots of an equation contained within a contour.
The result of the preceding paragraph can be at once applied to find the number of roots of an equation <f) (z) = 0 contained within a contour C.
For on putting f{z) = 1 in the preceding result, we obtain the result that ■^^ — = I . / X dz is equal to the excess of the number of zeros over the number
I 59 61] I'llI'. TIIKOKV ol' KKSIDl-KS. 93
I
i of poles of" ^{z) contained in the interior of C, eacli polo and zero being I reckoned according to its degree of multiplicity.
Example 1. Shew that a polynomial (p {z) of degree hi has m roots. Ut <t> («) = ao2"' + «,:"'-' + ...+a,„.
*l(f) _ L««o^"' - ' + •■■+ "j.. - .
For large values of z, this can be expanded in the t'onn
<t>'{z)_m A <t>(z) z^z^'^--
Thus if C be a largo circle wliose centre is at the origin, we have
I f <(,' (Z) . ^ TO /■ rfz^ ^
27rt J c4>{^) ~-27ri J c Z~ Hence as <^ (2) has no poles in the interior of C, we have
number of zeros of <b (,z) = ^ — ■■ I -^-7-% dz = m.
Example 2. If at all points of a contoiu- C the inequahty
|ai2* I >|ao + aiJ + ... + at_j2*-i + ai.n ;'*' + ...+«,„ 3™ is satisfied, then the contour contains k roots of the equation
For write /(2) = a,„3"' + a„_,j'"-' + ... + o',^ + rt„.
Then /(.)=«.^ (i-^"^-^--^"--^;\-r°-'^"+-^"°)
= aiZ* (1 + i/^) say, where 1 6' [ < 1 on the contour. Therefore the number of roots of f(z) contained iu C
- ^ ( -^^dz -2^jcf{z)'^'
_J_ [ ft _^ <^\^ ~2,rjjcV"^H-r dz )'''■■
But I — = 2ni ; and since | i' | < 1 we can expand ( 1 + 6^ " ' in the form
i_p+r2-r3+....
Therefore the number of roots contained in C is equal to k.
61. Connexion between the zeros of a function and the zeros of its derivate. ^
Macdonald* has shewn that if f{z) be a regular function of z in the interior of a
contour C, defined by an equation \f{z)\ — il where M ii a constant, then the number of zeros
of f{z) in this region exceeds the number of zeros of the derived function f'{z) in the same
region by unity. •
• Proc. Land. Math. Sac. xxix. (la'J8).
94 THE PROCESSES OF ANALYSIS. [CHAP. V.
For since f{z) has no essential singularity in the region, the number N of its zeros in the region is finite. Now if m be a .small number, the part of the locus 1/(2) =»!, in the interior of the contoiu- C consists of .\' closed curves surrounding the N zeros of /(s). As m increases, these ovals iiu-rciise, until two of them coalesce, the point at which they coalesce being a node on the curve corresponding to that particular value of m. When m has increased to its final value M, the N closed curves have coalesced into one closed curve, and therefore N—\ nodes have been pa-ssed through. Each of these nodes is
a zero of /'(z); for if /(2) = (^ + ?\|', where <fi and i// are functions of x and y with real
pi t ^ I
coefl&cients, then -^ and --— vanish at a node on the curve d)'- + \i/- = constant ; that is,
ox 01)
f'{z) vanishes. Moreover, two ovals cannot coalesce at more than one point, as f{z) is single-valued.
Hence the nvmiber of zeros of /' (:) inside the contour is (^ - 1).
The proof assumes the zeros of /(i) in the interior of C to be all simple : the case where f{z) has multiple zeros can be at once reduced to this, by dividing out the factor common to/(i) and f (s). If /' (z) has two zeros equal, two of the doulile points coalesce, that is, three ovals coalesce at the same point.
Similarly it can be shewn that the inimber of zeros o{ f'(z) in the region between the contours |/(')! = »Ji and |/(z)| = '«2 is equal to the number of zeros oi f{z) in the same region, if ,/'(^) is regular in the region.
Example 1. Deduce from Macdonald's result the theorem that a polynomial of degree 11 has n zeros.
Example 2. Deduce from Macdonald's result that if a fimction f{z), regular for real finite values of z, has all its coefficients real, and all its zeros real and different, then between two consecutive zeros ai f{z) there is one zero and one only of f (z).
Miscellaneous Examples.
1. A function (p (z) is zero for z = 0 and regular when ]z|^l. If f(x, y) is the coefficient of i in <^ (x+yi), prove that
Cl-2J'cos^ + ..^-^fc"^^' «i°^)^^=-'>(-^)-
(Trinity College Examination, 1898.)
o oi .a . [" sin ax , 1 e^ + l 1 n a ^
2. Shew that • j ^ gSJTTI '^•^' = 4 ^^^ " 2^ ' (Legendre.)
3. By integrating I e-^'dz round the perimeter of a rectangle of which one side is the real axis and another side is parallel to the real axis and at a distance a from it, shew that
I e-*^ con 'iatdt^^xjwe-o-'', and / e~*^- &miatdt=0.
J —to
ci iu i { l-rcos25 , . ... JT , 1-r
bhew that / , — ^ s^-,— 9 log sm 646= -r log — :— .
y 0 1 -2rcos25+>-2 ° 4 ° 4
MISC. EXS.] THE THEORY OK RESIDUES. 95
6. Shew that
r> sin 2a;
/:
-„ .vd.c= -' log (1 +«) if -!<«<!
(o l-2aco8:j:+a- 4((
and = ^ log (l + ^^) if ,'■' > 1 . (Caucl.y.)
6. Shew that
f° sin d>,.v sin<b.,s sin d)..r sinew , t . . .
I - -^ ^-^... 2-^=- cos n, .(•... cos 0,,,.C d.V=7i<Pi<Pi...Vn,
J „ X .V .V X i^riT^T
if a be difterent from zero and
o>|(^| + |<^.>l+ ... +!<^„l + !a, ! + ...+ |a,„|.
(Stormer.)
7. If a point ; describes a circle C of centre a, any one-valued function u=f{z) will descrilie a closed cur\'e y in the iz-plane. Shew that if to each element of y be attributed a mass proportional to the corresponding element of C, the centre of gi-avity of y is the
point r, where r is the sum of the residues of ^' at pol&s in the interior of C.
(Amigues.)
8. Shew that
dx TT (2(1+6)
/:
9. Shew that ^ [J^ v ^ "
dx _ TT 1.3...(2?t-3) 1 eiivi'**'^*^'-'*
,(a + 6.r2)«~2"-'6* 1.2...(7t-l) a""*'
/:
10. If /'„(.r) = (l-.i;)(l-.r2)...(l-.i"'->)...(l-.r2)(l-x^;...(l-.t-2"-2)...
.,.(l-u.-''-i)(l-.i,-2"-2)...(i_.i<n-iF),
shew that the series
converges when .!■ is not a root of one of the equations
and that the sum of the residues of f{x) contained in the ring-shaped space included between two circles whose centres are at the origin, one having a small radius and the other having a radius between n and n + \, is equal to the number of prime numbei-s less than n + l.
(Laiu-ent.)
CHAPTER VI. The Expansion of Functions in Infinite Series.
62. Darhoiixs formula.
Darboux has given* a formula from which a large number of expansions in infinite series can be derived.
Let f{z) be an analytic function of z, regular at all points z within a circle of centre a and radius r ; and let 2 be a point within this circle. Let <i> (z) be any polynomial in z, of degree n. Then if R^ denotes the expression
(_ 1)" (^ - «.)»+> r (j) (0/'"+" [a +t(z-a)] dt,
J n
where the integration is taken along the real axis of t, we have on integi'ation by parts
Rn = \\- l)" (^ - «)" </> (0/"" !« + H^- «)1
+ (- 1)"-' (z - <i)" I </>' (0/"" {« + t(z- a)} dt, or R„ = (- 1)« (z - «.)" (0(1)/ w (z) - 4, (0)/ '»' (a)}
+ (- 1 )"-' (z - a)"- 1 </)' («)/"" |a + t(z- a)} dt.
Integrating the last integral by parts in the same way, we obtain ie„= (- 1)« (z - a)" {4> (!)/<"> (z) - <i> (0)/"" {a)]
+ (- l)"-'(2-a)"-' {f (1)/'"-" {z) - f (0)/'"-" (a)} + ... -(^- a) (</>•"-" (l)/'(5)-<^'"-" (0)/'(a))
+ {z-a){ 4,'"^ (t)/' [a + t(z- a)} dt. Jo
Now <^'"' (t) is a constant independent of t, since ^ (<) is a polynomial of order n; and hence
{z - a) r <^i"' it)/' {a + tiz- 0)1 dt = 0" (0) [f(z) -/(a)}. Jo
* Liouville's Joimml (3), 11. (1876), p. 271.
62, 63] TIIK EXPANSION OF FUNCTIONS IN INFINITE SERIES. 97
Thus finally we have Darboux's foiimila <^"" (0) [f{z) -/(«)) = {z -a) {</.'"-» (1)/' {z) - <^"'- (0)/' (a)!...
+ (- 1 )" (2 - «)"+' I (/) (0/ '"+'» {a + < (^ - a)l (Z<. Jo
Taylor's expansion may bo derived from this formula by putting (^ (() = (<— 1)", and then making n tend to infinity: other new expansions maybe obtained by .substituting special polynomials of degree ?i for j>(t), and in the resulting formula making n tend to iutinity : in each case it must of course be shewn that i2„ tends to zero as n tends to infinity.
Example. By substituting Zn for n in Darboux'a formula, and taking 0 (<) = '" if - 1)", obtain the expansion
/(.-)-/(a)= i -" ^KHf, " "^" {/'"'(^) + ( - l)'"-"/''"(^)}, and find the expression for the remainder after n terms in this series.
63. The Bemoullian numbers and the Bernoullian polynomials.
Z Z
If the constants which occur in the expansion of ^ cot ^ |in ascending powers of z be denoted by B-^, B.,, B^, ... , so that
-coi^ = l-B,--B.,^^-B,~...,
then Bn is called the nth Bei-noulliaii number. It is found that
-Di = 6 > -^2 = 30 ' -^3 = 42 ' • • • •
The Bemoullian numbers can be expressed as definite integrals in the following way.
r*' sin itx dx '^ r^ We have —-^ — — = S e~"" sin pxdx
= S
P
„=i n-ir'+p-
1 t- .
= -^-9C0tzp
2/) 2^; ( 2! 4! j
Equating coefficients of ja-" ' on the two sides of this equation, and writing X = 2t, we obtain
B
n = 4« /
.' 0
f " (="-' rf«
A proof of this result, depending on contour integration, is given by Carda, Monatshefte far Math, und I'liys. v. (1894), pp. 321-4.
W. A. 7
98 THE PROCESSES OF ANALYSIS. [CHAP. VI.
Example. Shew that
2n r x^'-^dx
"~ tt'* (S^- - 1) j 0 sinh ar '
The Bernoullian polynomial of order n is defined to be the coefficient of
pi , e"^ 1 .
— in the expansion of t ~, ^ in ascending powers of (. It is denoted by
<f>n (^), SO that
e'-l~„ri nl ^^^•
This function possesses several important properties. Writing (^ + 1) for z in the preceding equation and taking the difference of the two results, we have
<e^ = i {</.,. (^+l)-<^„ (0)1 ^.
7» = 1 '''■
On equating coefficients of t" on both sides of this equation we obtain
«S»->=(/)„(^+l)-^„(2),
which is a difference-equation satisfied by the function <^„ (z).
The explicit expression of the Bernoullian polynomials can be obtained as follows. We have
, , ^ zH^ !?t^
and
t _ I e' + l _ t_ i«^^l~2e«-l 2
~ 2 i _* 2
« , i t = 2i"°*2i--2
2"^ 2! 4! "^
Hence
2! "*" 3! ■■■! 1 ^ 2! 4! "*"
»=i «! 1
From this, by equating coefficients of t^, we have
</,„(.) = .»-^^- + g) A.'- - g) 5,.- + g) 5.."-= ...,
the last term being that in ^ or 2- ; this is the explicit expression of the nth Bernoullian polynomial.
64] THE EXPANSION OK FUNCTIONS IN INFINITE SERIES. 99
The BernouUian numbers and polynomials were introduced into analysis by Jacob Bernoulli in 1713.
Kxample. Shew that
<^»w=(-ir«^«(i-4
64. Tike Maclaunn-Bernuullian expansion.
Ill Daibou.x's formula write 0 (0 = (/>„ (<), where 0,, {€) is the ?ith Bernoul- lian polynomial.
Now from the equation
4>n(.t^\)-4>„{t) = nt"-\ we have by differentiating k times
^„* (« + !)- </)„<*' (0 = « (n - 1) . . . (« - k) <»-*-'. Putting < = 0 in this, we have
</>„*(!) =<^„"-'lO). But the value of c/),,'*' (0) is obtained by comparing the expansion
</,„ {z) = <^,. (0) + z<l>,: (0) + |1 ,^„" (0) + . . . with the expansion
<^„ {z) = ^" - 1 z^-^ + g) 5,^"-= - g) 5,^"- + . . . .
Substituting the values of </>„*' (1) and (/)„*(0) thus obtained in Darboux's result, we find what is known as the Maclaurin-Bernoullian formula,
(z - a)/' (a) =fiz) -/(a) -~^{f' (^) -/' {a)\
- ^^-^f'^' / V«.(0/'"'+" (a + (^ - a) <} d<.
In certain cases the last term tends to zero as 7i tends to infinity, and we can thus derive an infinite series from the formula.
Example. l{f(z) be an odd function of z, shew that
^/'(^)=/(^) + f| (2^)V" W + ... +( - 1)" %^|"r /<--=' (-')
(2»-2)!
23n22n + 1 2/1 !
where (^„ (<) is the BernouUian polynomial of order n.
7—2
100 THE PROCESSES OF ANALYSIS. [CHAP. VI.
65. Burmann's theorem.
We shall next consider a number of theorems which have for their object the expansion of one function in powers of another function.
Let <i>{z) be a function of z, which takes the value b when z takes the
vakie a, so that
h=4> (a).
Suppose that ^ (z) is an analytic function of z, regular in the neighbour- hood of the value z = a, and that if)' (a) is not zero. Then Taylor's theorem furnishes the expansion
4>{z)-b = 4>' (a) {z - a) + '^\^f (^ - a)= + . . . , and on reversing this series we obtain
which expresses ^ as a regular function of the variable [<f>{z) — h], for values of z in the neighbourhood of a. If then f{z) be a regular function of z in the neighbourhood of a, it follows therefore t\\&tf{z) is a regular function of {(^ {z) — h] in this neighbourhood, and so an expansion of the form
f{z) =f{a) + a, !</. {z) - 6) + g (</> {z) - hf
will exist, which, as it is a power-series in 1^ {z) — h], will be valid so long as
[ <^ (^) - 6 1 < r, where r is some constant.
The actual expansion is given by the following theorem, which is generally known as Burmann's theorem.
If -^ {z) he a function of z defined by the equation
z — a
^{z) =
e function f{z) can fo the for
4>{z)-b'
then the function fiz) can for a certain domain of values of z be expanded in
fiz) =/(a) + I ^ i^M^ ^. [/' (a) (^ (an ; and the remainder after n terms in the series is
27nJ aJy
'4>{z)-b'
'>-\f'(t)(j)'(z)dtdz
L<^(0-6J <f>{t)-cf>{z)
where 7 is a simple contour in the t-plane, enclosing the point t = a.
65] THE EXPANSION OF FUNCTIONS IN INFINITE SEUIES.
To prove tliis, we have
1 i'-' f /' (t) <}>' (z) dtdz
101
1 C'tf'{t)<i,'{z)dtdz 27n]Jy <p{t)-b
<t>(t)-<f>(z)
But
'^[<f>{t)-b\"-^{<j>{t)-4>iz)\_
<f> (z) - bl"/' (0 f (z) dtdz _\<j>{z)- 6|*+' /• /' (t) dt
L-+1) L
j^(t)-b] <l>(t)-b 2-rn{lc+l) Jy{<}>{t)-b]''
{<t>{^)-br^ [f'{t){f(t)r^dt_{M±:bV^ d' rf'(a)l^(a)]>'^^] "i^rHW^i) jy ^ (r-^a)*+'^ - 2^i(kTlji du" ^-^ ^"^ ^^ ^"'■'^ ^-
Therefore /(z) =/(a) + 1 ^-^j^ |^, [/' («) !^ («))*]
t=i
Example 1. Prove that
27n J„ Jy |
[<i>(z)-bl l<p{t)-b] |
" (-1)""'<^»(^ -«)"«"'''""'' |
|
n=l » ! |
i |
""-^ /' it) <i>' (z) dtdz
<f>{t)-<p{z)
where
To obtain this expansion, vfrite
f{z) = z, ,i>{z)-b = {z-a)c''-''\ ir{z) = e-'-'\ in the above expression of Biirmann's theorem ; we thus have
n=l '*■ ■ V^^ J r=.a
But
= («-!)! X coefficient of Z""' in the expansion of e-"'l2'» + ') = (»i-l)! X coefficient of «"-■ in 2 ^ p
^ ' r=o(»-l-r)! (2r-n + l)!
The highest value of r which gives a term in the suinniation is r = 7i-l. Arranging therefore the summation in descending indices r, beginning with r=n- 1, we have
{ir.'i «"'"'-"'} =(-!)"-' {(2«a)"-'- "(»-j)("-^)(2„a)-'-^ + ...}
=(-i)"-'c„,
which gives the required result.
102 THE rUOCESSES OF ANALYSIS. [CHAP. VI.
Example 2. Obtain the expression
„ . , 2 1., 2.4 1 . „ z^ = 6\i\-z + ~ . 2^^'" ' + 3-5 • oSin'^H- ....
Example 3. Let a lino p be drawn tbrough the origin in the ^-plane, perpendicular to the line which joins the origin to any point a. If z be any point on the s-plane which is on the same side of the line p as the point a is, shew that
log2=loga + 2 2 r ( )
2in+l
66. Teixeira's extended form of Burrnann's theorem.
In the last paragraph we have not investigated closely the conditions of convergence of Burrnann's series, for the reason that the theorem itself will next be stated in a much more general form, which bears the same relation to the theorem just given that Laurent's theorem bears to Taylor's series : viz., in the last paragraph we were concerned only with the expansion of a function in positive powers of another function, whereas we shall now discuss the expansion of a function in positive and negative powers of the second function.
The general statement of the theorem is due to Teixeira*, whose exposi- tion we shall follow in the next two paragraphs.
Suppose (1) that/(^) is a regular function of 2: in a ring-shaped region A, bounded by an outer curve S and an inner curve s ; (2) that 0 (z) is a regular function everywhere inside S, and has a single zero a within this contour; (.3) that x is the affix of some point within A; (4) that for all points of the contour S we have
and for all points of the contour s we have
I ^(*-)i> 1^(^)1-
The equation
6{z)-e (x) = 0
has, in this case, a single root z = x in the interior of S, as is seen from the equation
1 r d'iz)dz _ 1 n ffiz) r e^)
27ri J s ^ (^) - 0 (^) ~ 2t7r LJ s ^ (^) ^ ^ is ^M^)
0' (z) dz
'5 e{z) '
of which the left-hand and right-hand members represent respectively the number of roots of the equation considered and that of the roots of the equation 6{z)=0 contained within S.
* Crelle's Journal, cxxii. (1900), pp. 97—123.
'2-mJ s
66, 67] THE EXPANSION OF FUNCTIONS IN INFINITE SERIES. 103
Cauchy's theorem therefore gives
^(^) = 2i
■f f(z)0'(z)dz r f(z)e\z)dz~\
Jse{z)-0{x) j,d{z)-e{x)j
The integrals in this formula can, ass in Laurent's theorem, be expanded in powers of 6(x), by the formulae
r fiz)d'(z)dz _ « rf{z)e'(z)dz
JsO{z)-0{x)-„Zo ^ '.Is 0"^'i^) ' We thus have the formula
f{x)=i Ane'H^)+ i ^.,
»=0 n=10 W
Where . 1 fAz)e'iz)dz
Bn=^\/i^)e-^{^)6'{z)dz.
This gives a development of f{x) according to positive and negative powers of 6 {x), valid for all points x within the ring-shaped space A.
67. Evaluation of the coefficients.
If the function /(2) has no singularities but poles in the region limited by the curve s, the integrals which occur in the preceding formula can be evaluated in the following way.
Let 6i, to, ... bk be the poles; and let c,, Co, ... Ck, c, be circles with centres 6,, b„, ... h^, a, respectively, and with very small radii.
Then A - ^ { f i^) 6- {z) dz _ 1 U\z)dz
^""STTiis e-^^{z) -27ri}sne«iz)
and ^''^'^ij -^(^^ ^"~' ^^) ^' ^^^ ^^
104
THE PROCESSES OF ANALYSIS.
[chap. VI.
6 (cc) Thus if dm be the degree of multiplicity of the pole 6,„, and if — — - be
denoted by ^, (x), we have
^" - «i
* 1
m = l Om!"
0" (*•) ) Jx=6„ '
and
* 1
Bn = - S — —
^—lf'(a^e-{x)(x-b„,Y- + '
I=6m
It may happen that a is also a pole of /(x). It is easily seen that in this case An is given by the formula
k 1 /I = V ^i_
m=l Km' W
cZ'" (f(x)(x-b„,)'^ + dx'" \ W(x)
(n + /3)!?i
#+" j/'(a;)(a;-a)3+'
da;3+"
^>"W
where /S is the degree of multiplicity of the pole a ; the formula for Bn must likewise be replaced by
* 1
Un — —
d
1
{^-n)\n
j^Jf'{x)e-{x){x-h,^Y
.+1',
x=b„
^?-n
dx^-
when
jAf'{x)0,''{x){x-af+'--]
The preceding formulae do not give the value of A,, ; this cau be found from the formula
1 r f{z)e'{z)dz , 1 {f{z)e'(z)dz
* 1 r f{z)e'{z)dz 1 r
which gives
i(am-l)!
• rf'"-' (/(a;) ^' («) (a; - h^Y daf^""'
0{x)
1 +/(«),
JX=bm
when a is a regular point iov f{x); and
A =
1
,=i(am-l)! 1
da;""-' I <9(a;) |
fl!=&«i
when a is a pole oif(x).
dfi \f{x)d'{x){x-aY dx» I (9, (a;)
OS] THE EXPANSION OF FUNCTIONS IN INFINITE SEIUES. 105
Example I. Shew that
^=2 Ir+x^-j + 274 [i+^) +27476 ir+W ^ - '
when - 1 < .<• < 1.
Shew that the second member represents - , when la^| > 1.
Example 2. If 6'''"' denote the sum of all combinations of the numbers ^ ill
22, 42, &,...{2n-2y,
taken m together, shew that
2-ihr^+„!„(2« + 2)!t2,i + 3 2« + l+-+ 3 T^'""^ '
the expansion being valid for all values of z represented by points within the oval whose equation is |8inz| = l and which contains the point ^ = 0. (Teixeira.)
68. ETpaitsion of a function of a root of an equation, in terms of a paravieter occurring in the equation.
Now consider the equation
d (a.) = {x- a) e, {j) = t,
where < is a number such that along the contour S we have |^ (^)| > |^1. and along the contour s we have | 6 {z)\ < \ti.
The equation $ (a) = t, regarded as an equation in x, will then have a single root in the ring-shaped region bounded by the curves S and s; we see, in fact, from the equations
and
2mJsd{z)-t~2^ilJse{z)'''^'js6-'{z) '^"
\j^e'(z)dz + l^j^e'{z)e(z)dz+ ...
= 1,
1 [ & {z) dz 1
•2iri},e{z)-t 2iTi = 0,
that the equation in question has one root in the interior of <S and none in the inteiior of s.
Then if the function f{x) is regular in the region limited by S and s, we see from the preceding articles that the formula
n = 0 n = l l-
where An and 5„ have the values already found, gives the expansion in powers of i of the function / (a;) of the root considered.
As an example of this formula consider the equation (x-a)cosec.r = (, and let
x-a
106 THE PROCESSES OF ANALYSIS. [CHAP. VI.
Then we find
, cos a
" sin a
1 c;n-n(ain''a)
1
Hence
sin a
1 cosa " <» c?" + i(sin"a) 1
x-a sin a n=\(n + \)\ n da"*^ isina'
and thus gives the expansion, in ascending jiowers of t, of , where x is given in terms
of t by the equation
x = a + tsmx. (Teixeira.)
69. Lagrange's theorem.
Suppose now that the function /(^:) is regular at all points in the interior oi S, so that the poles ft,, &„, ... 6^ do not exist. Then the formulae which give the quantities An and B,i now become
. 1 d"-' if (a)] , ^is
5„ = 0.
Moreover the contour s can now be dispensed with, and the theorem of the last article takes the following form :
Let f{z) be a regular function of z at all points in the interior of a contour S, and let 0 {z) be a regular function with no zero in the interior of S. Let a be a point inside /S, and t a number such that for all points z (m. S we have
\{z-a)e{z)\>\t\.
Then the equation (z — a) 0 (z) = t will have one root x in the interior of 8, andy(a;) will be given as a power-series in t by the expansion
JKx)-J (a.) +^^^ ^ , ^^„_^ -j ^. ^^^1 1 .
This result was published by Lagrange in 1768 ; it is usually stated in a slightly different form, to obtain which we shall write
the result may now be enunciated as follows :
If f{z) and <f> (z) be regular functions uf z within a contour S surrounding a point a, and ift he a quantity such that the inequality
\t<j>{z)\<\z-a\
(iOJ THE EXPANSION OF FUNCTIONS IN INFINITE SERIES. 107
is satisfied ut all points z on the penmeter of S, then the equation
z = a + t<f> (z),
regarded as an equation in z, has one root in the interior- of S: and if this root be denoted by x, then any regidar function of x can be expanded as a potver- series in t by the formula
/(.)=/(a)-fi^^^[/'(a)l<^(a)r].
This result is of course a particular case of the more general theorem given in § 68.
Example I. Within the contour siuTounding z = (i and defined by the inequality
\z{z-a)\>\a\, the equation
z-a — = 0
z
has one root z, the expansion of which is given by Lagrange's theorem in the form
Now from the ordinary theory of quadratic equations, wo know that the equation
z — a — = 0
z
has two roots, namely
and our expansion represents the former of these only — an example of the need for care in the discussion of these series. If however we regard the expansion as a power-series in a, and derive other power-series from it by continuation in the a-plane, we shall ultimately arrive at the series
- (-l)"(2ft-l)! g"
„ti n\ {n-\)\ a^"-" which represents the other branch of the function z.
Example 2. If y be that one of the roots of the equation
y = \-^zy'^ which reduces to unity wlien z is zero, shew that
y.= 1 + „. + " y 3)^,_^«Jn + 4)Jn + 5)^
, re(TO+5)(n+6)(n+7)^ , ft(w+6) (Tt+7)(re+8)(?t+9) ^ ,
so long as |«| < J.
Example 3. If x be that one of the roots of the equation
x = \-\-yx^
108
THE PROCESSES OF ANALYSIS.
[chap. VI.
which reduces to unity when y is zero, shew that
, _ 2a-l „ , (3a-l)(3o-2) , ,
the expansion being valid so long as
|3^|<|(a-l)''-ia-'-|.
(McClintock.)
70. RoucM's extension of Lagrange's theorem.
Consider now two functions /(z) and (/> (j), which are regular at all points
within a contour G, on the perimeter of which the inequality i ~!Vv-
satisfied.
< 1 is
Then we shall shew that if the equation f {z) = Q have p roots a^, a^, ... ap in the region contained hy C, the equation f{z) — a.tf)(z) = 0 will have p roots Oi', a^', ... ttp, in that region; and for ever ij function F {z) regular in the region we shall have
p p 'K f^n p /7"~1
r=\ r=\- 11 = 1 "• r=l ""r
where
ir{z)--
f(^)
We may note that this theorem reduces to that of Lagrange when
f(z) = z — a and p = l. The result stated may be obtained in the following way :
We have 1 F (a/) = A-, f ^C^)-^'' ^^ ~ "f/ 7 ^^
2m J c f{z)-i<i>{z)
F{z)dz\f'{z)-aci>'{z)]
/■(^) 1/(^)1
i/(^)l"
+
[f{^)Y[f{^)-^4>{^)].
w-1 dz\f(z)]
_ „„ (1(1)] "-' ^M ^ an \<}>(z)]\f'(z)-acl>'(z)- 1/(^)1 /(^) 1/(^)1 f{z)-acp{z)_ ■
When n is large, the last integral tends to zero : we thus have on the right-hand side a power-series in a, in which the coefficient of a" is
or
i'-.
d"-' ^F'(z){<l>(z)]''(z-arr[ _dz"-^ 1 {fW
70—72] THE EXPANSION OF FUNCTIONS IN INFINITE SERIES. 109
1 i;. (/"-'
or — i.
n! rli rf"r"~'
/" («,)<^(((,)|"
which establishes the theorem. Putting F{2) = 1, it is seen that the number of mots ((' is p.
71. Teixcira has published tlic following generalisation of Lagrange's theorem, the proof of which may be left to the student. Let
where <^, (;), ... <^i (j) are regular functions of;; in the interior of a contour A', and < is a point inside A'. Let a be a positive quantity, so small that the condition
°0i (^) z-t
z-t ■■' I z-t I
is satisfied along the contour K. Then to every value of .r which satisfies the condition \x\<a there corresponds a unique value of r in the interior of A'; and f{z), where / i.s a regular function at all points in the interior of A', can bo expanded in ascending powers of X by the formula
where the summation is extended over all positive integral solutions of the equation
and where
6 = a + /3 + y + ...+X.
Another form of this result is
f{z)=/{t)+ I I 7-^^.{/'(0<^.-I,^(0}, M=0 >'=0 (."+1) ■ dt
where the quantities (^,,^ arc obtaiued from the equations
01, !' = </>>'.
72. Laplace's extension of Lagrange's theorem.
Lagrange's result can easily be extended to a case- in which the given equation is of a somewhat more general type.
Suppose that the equation
z = y^ [a + t(l>{z)]
is given, and that it is desired to expand some function f{z) of a root of this equation in ascending powers of t.
If we write a-{-t<^{z) = u,
the equation reduces to u = a + t(j} [yfr (m)).
The problem of expanding /(:s) is therefore equivalent to that of expand- ing/'-</r(M)j, where « is given by the last equation; and this can be done by Lagrange's theorem.
110 THE PKOCES8ES OF ANALYSIS. [CHAP. VI.
73. /( further generalisation of Taylor's theorem.
The .series of Laurent, Darboux, Buriuanu, etc. may lie regarded as extensions in different directions of the fundamental series of Taylor. A generalisation of Taylor's theorem of a .somewhat different character to these, is furnished by the following result, the proof of which may bo left to the student.
Iff{z) and 6 (z) are regular fu7ictiona of z in the neighbourhood of the point z=x, and if e,{z) = j'j9{t)dt, 6^{z) = j'e,{t)dt, and generally
e„{z)=jy„_,{t)dt,
then, for values of z in the neighbourhood of the point x, f{z) can be expanded in a series of the form
f{z) = aoB (z) + ai^i (2) + a^e^ {z) + ...+ aj„ (j) + . . . ,
where
_d f/W1
' dx\e{x)]'
and generally
the number of differentiations in the last expression being n.
It is clear that Taylor's series is obtained from this expansion by putting 6 (z) = 1. Example 1. Shew that
Example 2. Shew that
da„_. ff [x)
* (Laurent, Journ. Math. Spec, 1897.)
Example 3. By writing 6 {z) = e', obtain the expansion of an arbitrary function of z in a series of the form
where a^, a^ are independent of 2.
Example 4. In the general result, shew that when .r = 0 we have
where
/(j) = 2-"2'' and e(z) = 2^,z\
\
(Guichard, Annales de I'Ec. Norm., 1887.)
73, 74] THE EXPANSION OK FUNCTIONS IN INFINITE SERIES. Ill
74. The expansion of a function in rational functions.
Consider now a function f{z), whose only singularities in the finite part of the plane are simple poles o,, a.j, «», ...: let c,, d, ... be the residues at these poles, ami let C be a cii-cle of very large radius R not passing through any poles, so that f(z) is finite at all points in the circumference of G. (The function cosfc z may be cited as an example of the class of functions con- sidered.) Suppo.se further that at all points on the circumference of C, the modulus of /(«) is less than M. where M is a quantity which remains finite when large values of R are taken.
Then „- . I ■'- — dz = sum of residues of • at points in the interior
zm J c 2 — X z - X
of (7
of (7
r ^r
where the summation extends over all poles in the interior of G,
But i-.f /M.. = 2_f/(£)^ + ^ f ^X^,,
27ri ) c z — x zin J c z Sttz } c^KZ—x)
„ Or. ttri 1 c z{z — x)' if we suppose the function ({z) to be regular at the origin.
Now R being supposed large, I y— — x is of the order ^ of small quantities, and so terids to zero as R tends to infinity.
Therefore on making R infinitely great, we have
0 = /»-/(0) + Sc„(^-i^^-£),
or /(^)=/-(0) + 2c,.|-l-+-J^[,
which is an expansion of/ (a;) in rational functions of a;.
If instead of the condition \f{z)\ < M we have the condition \f{z) | < AIR", where M is finite for all values of R and n is a positive integer, then we should have to expand
by writing c t-x
Z-X Z Z- "' 2™*'(Z-X)'
and should obtain a similar but somewhat more complicated expansion. Example 1. Prove that
cosec«=- + 2(-l)»( H ),
z \z-jiir nnj
the summation extending to all positive and negative values of n.
To obtain this result, let cosec z — =/('). The singularities of this function are at the points 2=7iir, where « is any positive or negative integer.
112 THE PROCESSES OF ANALYSIS. [CHAP. VI.
For points near one of these singularities, put z=nn + f. Tlien
/(i) = cosec(«;r + 0- -:^.= ^-^^ - ~ fl+^V nn + ( snif nn \ mrj
f V 3! + 5!-; n7ry^7>^)
(-1)" 1
= 1- positive powers of (.
z — nn 'iiTT '■ "
The residue of /(z) at the singularity nn is therefore ( - 1)". Applying now the general theorem
/(.-)=/(o)+2c„r-i--+ii,
where c„ is the residue at the singularity a„, we have
/(.)=/(o)+2(-ir f-J_+ J-l
(z — mr iln)
But
Therefore
/(0) = Lt2„i) - + (positive powers of z)-- =0.
cosec2=- + 2(-l)" H ,
z ' \_z-nn UttJ
which is the required result.
Example 2. If a is real and positive and less than unity, shew that c" _ 1 " 2^ cos 2wa7r — Anir sin inan
e*^ 1 For if /(2)= .— j- - 7 , the singularities o(f(z) are at the points z = 2mn, where
n= ±1, ±2, ±3, ... +00. For points z near z = 2inri, put 2 = 2ji7ri + f. Then
n2naTri
= — T — h a series of positive powers of f.
The residue at 2 = 2n7rt is therefore e-"""'. Also
/(0)=rL±^i±:::-r
Applying the general theorem
we have therefore
gIM 1 ±« /I 1 \
e'—lz - „=±i \z-2»tn- 2Mn^/
±» e^"'"" » sin2?jair
= a-H 2 s-^+ 2 .
„=±i 2 — 2niir „=i nit
EXS.] THK EXPANSION OF FUNCTIONS IN INFINITE SERIES. 113
But
2 ^51'""^ = _ l.log(l-e««") + r^.log(l-e-^")
1
Thus
e*— 1 2 „^±iz-2mn „=i\z-2n{ir z + ZninJ " 22 cos 2najr - 4«7r sin 2najr ^
Example 3. Prove that
1 1 1 „ i . ^1^ 1
e3>r_e-3T(3,r)*+i^ For the general term of the series on the right is
{-lyr 1
which is the residue at either of the four singularities r, - r, ri, - ri, of the function
(ttV - ^x^){e" - e-'")siu JTJ '
Tlie singularities of this latter function which are not of the type r, -r, ri, —ri, are at the points
+ \/i X + \J -i X
2 = 0, .- =
^■2 t' \/2 t'
At 2 = 0 the residue is .,
at either of the four points 2= ~ — ^ — - , the residue is
\/2 ^
n . 2ir' \eV2 - e' Vs 7 sin \*^ V2
Therefore
2jri j c (t*^
itzdz
♦2* - J.Z:*) (e^« - e - «) sin nz '
where C is an infinite contour. But at points on C, this integrand is infinitely small
compared with - ; the integral round C is therefore zero. z
W. A. 8
114 THE PUOCESSES OF ANALYSIS. [CHAP. VI.
1 » (-l/r 1 -J
"" 2n.v''^rll e'^-e-'"' {r^y+i^* / Vfe .n/<?W ^v^ -i"l^\
-1
7ra;2 {e« + e-« - e« - e-«}
1
"7r.^2(e«-2cosa;+e-«)'
which is the required re.sult. Example 4. Pi'ove that
, , 1 3 5
sec.^• = 47r „ — 7-5 — ?n — 7~9 + ;
...)^
Example 5. Prove that
cosech..= l - 2.. (^^, - -^^ + ^^ ...) .
Example 6. Prove that
■sech :>: = in {^^,._^_^^i ' 9^2 + 4^2 + 25,7^4^ -) ' Example 7. Prove that
cotha; = -+2a- ( - - — s+j— „-, o + n-,-; — 5+...). Example 8. Prove that
» « 1 TT*
2 2 j—^, — .,,, o , ,„■ = —; coth n(t coth tt?).
(Cambridge Mathematical Tripos, Part I, 1899.)
75. Expatision of a function in an infinite prodact.
The theorem of the last article can be applied to the expansion of functions as infinite products.
For let f {z)\)Q & function, which has simple zeros at the points a,, aj.aa, ... where Limit | a„ | is infinite ; and suppose that f{z) has no singularities in
n=oo
the finite part of the plane.
Then clearly /' {z) can have no singularities in the finite part of the plane, and so"'^^ can have singularities only at the places ftj, a^, a^
75] THK EXPANSION OF FUNCTIONS IN INFINITK SEKIES. 115
Now for values of 2 near Hr, we have hy Taylor's theorem /(z) = (2 - «,)/' {aA + ^'-^f" («,) + . . .
and /'(z)=f'inr) + {z-ar)/"(<ir)+ ■■■■
Thus we have
• ./- = \- a constant + positive powers of (z — uA
/(z) Z-Ur
f'(z) At each of the points <(,.. tlie function ■ has therefore a simple pole, with
the residue + 1.
f (z) If then ^^^^ has at iiitinitv the character of the functions considered in /(2)
the last theorem, it can be e'xpanded in the form
/•(^)_/'(0) |T[ ^ I ^
/{z) /(O) „t, \z-a„ an
Integrating this expression, and raising it to the exponential, we have
f(z) = ce^^o^ n I-- e«..l
where c is a constant independent of z.
Putting z = 0, we see that/(()) = c, and thus the general result becomes
/(^)=/(0)e/t"> n 1-f e"n .
This furnishes the expansion, in the form of an infinite product, of any function _/" (2) which fulfils the conditions stated.
Thi.s theorem i.s a case of a general theorem on the factorisation of functions, which is due to Weierstrass, and which will be fouiul in Forsyth's Theory of Functions, Chapter v.
Example 1. Consider the function /(i)= '-, which has simple zeros at the points
z
r»r, where r is any positive or negative integer.
In this case we have /(0) = 1, /'(0) = 0,
and so the theorem gives imraeiliately
•^'%l(-4)-i'
/' (2) since the condition relative to the behaviour of 7.7^^ at infinity is easily seen to be
fulfilled.
8—2
116 THE PROCESSES OF ANALYSIS. [CHAP. VI.
Example 2. Prove that
(-©')l-(J:JH-UtJ)l-G/^J}{'HsiJ)
eosh k - cos x
1 - cos X
(Trinity College Examination, 1899.)
76. Expansion of a periodic function in a series of cotangents. Another mode of expansion, which may be applied to periodic functions whose poles are all simple, is that indicated in the following example.
Consider the function
cot {x — O]) cot {x — fl„) . . . cot (x — a„).
This is a trigonometric function of a-, having poles at the points a,, a,, ... a„, and also at all other points whose affixes differ from one of these quantities by a multiple of tt. There is clearly no loss of generality in supposing that the real part of each of the quantities Ui, a,, ... a„, lies between 0 and v.
Now let ABCD be a rectangle in the ^-plane whose corners are the points A{z = — ioo), B {z = TT — ioc ), C{z = 7r -h iao ), and D{z=iao); and consider the integral
r — ; I cot (z — Oi) cot {z — ttj) . . . cot {z — an) cot (z — x) dz
taken round the perimeter of the rectangle.
The integrals along DA and CB are equal but of opposite sign and cancel each other. Along CD, each of the cotangents has the value - i, so the
integral along CD is — k^- Similarly the integral along AB has the value £i
— . The whole integral has therefore the value
2 ■
The singularities of the integrand in the interior of the contour are at the points ^= Oi, flj. ••• f'jt) *! S'ld clearly the residue at a^ is
cot iflr — «i) cot {a,. — ((j) . . . cot {Ur — «r-i) Cot (tt^ — flr+l) • • •
cot (a,. — a„) cot (a,. — x), while the residue at x is
cot {x — a^) . . . cot (x — a„).
Since the value of the integral is equal to the sum of all these residues, we thus have
1 + (— 1)" . *■="
^ — i" = cot (« — Oi) ... cot (X — On) + ~ cot (ttr — a,) ...
cot (Or — a„) cot (ttr — x).
76, 77] THE EXPANSION OK FUNCTIONS IN INFINITE SEUIES. 117
Thus if II be even, \vu have
r=n 2
cot (x — a,) . . . cot (.«• — «„) = 2 cot (Ur - «i) . . . cot («r — ('„) COt (x — U,) + (— 1 )^,
r = l
ami if It be odd we have
r = n
cot (j — a,) ... cot(j; - a„) — 1 cot (Wr — <'i) ••• cot (ar — a^) cot {x — Ur).
r = l
This method of decomposition into a series of cotangents is of very general application to periodic functions ; it may be regarded as the trigono- metrical analogue of the decomposition of a rational function into partial fractions.
Example. Prove that
8iii(j.--fei) s,\n{x-b^ ... sin {x-b„) _ sin (a^-b^) ... sin (a,- 6,) ^ , ._ % 8in(.r-o,)sin(ar-aj) ... sin(j;-a„) sin (01-02) •■•sin (a, — a„) '
+ «i»(^2Z_^)^:iiL'L(«2 -1») cot (:.• - a.,) sin (a, -a{j ... sni [a^ - a„)
+
+ co.s(a,-|-«2 + ...+(f„-/<, -h.^- ... -t„).
77. Expansion in inverse factonals.
Another mode of developn;ient of functions, which although investigated by Schlorailch as long ago as 1863 has hitherto not been much used*, is that of expansion in inverse factorials.
Let I be a line drawn parallel to the imaginary a.\is in the 2-plane ; and draw a circle of large radius, having its centre at the point where I cuts the real a.xis.
Consider a function f{z), which has no singularities within the semi- circular area which is bounded by I and this circle and which lies on the positive side of I; let 7 be the semi-circular arc which bounds this region. Suppose moreover that at all points of 7 we have the inequality
\f(.^)\<M
satisfied, where M is finite however large the radius of 7 may be chosen.
Then if ^ be a point within this semi-circular region, we have
f^--uh\y-^-
Nov
)dt
■2)'
r/(t)dt^rf{t)(H t zf{t) ,
!y t-z ]., t .'ytU-
y " • Y'
* References to some recent work are Riven by Kluyver, Comptes lieiidiu, cxxuv. (1902), p. .587.
118
THE PROCESSES OF ANALYSIS.
[chap. VI.
Jyt(t-.
But
which is infinitesittiiil wlien the radiu.s of 7 is infinitely great. Thus
•' ^ ' 2.iri}y t 2TriJ, t-z '
if we now suppose that the direction of integration along I is from — I'oo to + too.
Now if n be any positive integer and z be not equal to 0, — 1, — 2, etc., we have the identity
1
+
t
+
t{t + \)
.+ ... +
t{t+\)...(t + n)
z-t z ' z(2+l) ' z{z+\){z-\-T) ^(2 + l)...(2 + n)(2-i)'
on substituting this in the second integral we have therefore
J/ \ "1
z z {z -\-Y)
+ ...+
where
z{z-\-\) ...{z-Vn) ^ 1-Ki] iz{z ^\) ...{z^ n){z -t) '
''^=hi\.
f{t)dt
27ri J y t
(h = ^ ■[ f{t)tdt,
an+i = H-- / f{t)t(t + \)...{t + n- \)dt. Now the product
can be written
t{t + l) ...{t-\-n) z{z+\)...{z + n)
-n
Zr=\
1 +
1 +
and it diverges to zero or to infinity when z tends to 00 according as the real part of t — z is negative or positive, as can be seen by comparing it with the product
n 1 + -), .
/,
dt
77] THE EXPANSION OK FUNCTIONS IN INFINITE SERIES. 119
which has the value {n+l)'~'. But the real part of t — z is, in the case under consideration, negative ; aii(i so the product
t{t+ l)...(t + n) z(i + l)...(z+n)
is infinitesimal when » is infinite.
Smce/{t) is finite along I, and | - — -. is finite, we see that
J i\e — t\
/(()<(<+ 1) ■■■(< + «)
2{z+l)...{z+ n)(z-t)
is infinitesimal when n is infinite.
We can therefore expand f{z) in the form
^("^ = "« + 7 + .(.-TT) + .(.+ i)(. + 2) + --
the coefficients a being given by the above equation ; and this expansion is valid (or all values of z whose real part is greater than the real part ot z at any of the singular points oif{z), e.xcept for the points
z=0, -1, -2
Example 1. Obtain the same re-sult by using the equalities
J c 2-t J c Jo
Example 2. Obtain the expansion
log(l+l\=l- °i_ + ^
[I where a„= i t {I -t){-2-t) ... (n-l-t)di,
and discuss the region of its convergency. (Schlomilch.)
Miscellaneous Examples.
1. Let e-'^P^ denote the »ith derivate of e"^, so that
i>(,= l, P^=-2z, ^.^=42=- 2, etc.
Shew that if f{z) is an arbitrary function, then/(z) am lie expanded in the furrn
/{z) = aaPo + aiPi + aiP^+...,
1 f
where <'»=r"^— ;; ^ — 7- e-''P„{x)/{x)dx,
2. 4. 6 ... 271 vjt^ -"
and find the region of convergence of this series. (Hermite.)
120 THE PROCESSKS OF ANALYSIS.
2. Obtain (from Darboux's forimilii or otherwisL') the expansion
(z-af
[chap. VI.
2!(l-r)2
{/"(2)-ry"(a)}
+ .
+ ^(l-r)" ■!/""(--) -'•■/"■'(a)}
find the remainder after re terms, and di.souss the convergence of the series. 3. Shew that
/(j; + A)-/(x) = ^!/'(r + A)+/' (x')}
+(-i)"-''-''-'(-;;,f""'^|-l!/"(.'^+^)+(-i)"/"wi
+ ( - 1)» /'»" ' J\„ (0/"*' (■^• + '^0 dt, where y„ (.^) = -_^i_^ :,.,.+i (I -:r)"+i ^ {^-i (1 -x)-h}
77711 J D
and shew that y„ (x) is the coefficient of ti'. t" in the expansion of {( 1 - tx) (1+t- tx)} - i in ascending powers of t.
4. By taking
,, I rd" f(l-)-)e^"n
in Darboux's formula, shew that
f{x + h) -f{x)= - a,h |/' {x + h) - J/' (:r)|
. + ( - 1 )» A" + • r <^„ (<)/»*' (.i- + ht) dt,
where
l_,-g-M"
MISC. EXS.] THE EXPANSION OF FUNCTIONS IN INFINITE SERIES. 121
&. Shew that
+
+ ( _ 1 )»-i 2^» (2^» - 1) (i::")^ ^^p„ . „ („) +_^,2„ . 1, (,),
27i!
6. Prove that
/ ih) =/ (^i) + Ci (-'2 - ^i) /' (h) + 0, {^, - z,) V" (^i) + f'a (^2 - ^i)' /'" (^2)
+ Q('2-^l)V"'('l) + -
I where t'„ is the coefficient of .:" in the expansion of cot ( 1 - 5 ) 'n ascending powers of t.
(Trinity College Examination.)
7. If .r, and .r, are integers, and (p (z) is a function which is regular for all values of z (finite or infinite) of which the real part lies between .1:^ and .1:^, shew (by integrating
f <(> (z) dz J e2TU - 1
round a rectangle whose sides are parallel to the real and imaginary axes) that
i</)(j.-,) + <^(j:, + l) + <>(.r, + 2)+.. +</)(.r2-l) + i(^(:i-2)
Hence by applying the theorem
/■« „2n-l
where Bi, B^, ... are the Bemoullian numbers, shew that
0(l) + <^(2) + ...+<^(«) = C+i<^(n)+p<^(.-)rf2 + ^i ^~^^2,.',^''""'<^'-'"-"(»). (where C is a constant not involving n) provided that the last series converges.
8. Obtain the expansion
X "-• , , 1 .3...(2«-3)x*
u= + 2-1)"-' \ ^'jr-
2 „_., m! 2"
for one root of the equation x = 2u + m', and shew that it converges so long as | x | < 1.
122 THK PU(JCE.S.SES OF ANALYSIS. [CUAP. VI.
9. If <S' " , , denote the sum of all combinatiou.s of the numliors
^Tt + l
P, 3'\ 52, ... (2n-l)- taken m together, .shew that
2 .sin2^„.„ (2w + 2)! l2« + 3 '^2(«+i) 2« + 1 ^ •■' ^ 2("+i) 3/
10. If the function f{z) is regular in the interior of that one of the ovals whose equation is | sin2| = C (where 6'^ 1), which includes the origin, shew that f{z) can, for all points z within this oval, be expanded in the form
/(2)=/(0)+ 2 ^ ^"^ ^, 2. f ^) ^.^^^
71=1 ^"- ■
i
+ g/'--"(0)-f^;'^../'--"(0)+-+C../-(o)„i„,„.,,
ti-o (2» + l)!
where S^ is the sum of all combinations of the numbers
22, 42, 62, ... (2»-2)2
taken m together, and <Sl'"!. , denotes the sum of all combinations of the numbers
12, 32, 52,... (27(-l)2, taken m together.
11. Shew that the two series
„ 223 22^
^^+3^+F +
and
22 2^ / 22 Y 2.4/ 22 Y
1-22 1 . 32 {j^^j "•" 3T52 \i^y
represent the same function in one part of the plane, and can be transformed into each other by Burmann's theorem.
12. If a function f{z) is periodic, of period 2Tr, and is regular in the infinite strip of the plane, included between the two branches of the curve | sin 2 | = C (where C> 1), shew that at all points in the strip it can be expanded in an infinite series of the form
/(2) = ^0 + ^j sin 2 + ... + /!„ sin" s +
+ C0S2(5, + jS.3sin2 + ...+ZJ„sin«-"2+...);
and find the coefficients ^-1 and B.
13. If <p and / be connected by the equation
of which one root is a, shew that
F (.v) = F fF -\ f .'
^ ' \ <t> -^ ^ 2 ! 0'3 I 0" {pF')' 1 1
"3! <^'«
^' (02)' (/3/")
0" (02)" ipF')'
0"'(0T'(/3/")"
1
17172+-
;
MISC. EXS.] THE EXPANSION OK FUNCTIONS IN INFINITE SERIES, where F, /, /", etc. denote
'■w, /<..., %^'
123
14. If 11 function H'(o, b, x) lie detineii by the series
\V{a, b, x) = x+ 2, ■>^' + ^ ~^, ^'^•'+-
1
which converges so long as
H<
\i>r
show that
^ir(a, b, a-) = }+{a-b)W(a-b, b, x);
and shew that if y = If (o, b, x),
then x= W{h, a, y).
Examples of this function <are
W{\, 0, ..-) = *'- 1,
ir(o, 1, .i-)=iog(i+-^-)- {\ + xY-\
15. Prove that
W{a, 1, x) = ^-
1 .. 1 , ;(-i)"^«
2 a.if' " °
where
G'.= l
6a,
3ai 5a.,
0
■2a, 4a,
0
0 3a„
(2n-2)a„_i (n-l)«a
na„ C«-I)«„.,
and obtain a similar expression for
16. Shew that
I 2 a^A
In-U )
(Jezek.)
(Mangeot.)
2 ttrlf 0
r+1 8a,
where S^ is the sum of the rth powers of the roots of the equation
0
(Gambroli.)
17. If /,(z) denote the «th derivate of /(z), and if /_„(i) denote that one of the reth integrals of f{z) which has an n-ple zero at z=0, shew that
f{z + x)g{z + x)= 2 f„{z)g-n{x)\ and obtain Taylor's series from this result, hy putting ^(2)=!.
(Guichard.)
124
THK PROCESSES OF ANALYSIS.
[chap. VI.
18. Shew that, if x bo not an integer, the series
{x+mfix + nf
in which m cand n receive in every possible way unequal values, zero or integers lying between +/ and -/, vanishes when /increa.ses indefinitely.
(Cambridge Mathematical Tripos, Part I, 1895.)
19. Sum the infinite series
n=-,i\{-)''x-a-n 71/'
where the value n = 0 is omitted, and p, q are positive integers to be increased without
limit.
(Cambridge Mathematical Tripos, Part I, 1896.)
,„ „ XncoH.Zn)dx
20. li F{x)=e''> , shew that
F{x) = e^
and that the function thus defined satisfies the relations
1
F{-x) =
F(xy
Further, if
shew that when
21. Shew that
F{x)F{l-x) = 2s\nx-,T.
V'W = « + J + ^.+ ...
= -j'^og(l-.-)c;(log2),
(Trinity College Examination.)
[■+©"] [-G-i-.r] [-G.-VJ] [-U^.)"] [-G.VJ]
n{l -2e-i>« cos (^ + 0„)+e-2<»<;}i{l- 2e-»i7COS (a;-0„)+e-2»i'}i
i 5 -it cos =
2''(l-cosaO^«
where
a„ = ^sin
2(,-l
MISC. KXS.] THE EXPANSION OF KUNCTIONS IN INFINITE SERIES. 125
2(7-1
^„ = X-co8 • rr,
n
»iid 0<j.-<2»r. (Mildner.)
22. If |x|< 1 and a is not a jHwitive integer, shew that
where C is a contour in the <-pl.ine enclosing the points 0, x. (Lerch.)
23. If 4>j{2), <t>-,(^)y •■■ •''•re any polynomials in z, and if F{z) be any function, and if ^i (^)i '^2 (')) ••• ^ polynomials lielinod by the equations
J a z — x
J a z — x
f F{x) 4,, (.r) 4,, (.r) ... <^„.., (a-) '^•" ^'l^^^-M dx = ^^(z), J a - — J-
«hewthat f IM^ = pM -^ J^
<^l (2) <t>2 («) *3 (^) <^,(.')0,(2)...^„(Z)
.^,(^).^.(')-0..(J>^'^-^-^^'^-^-'^-<-^iS-
24. A system of functions />„ (z), pi {z), p., (2), ... is defined by the equations
i'o (2) = 1 > P» -M (-') = ('■ + «n'" + K)P„ (Z),
where a„ and &„ are given functions of n, which for h = cjo tend respectively to the limits 0 and - 1.
Shew that the region of convergence of a series
2e„i>„(j), where e, , e.,, ... are independent of z, is a Cassini's oval with the foci +1, - 1.
Shew that everj- analytic function /(c), which is regular in the interior of the oval, can for points in this region be expanded in a series
f{z) = S{c„ + zc„-)p„{z), where
126
THE PROCESSES OF ANALYSIS.
[chap. VI.
the iiitegi'als beiny taken round the boundary of the region, and the function.-) y„ (z) being defined by
?o (^) =.2+4+ 6„ • ^— ^^^ = ^+»„.U6„. , !?" (^)-
(Pincherle.)
25. If I'n (.!■) be the coefiiciont of — . in the expansion of
in ascending power.s of j, .so that
3v' — h^ Po=h A='-. /'2= ' , etc.,
shew that
(1) Pn{x) is a homogeneous polynomial of degree n in .r and li, rIP
(3) I" P„{.v)dx^O («>1),
(4) If y = «o/'|) (a;) + aiP,(.r) + a2P2W + ---> where a|),aj, a.,, ... are real constants, then the mean value of -j-^ in the interval from .r= - /; to .)■= +/( is a^. (Leaute.)
26. If P„ (.r) be defined as in the preceding example, shew that /-„„ = ( - 1)".2 ^,,„. (^oos -^- - ^^,„ cos ^ + 3.„„ cos ^^ + ... j ,
22m + 1 «>" -f- + gSSTi s"' ^- + • • ■ ) ■ ( Appell.)
I
CHAPTER VII. FouRiKR Series.
78. Definition of Fourier series ; nature of the region within ivhich a Fourier series converges.
Series of the type
Uo + rti COS z + a^ cos 'Iz + as cos 'iz + . . . + 61 sin 2 + b., sin iz + b, sin 3i + . . . ,
where «o. «!> «3. "a- ^1. ^■^> f>3, ■■■ i^i'e independent of z, are of great import- ance in many analytical investigations. They are called Fourier Series.
We have already seen that the region within which a series of ascending powers of z converges is always a circle; and the region within which a series of ascending and descending powers of z converges is the ring-shaped space betiveen two circles ; we are therefore led by analogy to expect that series of the Fourier type will likewise converge within a region of some definite character.
To investigate this question, write e'' = f.
The series becomes
«o+ 2 t + --- + — 2 — r+... + — r> — 5 +•••+ 2^'^
This is a Laurent series in f ; it will therefore be convergent, if at all, within a ring-shaped space bounded by two circles in the f-plane ; that is, it will be convergent for values of f satisfying an inequality of the type
a<\^\<b,
where a and b are positive constants.
Now let
z = X + iy ; then
128 THE PROCESSES OF ANALYSIS. [CHAP. VII.
SO \^\ = e-'->,
and therefore the inequality becomes
log (I < — 1/ < log 6.
This inequality defiiies a belt of the ^-plarie, bounded by the two lines y = — log a and y = — log h ; hence tlte regiun of convergence of a Fourier series is a belt of the z-plane, bounded by two lines parallel to the real axis.
It may however happen that the Laurent series in f is divergent for all values of f, in which case the Fourier series is divergent for all values of z; or, (and this is the most important case for our purpose,) it may happen that a = 6, so that the region of convei-gence of the Laurent series narrows down to the circumference of a single circle in the f-plane ; in this case the region of convergence of the Fourier series narrows down to a single line parallel to the real axis in the plane of the variable z.
If now the coefficients a^, «i, a„, ... b^, b^, ... are all real, considerations of symmetry shew that if the Fourier series is divergent for a value z = a+ ib, it will also be divergent for the value z = a—ib; so if in this case the region of convergence narrows down to a line, that line can only be the real axis in the 2r-plane.
Hence a Fourier series with real coefficients may converge only for real values of z, and diverge for all complex values of z.
An example of this class of expansions is afforded by the series
sin 2 — 2 sin 2^ +3 sin 82 — ^ sin 4^ + . . . . Writing this in the form
2iv 2 3^ •••; 2i\ 2 ^^ ■
we see that it diverges when z is not purely real ; when z is purely real and not an odd multiple of it, the sum of the series is
i.log(l+e-)-llog(l + e--), or 2t^^^^"'
or ■ • 2 ^ "*" ^'"''
where k is some integer, as yet undetermined.
Now when ^ = 0 the sum of the series is seen directly to be 0 ; when z=a< tte sum of the series is tan~' 1, or j ; when 2 = — — the sum is
78] FOURIER SERIES. 129
— tan"' 1, or — - . In this way we see that tvhen z lies between — tt and + tt,
4
the integer k is zero.
But k is no longer zero when z is greater than tt ; for each term of the series is cleariy unaffected if s + 27r be written for z : hence the sum of the series must be the same for z + 2v as for z ; and hence when n <z < Stt, the sum of the series is z — v. so that when z lies between ir and Stt, the integer k is — 1.
Proceeding in this way, we see that the sum of the Fourier series is -z + k-rr, where k is an integer chosen so as to make ^z+k-n- lie between
— 5 and +-„• This is important as shewing that the sum of a Fourier- series
is not necessarily a continuous analytic function. It is clear however that the sum of a Fourier series can have discontinuities only in the case in which the region of convergence nan-ows down to the real axis ; in the other case when the region of convergence is a belt of finite and infinite breadth, the Laurent series in ^ represents an analytic function, and therefore the Fourier series in z does also.
Krample. Shew that the scries
cos ' - 5;, cos 2: + .J2 cos 3z- ...
■ n- 1 converges only for real values of z, and that when — tt < c < + rr its sum is — - 7 z^.
For when z is real, the series is absolutely and uuiformly convergent, as is seen by com- paring it with the series 1 +92 + ~2+ ••••
When z ia complex, we have (putting z = j; + i>/)
— , cos nz = ;^, {c' (« + nirt + e'l - « - "«)} ; n' 2n-
g-ny g + nv
now either -^ j or j is infinite for « = x , so the terms of the series are ultimately
infinitely great and the series diverges.
To find the sum when z is real, it has been shewn that when — n<z<Tr we have
ii = sin J- J sin 22 + J sin3z....
This series is uniformly convergent in the interval (though not at its extremes - n and n) and so can be integrated.
Thus c -ic-=cos z - gj cos 2j+ ^j cos 3z - ...,
where c is a constant.
W. A. 9
130 THE PROCESSES OF ANALYSIS. [CHAP. VII.
To find c put 2=0, which gives
«-l 22 "'■32 12'
whence the result.
79. Values of the coefficients in terms of the sum of a Fourier series, when the series converges at all points in a belt of finite breadth in the z-plane.
The connexion between the coefficients a„, a,, a.,, ... , b^, h,, ... of a Fourier series, and the sum of the series, can be easily found in the case in which the series converges in a belt of finite breadth in the 3-plane. For in this case, as we have seen, the sum of the series is an analytic function of z. Let it be denoted hyf(z), so that
f{z) = ao+ a^cos z + a^cos 2z + ... +6, sin z + b.,sm2z + ... .
Writing ? = e", the series becomes
a,-ibi ,ar-ibr^ , a^ + ibx y_, , »r + iK ^_
and by Laurent's theorem the coefficients in this expansion are given by the equations
where C is any circle in the ?'-plane, surrounding the origin and contained within the ring-shaped region in which the expanded function is regular. Now if the quantities ar and b, are all real, we see as before by symmetry that the real axis must be contained in the region of convergence in the 2-plane, and therefore the circle of radius unity must be contained in the region of convergence in the f-plaue, since this circle corresponds to the real axis in the 2:-plane. We can therefore take C to be a circle of radius unity, with the point ^ = 0 as centre.
Now writing f=e" in the integrals, we have
■7r(a,-ibr)==!'f(z)e-"'dz,
Jo
IT {Qr + ihr) = f{z) e^'dz, Jo
]d so
1 f'" a. = - f(z)cosrzdz (r>0),
■n-J n
6r = - /"(2) sin rzdz,
TtJ n '
79, 80] FOURIER SERIES. 131
1 [-' and «„ = 2 I J i^) ^^•
These equations give the vahies of the coefficients a,,, a,, a,, ... , t,, h„, ... , of the Fourier series, in terms of the sum f{z) of the series, in the case in which the series converges over a belt of finite breadth in the c-plane. We shall see in the ne.Kt article that the same formulae hold good in the more extended case, in which the series converges only for real values of z.
Example. Shew that the function - — _, — m ca", when ^•<1, be expanded in a
X — jiA- COS * ^p A«
Foiu-ier series of sines of umltiples of ;, valid for all points z situated in a belt, of width - 2 log k; parallel to the real axis in the *-plane.
For we have
sin z _ ^ i 1
l-2y(- cos z + /•- " 2a- ( r^e'
1
l-y!-e-
« 1 and this can l>e expanded in the form 2 ., /"(e"" — e"""), provided ^le**! andifcle""! are
less than unity. This can only hapjren when their product k- is less than 1, i.e. when
-l<;t<l.
When this condition is satisfied, on putting z = x- + ii/, it is clear that we must have
!e""''i<T and>y(:, i.e. we must havo-y lying between log Ij] and log^, i.e. z must be
within a belt of width - 2 log k, parallel to the real axis. When these conditions are satisfied the expansion is valid, and so
sin 2 2 ;„ 1 •
, — ?ri — ,.>= 2 A-"~'sm7i3. 1 — 2* cose + A- „=i
80. Fourier's Theorem.
We have already said that the most interesting cases of Fourier's series are those to which the investigation of the last article cannot be applied, on account of the iact that the series converges only for real values of z. It is therefore necessary to undertake another investigation, in which the assumptions of the last article are no longer made. The result to which we shall be led is known as Fourier's theorem, and may be stated thus :
If/{z) be a quantity which depends on a variable z, and which is finite and has only a limited number nf maxima and minima and of finite discontinuities in the interval 0 < z < 'Iw, then the sum of the series
a„+ ^ (a m cos mz + bm sin mz),
m = l
9—2
132 THE PROCESSES OF ANALYSIS. [CHAP. VII.
where
1 /■'" Om = - ( f{t) COS mtdt,
TTJo
1 P"
hm= ~ \ f (t) sin mtdt, Try 0
represents /{z), at every point in the interval 0<z<2Tr /or which f(z) is continuous ; and at every point in the interval 0 <^< 2Tr for luhich f{z) is discontinuous, the sum of the series is the arithmetic mean of the two values of f{z) at the discontinuity.
The discussion of Fourier's theorem given below is a modification of what is known as Cmichy's second proof, which was originally pulilished in 1827 in the second volume of his Exercices de MaMinatiquos, and is reprinted in his Collected Worka, Second Series, Vol. vir., p. 393).
This proof, (which in its original form was in some respects imperfect,) seems to have been little used by the mathematicians of the nineteenth century, who in the discussion of Fourier's theorem almost universally followed the exposition of Dirichlet (which is also reproduced later in this chapter) ; the importance of /Cauchy's proof was shewn by A. Harnack in 1888. It may be observed that the restrictions placed on f{z) — as to its having only a limited number of maxima and minima, etc. — are sv,ffi,cient but not necessary for the validity of the expansion.
To establish the theorem, we write the first 2k + 1 terms of the expansion in the form
1 /•Sir 1 w=k f'lrr
^- f(t)dt+- S /' it) cos m {z - t) dt, •^ttJo 7r,„ = iJo'
or
m=k Y
. = k 1 fin
S -|- e'^'^'-ofiDdt,
or Uk + Vk,
m — k 1 m=-k
m — k 1 rz
where t^*= S ^ e"" «" "/(O £^«.
n= 2 ^ \ e"^'^'-'^f{t)dt.
m=-k ^TT Jj
We shall now investigate the behavi(jur of the quantity U^ when k, though finite, is a large number.
Let (p (f) denote the quantity
g2»-f
}~-^re<^'-"f{t)dt.
80] ForuiEU SEHIES. 133
Then <f> (f) clearly has a definite value corresponding to every value of f, fxcopt the exceptional values ^=0, ±i, ± 2i, for which 6*^= 1 ; moreover, it is easily seen that the quantity
k '
tends to a definite limit when Sf tends to zero, independently of the way in which S^ tends to zero (still excepting the points 0, ±i, ±2i...); hence <}>{^) is an analytic function of f, having poles at the points 0, ±i, ±2i ...; and the series U^. is clearly the sum of the residues of <p {^) at those of its poles which are contained within a circle C^ in the ^-plane, whose centre is
at the origin and whose radius is (^" + 5 ) •
Hence ^a = ./,• I 't>iOd^.
Write
Thus ^"^Lj^'f'^^^
de.
Now we can write
"■ f - A "■ , t - t 3»r , _ 1 -St , , _ 1
^"^LW + + + + \K<^{K)dd
^-t-i ■^+k-i 'j-k-i ■'^+k-
= I^ + I^ + I^ + I. + I, say.
At points in the range of integration of /, and /j, the real part of f is positive and at least of order ^•"i ; and so for these integrals we have
In this expression, as k tends to infinity, :j :::j_> tends to the limit unity,
and fe^i^-'-'"' tends to the limit zero: thus f<^(f) tends to the limit zero, since the range of integi-ation from 0 to ^ is finite ; and hence as /, and /^ are the integrals of ^<p (f) taken over finite ranges, we see that /j and /j tend to zero as k tends to infinity.
Considering next the integrals /, and /j, we observe that the quantity ^^_. is never infinite when Q <t <z < 27r, and so f<^ (f) is never infinite ;
134 THE PROCESSES OF ANALYSIS. [CHAP. VII.
and thus, .since Z, and It are integrals of ^(/) (f) over ranges which become infinitesimal a.s k tends to infinity, it follows that /j and /< tend to zero us k tends to infinity.
Consider next the integral /„ or
.7 + *"'
where r<^ (?) = ^^ jje<''-^^f(t) dt.
In the range of integration of /,, the real part of ^ is negative and at
least of order kK The factor „-; — - therefore tends to the value — 1 as A;
tends to infinity ; denote it by — (1 + a^.), where a^. tends to zero as A; tends to infinity.
Now r^ei''-'>f(t)dt = J, + J„
Jo
i_
where /, = I "* * fef ''-" /(<) dt,
■ 0
and J,=j' ^ ^e<''-'^fit)dt.
«-i — ; log*
Considering first Jj, we see that within its range of integration the
quantity ^(z — t) has its real part always negative and at least of order
k^ .
■r — T, which tends to infinity with k; hence the quantity fe^''~" tends to
zero as A; tends to infinity ; and therefore as the range of integration in Ji is finite, we see that Ji tends to zero as k tends to infinity.
Consider next J^. Writing v = ^ {z — t), we have
flog*
/,=/;V(-j)c^^.
and writing e" = w, this becomes
i
j-glog* / \oo;w\ ,
— log w
Now as k tends to infinity, e'""* and y- tend to the limit zero. Let /(z — 0) denote /(«) if ^ is a point at which the function /(z) is a continuous
80] FOURIER SERIES. 13-5
function, and at those points at which /"(;) is a discontinuous function let /(^ — 0) denote that one of the two values oi/{z) which is continuous with the value of/ for values smaller than z. Then since there cannot be another discontinuity within an infinitesimal distance of 2, we can write
where ;; tends to zero as k tends to infinity ; and so
i _J_
J„=/(2-0)| dw+j -qd
log ft
= -/(.--0) + e'"^' 7(^-0)+]^^ 7,dw,
or j; = -/(s - 0) + e,
where e tends to zero as k tends to infinitij.
Thus
?</>(?) = - (1 + at) {/,-/(^- 0)4-6),
where a,., /,, and e, each tend to zero as k tends to infinity. We can write this ^<^ {i,) =f{z — 0) + T, where t tends to zero as k tends to infinity ; and this is tnie throughout the range of integration of the integral 1,.
Thus
a=4f {f{z-(i) + T]dd,
or /j = ^ f{z — 0) + (7, where a tends to zero as k tends to infinity. Hence finally
where cr, /], /j, It, /, each tend to zero as k tends to infinity ; which can be written
U, = \f{z-0)+u„ where Wi tends to zero as k tends to infinity. Similarly we can shew that
Vk = \f{z + 0) + vu, where V), tends to zero as k tends to infinity, and where f(z + 0) denotes f{z)
136 THE PROCESSES OF ANALYSIS. [CHAP. VII.
if « is a value for which the function/ (2:) is continuous, and denotes the vahie off (2) for values slightly greater than 2 if ^ is a value for which the function /(z) is discontinuous. Hence the sum of the tirst (2A.- + 1) terms of the Fourier series is
where wj and vt tend to zero as k tends to infinity ; the sum to infinity of the series is therefore
l[fiz-0)+f{z + 0)], which establishes Fourier's theorem.
It must be observed that the sum of the series coincides with /(z) only for values of z between 0 and 27r ; outside these limits the sum S (z) of the series can be found from the circumstance that S (z + 2n-7r) = S (z), (a result which is obvious, since all the terms are periodic) ; while /{z) may of course have any values whatever when z is not included between the limits 0 and 27r.
Example. Take a function /(j) such that
f{z) = j from 2=0 to z = i!,
and /(s) = -- from 2 = 7r to 4=2n-.
The corresponding Fourier series is
Oq + 2a,„ cos mz + 26,,, sin mz,
1 /■2ir
where Om = - f (t) cos mtdt,
"•jo"
6„,= - I /(t) sin mtdt, 1" J 0
1 rz"
These integrals give
1 /"ir 1 /•2)r
00 = 0, (im = -7 I CQsmtdt — j I coii7ntdt = 0,
1 /"rr 1 [in 1
&m = 7 I tiinmtdt- I sin »!Zcfo = -— (1 -cos «i7r). ■i J 0 -i J TT -im
Therefore 6„, = 0 if m is even, and 6„,= — if m is odd ; and so we liave J, . sin z sin 3z sin 5z .
which is the required Fourier expansion.
4
81] FOURIEK SERIES. 137
This series eau bo summed by clomentfiry motliods in the following inauuor. We have
4(--f--)4('-f--)4(--'r--)-i(--"-'r--)
4i °°(l-e«)(l + e— ) 4i "S*^ 4+ 2 '
where r is an undetermined integer. It is clear from the above that r actually has the value zero when 0 < * < rr, and unity when n <z<2ir.
81. The represmtation of a function by Fourier senes for ranges other than 0 to 27r.
Suppose now that the range of values of z, for which it is required to
represent a function f{s) by a Fourier series, is not the range from 0 to 27r,
bat from a to b, where a and b are any given real numbers. To extend
Fourier's result to this case, we take a new variable z defined by the
equation
b — a , z = a -t
and write
b — a
z = a+^z,
fla+^^z'^Fiz').
Then F{z') is a function whose value is given for all values of its argument . z' between 0 and 27r.
I
Therefore by the previous result we have
F{z') = ~ r''F(t')dt'+- I r cos m(z'-t')F (t')dt',
or writing
»b — a ,, * = «+ 2-rr*' we have
This last result may be regarded as the general form of Fourier's theorem.
aim ^ fi-mi
Example. To express the fuuction — ^ 1^^,, as a Fourier series, valid when
-»r<;<7r. Here a= — rr, h = n.
138 THE PROCESSES OF ANALYSIS. [CHAP. VII.
The formula therefore becomes
/W = 9l I' /W<^«+- 2 f" COS n{z-t)f(t)dt.
^rr J -„ TT n=l J -ir
Since in this case /(<)= -/( - 1), this reduces to
2 "= /* )i e"** _ ^ ~ *"* f(z) = - 2 sin wi I „ sinTi^rfi
= 2 sinni I ^ ., '^ r '(It
n=l Jo nl {€""• - «-"")
_ " ainnz (-£(>» + in) >r_g-(m + in)ir g(,m-in)ir _ g-{m-m)jr\
n=i»n' («"'"■-«"""') 1 111 + in m-in J
_ I (-I)"sin nz ,' 1 1_\
n=i ffi \TO+tVi m — in)
which is the required expansion.
82. The Sine and Cosine Series.
We proceed to derive two particular cases of Fourier's theorem which are of frequent occurrence.
Suppose that a function f(z) is given for a range 0 to Z of values of the variable z, and that we require a series which shall represent f(z) for these values of z, and which shall have the value /(— z) for values of z between 0 and — I.
To obtain a series of this character, we write in the preceding result a = — I, b = I, J (— z) =/(z). Thus we have
or
/ (.) = ]j\fit) dt + \ l^ COS ^7^£ cos ^y (0 dt,
which is called the Cosine Series.
If on the other hand we require a series which shall represent /(^) for values of z between 0 and I, and shall have the sum — /(— z) for values of z between 0 and — I, we write in the general result a = — I, b = l,f{— z) = —f{z), and thus obtain
f{z) = - 2sm— ^ sm -.-f{t)dt,
which is called the Sine Series.
J^. Or '
82] FOURIER SERIES. 139
Example 1. Expand — - sinz in a cosi no .series, valid when 0<2<jr.
When 0 < 2 < n-, we have by the formula just obtained
ir — z If"" 1 "^ f^
' :=- j (it- t) sin tdt+ 2 cos wjr I {n - 1) ain t cos mtdt ^^ J 0 ^ «=\ Jo
, 8ini =
= - «- cosM-;— / COS tdt
n\_o 2 J 2n- ./ 0
+ s- 2c087/iij| ()r-r)sin(m+l)^/f- I (»r- ?) sin (»i- 1) ^rfA
^^ m=l [Jo Jo I
1.1 f (t-<)cos2«"] , 1 " p {w-t)coam + lt (■n-t)cos7n-lt~\
2 2jr Lo 2 J 27rm=2 Lo m + l m-1 J
1 1 1 - r TT ff 1
2 4 2jrm=j \_m + l m-lj
11 " cos mi
2 4 „=2(m-l)(m + l)
The required series is therefore
11 1 o 1 o 1
- +-C0SZ- ,-7, cos2j — rr — , cos3z-- — ^ cosiz-....
24 1.3 2.4 3.5
It will be obscr\'ed that it is only for values of z between 0 and n that the sum of this scries is proved to be -^— ' sin z ; thus for instance when z has a value between 0 and — n,
the sum of the series is not ^ sin z, but — sin z ; when z has a value between n
and 2n-, the sum of the series happens to be again sin z, but this must be regarded
as a mere coincidence arising from the special function considered, and not from the general theorem.
I
JSxample 2. To expand g in a sine series, valid when 0 < z < tt.
We have
Trz(jr-2) 2 - . ["ntin-t) . .,
— -^ ' = — 2 Bin mz I — ^-^i; smmtdt
= 2 sm 77IZ I , sm mtdt.
m=l ./ 0 4
But I -^— — -'sin OTttft= V- ' +7—/ (n-2t) cos mtdt
y 0 4 Lo 4 OT J 4m y 0 '
f' (it - 2t) ain mt~\ 1 /"» . = L^-^-J-*-2^'./o'""'"""
_l-(-l)'" 2m' ■ Therefore
irz(ir — z) . sin 32 sin5z
140 THE PUOCESSES OF ANALYSIS. [CHAP. VII.
Here again the .sum of the Herie.s ia - only when z hcs between 0 and jr. Thus when z lies between tt and 27r, the sum of the series is '-— = . The sum of the
o
series for values of z beyond the limits 0 and n can be found at once from the equations
S{z)—-S{-z) and S{,z + 'in) = S{z), where S{z) denotes the sum of the series.
Example 3. Prove that, when 0 < z < tt,
TT (rr-22)(7r'* + 27r2-22''^) COS 32 cos 52
^ 96 ='=°^^+ 3^ + 5-*- + --
For when 0 < z < tt we have
7r(7r-22)(7r2 + 27r2-222) 2 - /"-r tt (tt -2<) (7r2 + 2ff<-2<2) ,
96 TT m=i .' 0 96
(integrating by parts) = 2 cos 7)iz I — j — sin mtdt
(integrating by parts) = 2 cos mz I — — 5- cos mtdt m=i J 0 4ot-
(integrating by parts) = 2 cos m2 / - — ain mtdt m=i J 0 2?H
" l-(-l)'"
= 2 ;^ — j-i- C0S»i2
m=i 2m«
cos 32 cos f)z
= C0S2+-^- + -^,-^ + ....
Example 4. Shew that for values of 2 between 0 and n, e" can be expanded in the cosine series
2s,,, ,n/1 cos 22 cos 42 \ 2s,, _ /cosz 00382 \
and draw graphs of the function e" and of the sum of the series.
fr (tt — 22) Example 5. Shew that for values of 2 between 0 and n, the function can
be expanded in the cosine series
cos 32 cos 52 C0S2 + -^+ ,., +...,
and draw graphs of the function "-—, ■ and of the sum of the series.
83. A Itemative proof of Fourier's theorem.
Another proof of Fourier's theorem, based on an entirely different set of ideas, is due to Dirichlet*.
* Collected Works, Vol. i. pp. 133—160.
83] FOURIER SERIES. 141
Consider first the sum of a limited number of terms of the series a^+ 2 [amCoanu+bmSimm],
IR = I
where
1 fi"
^n J 0
1 /■«"• am = - I f {t) COS mtdt ()H = 1, 2, 3, ...),
TT J 0 '
b„ = - \ f{t) am mtdt ()n = l, 2, 3, ...),
TT jo
imd where z is supposed to be a real variable.
Since
a„, cosn!2 + 6,„sin ww = - I f(^t) cos m{t — z)dt, TT .1 II •
we have the sum to (2to + 1) terms of the series expressed by the formula
1 f-"
S„ = - I {^ + coa{t-z)+coa2 {t- z) + ...+coam{t- z)}f{t)dt
/■2T8in(2»J + l)^^ Jo sin —
■^ J z am a
~2
= i r"'^^in(2^ + l)^ ^^
TT _/ 0 Sin ^
We have therefore to inve.stigate the value to which integrals of this class tend as m tends to infinity. Consider in general the value to which
/= I ; q>(z)dz
J 0 amz
tends when I; supposed to be an odd integer, increases without limit.
First suppose 0 < A <- , and suppose that, for values of z within this range, (p (z) is continuous and positive, and that <f) (z) continually decreases as z increases.
Let -T- be the greatest multiple of 7 in A, so -7^ <^<r+l t .
142 THE PROCESSES OF ANALYSIS. [CHAP. VII.
Then
IT Stt (rt+l))r rrt
, f' f' f k [T [h smkz
i= + +...+ +...+ + ~<i,{z)dz.
J 0 J " J 7Vr J (r-l)ir j rit HIU Z ^ ^ '
k k k T
(n + l).r
/ , s„ r * sin ^2 , , , ,
Now write
(n + l).r
r * .sin hi , [mr \ ,
k
TlTT
where y = z- , .
The integrand in this last integral is clearly positive throughout the range of integration, and u„ is therefore po.sitive. Moreover, under the suppositions already stated, the quantity
8i"(^+y)
decreases as n increases, and it therefore follows that ?i„ decreases as n increases.
Also the well-known theorem of Mean Value shews that u„ can be represented in the form
, A sin hi ,
where .„= ' .dy,
and p„ = (t>r^ + 6j,
6 being some quantity between 0 and t. Clearly v^ is positive, and decreases as n
increases.
Now we can write
_ - /"* sin kz , , ^ , ^=-'+J^sin7*(^)'^^'
k
where ./=m„ — Mj + Mj- M3 + ...+( -l)''"'^,-! •
Since u„ is always positive, and decrea.ses as n increases, we have
J<lC^-u^ + u.,-...■\■l^,^,
r-1
where m is any number less than —^ .
This gives
J < I'oPo - "iPl + ^".P-1 - ••• +''2mp2m
< "OPO - ("1 - "2) P2 - ("3 - "4) P4 - ••• - (''2..1- 1 - "2"!) P2>n
- "1 (Pl - P2) - ".■) (Ps - Pi) - •• • - "am - I (P2m-1 - P2ni)-
83] FOURIER SERIES. 143
As p„ decrcsjuscs with iiicrcjuse of it, the terms in the hist lino are negative, and axn be removed without aftecting the inequality.
Thius
•'< "OPO - ("l - ''a) P2 - ("3 - ""l) P4 - • • • - ("ii.. - 1 - "j"!) P2.H < "oPo - ("l - "2) Plfn - ("3 - "*) Pirn---- ("'•.i - l - >'2.n) Pjm
-("1 - "•>) {pi- Pirn) -{"3- "a) {pi-P'im)---- - ("-an-j- "atn-a) (P2...-2 -p-.'m).
The terms in the hust line are again negative and can bo removed. Thas
•^<''o(Po-P»fn) + (l'o-''l + l'2-... + f.,.m)P2.n-
AVe also have clearly
J> Uo-Ui + V,- ... - U^m.l,
or ./>i'opo->'iPi + >'..p>— •••->' .mi + i Piin+:,
which in the same way gives
•^>P2m(>'o-"l + ''2-"'->'2m..l)- .
Thus J is intermediate in value between the quantities
"0 (Po - Psm) + (>'o- "1 + "o - ... + fjm) Plm and P2m(>'o-''l + ''2--"-''2m+l)-
Now let i: tecome infinitely great, and let the quantity vi likewise become infinitely great, but in such a way that , tends to the limit zero. Then the quantities p,) and p.„„ tend to the limit <f> (0) ; and the quantity
"0 - "1 + "2 -••■ + ■' -im (2m + l)ir
* sin iy
I.
0 smy
in t
dy
dt
k sin T k
can, since k is infinitely large compared with m, be replaced by
f(2m+l)irsin t
1:
' dt\
t
and this, when m becomes infinitely great, tends to the limit - .
We see therefore that J is intermediate in value between two quantities, each of which
tends to the same limit, namely - (f> (0). J therefore tends to the limit ^ <^ (0) ; and
therefore /, which differs from ./ only l>y a vanishing integral, likewise tends to the limit
5 ^ (0) as it becomes infinitely great. This result may be called Dirichlet's lemma. To
complete the lemma, however, it will l)e neces.sary to shew that it is still true when a number of the restrictions imposed on (p (z) are removed.
(1) It was a.ssimied that (l>(z) was positive and steadily decreasing throughout the range.
(a) Suppo,se that (f> (z) is constjint. This does not invalidate any of the preceding proof, ao the theorem still holds if <^ (z) is constant.
144 THE PROCESSES OF ANALYSIS. [CHAP. VII.
O) Siippase that <^ («) is negative, or partly positive and partly negative, but still steadily decreasing; then choo.se a constant c so that c + (f>(z) is positive through the range; then the theorem applies Vioth to c and to c + <f>(z) and therefore on subtraction to (f>(z) alone.
(y) Suppose that 0 (z) increases steadily throughout the range. Then the theorem is true for { - (f> (z)} and therefore for <f> (2).
Therefore the theorem is still true if <f> (z) is finite, continuous, and steadily increases or decreases throughout the range.
(2) Instead of taking the integral between 0 and h, take it between g and h, where 0 <g <k^^. We assume that the value of ^(2) is only known for values of z from
g to /(.
Take a new function <pi (2), defined as being equal to cji (g), a constant, for values of 2 from 0 to g, and equal to (f> (z) for values of 2 from g to h. Then the theorem holds for (^j (2).
Also
^i?i* 11 l>t' "^^ ^'^ ^' = 1 -^i (0) = i <^ (^)'
and
Therefore
by subtraction.
Limit I ^'^-^ 0, (2) c?2 =1 01 (0)=| 0 {g). *=« J 0 sin i ^ /,
T . ., /"''sinifci , , , , Lmnt I ^ — 0 (2) dz=Q, t=„ y (, sm 2 ^ ' '
(3) Now assume there are a limited number n of maxima and minima within the range 0 to h.
Let them be at the values Oj, a^, ... a„, of z. Then
fh fa, fa, fh tiinh , , , ,
jo jo J a, j a„ sm 2 ^
On applying the theorem to each of these integrals in succession, it is clear that the theorem holds for the whole integral.
Therefore the theorem is still true if 0 (2) is finite, continuous, and has not more than a limited number of maxima and minima within the range.
It must l>e noted that these conditions still exclude such functions as e.g. (2 - c) sin
2 — c
where 0<c<h.
(4) AVe shall now no longer restrict A to be less than - . Take Q < h^ir.
Then(a)let|<A<7r.
Write /i = TT - h', where 0 < A' < ^ . Then
Limit/= —. (b(z)dz+l --. <b{z)dz.
A:_« / ,1 sm 2 ^ ' ' j sm 2 ^ ' '
83] FOURIER SERIES. 145
Writing : = n -■ ( in the liitter integnil, we have
IT IT
Limit /= / -r- <b{:)d:+l . ^(b(iT-C)df.
' Since <^ (n- - f ) .'wvtisfies the conditions stated, we see that wlicii h' > 0 the second integral
is zero.
Therefore Limit /= - 0 (0).
(^) Let /i = ir. Then nil the iibove reasoning apphe.s, except that now A' = 0, so
Liniit/=5 0(O)+^0(7r),
which, in order to guard against uncei'tainty in the case in which the function <f) is discontinuous at 0 and n, is often written
where < is a vanishing positive (juantity.
(5) Next, suppose that the function <f> {z) within the range has a finite number of dis- continuities, in the form of abrupt but finite changes of vahie. Divide the range into various jmrtions, so that e;ich of them ends at one discontinuity and begins at the next, and divide each of these into others each beginning and ending at a point of stationary vahic. The above theorems apply to each of the portions, and therefore each integral is
zero except the tii'st, which is equal to - <f) (f), and possibly the last, which when /i = n has the value ^ <^ (tt - «).
(6) Finally, consider a function <f>(2) which becomes infinite for z = c, but in such a way that the value of I (p (z) dz tends to a definite limit as z approaches c from either lower or greater values.
Then
Cc-e fc fci-t /■/» sin h , , ^ ,
1 where € is a small positive quantity.
In the second integral, a quantity f can be chosen intermediate between c and c- f, such
that the integral is equal to ' W <f) (z) dz ; on taking e small this vanishes ; and
sni f J c-€
.similarly the third integral is zero.
On making it infinitely large, the fourth integi-al tends to zero. Therefore the theorem holds in this case also.
(7) Thus we have, summarising the results obtained, the theorem that the limit when k tends to infinity of \ -.-— <l> (z) dz is - <f> (t) if 0 < h < n, <uid is
J 0 Sill Z A
|{<^( + 0 + <^(^-0}
W. A. 10
14C THE PROCESSES OF ANALYSIS. [CHAP. VII.
if h = iT ; where ( is a vanishing positive quantity; provided that <^(i) is everywhere finite, and hot only a limited number of finite discontinuities and maxima and minima between the vahies 0 and h of t/te variable z; and this is still true if <^{z) has a limited number of
tingularities of specified type, namely such that I <f> {z) dz is finite.
Thin res\ilt may bo called Krichlefs lemma, the CDiKlitioiis jiLst stated being referred to as DirichleCs conditions.
We can now return to the o.\paii.sion which wa.s found to repre.sent the sum of the first (2m + 1) terms of the Fourier series.
We had S,„ = I^ + L,
Z
where I,= -j^ .__^„_L/(, + 2^)rf5,
IT J 0 sm 5 -^ ^
If 0<£;<27r, and/(2) satisfies Dirichlet's conditions, we have by Dirichlet's lemma Limit 7i = i/(j + €),
and
Limit I.i = ^f{,z-f),
and so
Limit ,?,„ = ! {/(.- + f)+/(2-f)}.
If i = 0, we have
Limit /, = i (/(f)+/(27r-e)j, Limit /2 = 0,
and so
Limit 5m = i (/(f) +/(27r -f)}.
m=oo
If £ = 2jr, we have
Limit /i = 0, Limit /o=J{/(f)+/(27r-0},
and so
Limit 5„. = i{/(e)+/(2,r-f)}.
Thus we finally arrive at Fourier's theorem, namely that the sum to infinity of the series
CO
aQ+ 2 (a^cosmz + b^ammz)
is f{z) at points z for which f is contijmous, and is the arithmetic mean of the two values of f(z) at points z for which f is discontinuous : it being assumed that f{z) satisfies Dirichlet's conditions.
Example. Prove that in the limit when n becomes infinitely great
"»sin(2« + l)e _„.,. . ,, , r 0 sni 6 z i >
a being a real positive constant.
(Cambridge Mathematical Tripos, Part II., 1894.)
/:
84] FOUUIER SEUIES. 147
84. Nature of the convergence of a Fourier series.
The proofs of Fourier's theorem which have been given establish the result only for the ease in which the sequence of the terms in the series
S (a,„ cos mz + b,,, sin mz)
is that in which m takes the ordeily succession of values 1, 2, 3, 4, ... .
The question now arises whether the order of succession of the terms can be deranged without ati'ecting the value of the sum of the series ; in other words, we have proved that the expansion of a function by Fourier's theorem is a convergent series : we want to find whether it is absolutelij convergent, or only semi-convergent. The question has also to be considered whether the series is uniformly convergent or non-uniformly convergent in the neighbour- hood of a given value ol z.
We shall first shew by considering special cases that there is no general answer to these questions.
Consider the series
sin a — - sin 2^ 4- 5 sin ds — . . . ,
which represents - z when 0 <z <tt, and ^z — tt when tt < 2 < 27r ; this series
is semi-convergent for all real values of z, since sin nz is finite for all values of n when z is real, and so the modulus of the general term bears a finite ratio to the general term of the divergent series
1 1 1 1+2 + 3 + 4 + -"-
In this series, therefore, the value of the sum will be modified if the order of
succession of the terms is changed.
Moreover, we can shew that the series is non-uniformly convergent at its
discontinuity tt. For the sum of the first ;; terms is
sin 2z (— 1)"~' sin nz siu, ____ + ...+ ^ ,
or
(cos < — cos 2< -(- . . . -h (- 1 )"""' cos nt) dt,
1:
'0 or
1:
1 (- 1 )"-' cos {n+l)t + cos nt
dt.
2 2 1 -f cos «
The term I - dz represents the sum of the whole series ; so the remainder after n terms, when — tt < 2 < tt, is
Jt„ = (-l)
cos(n+ ] t
T r
2 cos 2
dt.
10—2
148 THE PROCESSES OF ANALYSIS. [CHAP. VII.
Writing z = 7r—r), t = -7r — u, this can be written
" sin ( ?i +
Rn = —
2 sin ^
■du.
Write (n+^\u = v. The equation becomes
f(»+i)'
Rn = —
Sin V
-dv.
However great n may be taken, if 77 be taken so small that (n + - ) 7/ is
infinitesimal, this integral tends to — I or — - , and so is not infini-
tesimal. It follows that the series is non-uniformly convergent in the vicinity
of ^ = TT.
Consider next the series
1 o 1 r
cos 2 + ^ COS 6Z + —COSOZ + ... ,
O O
I,- 1 i "■ (tt - 2^) , „ j7r(22-37r) ,
which represents — ^^—5 when 0 < 2 < tt, and ' when tt < ^ < 27r.
o o
This series is absolutely convergent for all real values of 2, since the moduli
of its terms are less than the corresponding terms of the convergent series
l + 3, + 5,+ ....
In this series therefore the order of succession of the terms can be changed in any way, without altering the value of the sum of the series ; and since the comparison series is independent of z, the series is also uniformly convergent for all real values of z.
Returning now to the general Fourier series, we can discover the nature of the convergence by a consideration of the coefficients in the series, which can be made in the following way.
We have shewn that if
f{z) = ao+ S {cim cos mz + h,n sin mz),
m—l
then
1 ["" am = - ( f(t) COS mtdt.
84]
KOl'KIEK SEKIES.
149
Suppose that (as in most of the examples we have discussed) the range 0<2<27r can be divided into other ranges, say 0<2<Ar,, k-i<z<k„, ... , k„ < 2 < Stt, which are such that in each of these smaller ranges f(z) is an analytic function of z, regular in the range, {/(z) will not necessarily be the same analytic function in the different ranges.) Thus i{ f(z) has the value ^ for 0 < z < TT, and has the value — z for t < z < 27r, we should have n = 1 and k\ -• tt. Then
1 r*' 1 /■*-•
«.n= t {t) COS mtdt -¥" \ f (t) cos mtdt+ .
71".' II • TT.'x,"
1 r^"
i /(OCOS
TT./ i-.
-^J K
mtdt.
Each of these integrals can then be integrated by parts ; we thus obtain
*' 1 , , , sin i)it
^ J V') 0 TT" in
+
*» 1 . , . sin mt
+ ...
1 /■*■' 1 r*'
■ — I /' (t) sin 7ntdt f ' (t) sin mtdt— ... ,
or
in m
where
A = ^ [sin mk; {/{k; - 0) -/(k; + O)' + sin mk, {/{k, - 0) -f{k, + 0)} + . . .],
and where b,„' is the coefKcient of sin mz in the Fburier expansion of/' (z) — an expansion which will exi.st, since/' (z) is a function of the same character as/(z), though the terms of this expansion will not always be the derivates of the corresponding terras of the Fourier series for/(^).
Similarly
m VI
where
B = -- [-/(+ 0) + cos mk, {f{k\ - 0) -f{k, + 0}] + cos ink:, [J {k, - 0)
-/(^•, + 0)1 + ...+y(27r -())], and where a,„' is the coefficient of cos mz in the Fourier expansion of/' (z).
In the same wav we have
'" " m m
where
A' = - [sin mki l/'(/:,-U)-/'(A-, + 0)} +sinjnA-s{/'(i-,-0)-/'(A,+0)j + ...].
160 THE PROCESSES OF ANALYSIS. [cHAP. VII.
and
m m where
B' = -- [-f'(+0) + co^mk {/'{k, - 0) -/' (A + 0)1 + ... +/'(27r -0)], 7r
Om" and bm" being the coefficients of cos mz and sin mz respectively in the Fourier expansion off" (z).
Thus
_A B^_a^
, B A' 6,„" «i III' m'
The conditions for the absolute convergence of the Fourier expansion of f{z) are therefore expressed by the equations
^ = 0, 5 = 0;
for if these equations are satisfied, we have
B' + arn" „„ , , A' - bm"
^m = i — and b,n = -„ — ,
and the terms of the Fourier series are comparable with those of the con- vergent series
1 111
l + 2-= + 3i + 4^ + --
Now in order that we may have ^ = 0, B = 0, for all values of m we must have
f{h-Q)=f(k, + Q),
/(A-,-0)=/(A:, + 0),
f{K-0)=f{kn + 0), /(27r-0)=/(0).
That is to say, if a Fourier series is absolutely convergent for all real values of z, the function represented by the series has no discontinuities, and has the same value at z = 0 as at z = 27r.
If these conditions are satisfied the Fourier series is not only absolutely, but is also uniformly convergent. For its coefficients a^ and 6,„ are in this
case of the order — - , and so the series of constants m'
Iflol + \a,\ + \b,\ + \a,\ + \b,\+...
85] KOURIEH SERIES. lol
converges ; but the moduli of the terms of the Fourier series are less than the corresponding terms of this series, and consequently the Fourier series is uniformly convergent for all real values of ^.
Kvample 1. Shew that in general, when tlie Fourier series converges only for real values of z, the quantities «,„ and /j,„ can he expanded in infinite series of the form
m of which the terms
C| t'o Co c,
/H MJ- my m*
A ff , li A'
.1 and - + „
m m- M m-
found above are the initial terms ; but that when the Fourier series converges within a belt of finite breadth in the z-plane, all the coefficients Cj, c.^, c^, ... vanish, and this expansion liecomes illusory.
E.xample 2. Let f{z) Ix; a function of z, which is regular for all real values of z between z = 0 and 2 = >r, and which is zero at r = 0 and z = jr. Prove that if /(z) is ei]iande<l in a sine scries, valid >ietween r = 0 and z = tt, the series will be absolutely and uniformly convergent for all real values of z.
Example 3. f (z) is a function of z which is regular for all real valvies of z between 0 and jr. Prove that if it is expanded in a cosine series, valid between 2 = 0 and z = w, the series will be absolutely and uniformly convergent for all real values of z.
85. Determination of points of discontinuity.
The expressions for n,„ and b,„ which have been found in the last paragraph can be applied to determine the points at which the sum of a given Fourier series is discontinuous. This can best be shewn by an e.xample.
Example. Let it be required to determine the places at which tlie sum of the series
sin 2+ J sin 3c+^sin5i + .,, is discontinuous.
For this series we have
, _ 1 - cos mn "" 2m •
Comparing this with the formula found in the last paragraph, we have
vl=0, B = i - ^cos Hiw,
A' = B' = a„" = l)„," = 0.
Hence if /■,, /j ... are the places at which the analytic character of the svun is broken, we have
O = A = l[^mnd-ilfii-^-O)-f(t^+O)\ + smnd:,{f{l-^-0)-f {1:^ + 0)] + ...]. Since this is tnie for all values of ?/j, the quantitie?^ /-,, /(■.^, ... must be multiplea-oLu. bxit
152 THE PROCESSES OF ANALYSIS. [CHAR VII.
there i.s only one multiple of n in the range 0<z<2jr, namely n itself. So /■i = 7r, and ^■31 ^'31 ••• do not exist. Substituting /■, = 7r in the equation Jl = i-^coHmn, we have
i-iC0SW(7r=---[- /( + 0) + COS/;i7r [/(7r-0)-/(,r + 0)}+/(27r-0)].
Since this is true for all values of m, we luivo
i=-i{/(2^-0)-/( + 0)!,
and _i=_i//-(^_0)-/-(ff4.0)!.
7r
This shews that there is a discontinuity at the point z = 7r, such that
and that
.f{^-0)-f{n + 0) = -,
/(2^_0)-/( + 0)=--.
Example. Find the discontinuities in value of the sum of the series
sin 0 - ^ sin 2j + J sin 4j - 1 sin 5z + 1 sin 72 - J sin 81: + ^ sin IOj + . . . .
86. The uniqueness of the Fourier expansion.
We have seen that if / {z) is a quantity depending on z, and satisfying certain conditions as to finiteness, etc., then the series
tto + S (a„i cos im -I- i,„ sin mz),
m=l
1 r-"" where o,„ = - f{t) cos nd dt (m > 1),
"^J I) '
bm = — j f{t) ^in mtdt,
TT./o ■
«o = ^ /(0f'^
has the sum/(0) when 0 •$ 2 $ 27r, except at the isolated points at which /(0) is discontinuous.
The question arises whether any other expansion Co + 2 (c„, cos mz + fZ,„ sin mz)
m = \
of the same form exists, which also represents f{z) in the interval from 0 to 27r; in other words, whether the Fourier expansion is unique.
We may observe that it is certainly possible to have other trigonometrical expansions
of (say) the form
=» / mz „ mz\
0^+2 ( a„,cosy + /3,„cos -
I
I
stl] FOURIER SERIES. 153
which represent /(«) between 0 and -In; for write 2 = 2f, ami consider a function <t){(), uliich is Niich that </>(f) = /(2f) when 0<f<7r, and 0(f) = y(f) when ir<(<-2n, where g{() is any other function. Then on expanding <^(f) in a Fourier expansion of the form
ao+ 2 (o„,cosmf + /3„, cosmf), m=0
this expansion represents /(e) when 0<z<2n ; and clearly by choosing the function g {() m difl'ercnt ways an infinite number of such e.xpansions can be obtained.
The question now at is.sue is, whether other series proceeding in sines and cosines of i'ti-gnd nudtiplcs of z e.\ist, which difl'er from Fourier's expansion and yet represent f{z) li'twecn 0 and 2jr.
If it were possible to have a distinct expansion
/ (^) = c„ + S (c,„ cos mz + d,n sin niz),
m = l
then on subtracting this from the Fourier expansion we should have an expansion
(oo - Co) + i {(a„, - Cm) cos mz + (&„, - dm) sin 7nz}
m = l
whose sum is zero for all values of z between 0 and 27r, except possibly a certain finite number of values (namely the discontinuities).
The investigation therefore turns on the question whether it is possible for such an expansion as this last to exist. We shall shew that it cannot exist, and that consequently the Fourier expansion is unique*.
Let Ai, = -^a„,
•4,n = «m cos mz + h,n sin mz (m > 1) ; and let
be a convergent (not necessarily absolutely convergent) series for values of z from 0 to 27r, so that the limit of a„ and b„ is zero for n = oo ; and suppose that (except at certain exceptional points) its sum is zero.
Then the series
F(z) = A/{-A.-^-...-^_-...
converges absolutely and uniformly for this range of values of z, as is seen by comparing it mth the series S — ^ .
We shall fii-st establish a lemma duo to Riemann-f, which may be stated thus:
* The proof is due to G. Cantor, Journal Jiir Math. Lxxii. t Collected Worki, p. 213.
154 THE PROCESSES OF ANALYSIS. [CHAP. VII.
The quantity
„ _ F(z + 2oL)+F{z- la) - IF {z) ^ W
tends to the limit f{z) as a tends to zero, if at z the series 2 converges to the sum f(z).
For the term involving «„ in R is
— -T-zr , ("" <^os n (z + 2a) + «„ cos n (z — 2a) — 2a„ cos nz}, torn
a„ cos nz sin- na
J ..,,,, ^ . , . , . 6„ sin nz sin' na and similarly the term involving o„ is p^ .
As F(z) converges absolutely, we can rearrange the order of the terms, and so can write
n . . /sinaV < /sin 2a\-
Now considering the series 2, we can write
Ao + A, + A, + ... + An-, =f(z) + €n,
say, where z being given, and any small quantity 8 being assigned at will, we shall have | e„ [ < 8 for values of n ^ some integer m.
Now An = €n+i — e„ for all values of n. Therefore substituting, we have
R=f(z)+Xen\^^^-^i'-\'''"H n=i II in-l)a ] [ na ]
Divide the series on the right-hand side of this equation into three parts, for which respectively
(1) l^n^7n,
IT
(2) 7rt + 1 :$ ?i < s, where s is the greatest integer in - ,
TT
a
(3) s+l^n.
The first part consists of a finite number of terms, each tending to zero as « tends to zero, so the first part is zero.
Considering next the second part, the quantities are of the
,. sin .« , „ , . . , . r.
lorm where 0 < a; < tt ; this quantity decreases as *• increases irom
0 to TT, so the sum of the moduli of the terms in the second part is less than
sin may , sin sa ^
mi j \ SOL )
which tends to zero when h tends to zero.
-
86] FOURIER SERIES. 155
Considering next the third part, we can write the nth term in the form
/sin n — 1 ay /sin n — 1 aVl 6„
(sin' n — la — sin' no),
or
sm' n-lof 1 11
a' In - 1 ■ «-J
sin 2n — la sin a
/i-'a=
so, as I 6„ I < S, its modulus is less than
a'\n-V
IN 8
Thus the whole sum of the terms in the third part
S 1 S
a' s" a
1 1
s+1' s + 2-
^ sV "*" 2 is x''^ s'a' "'" sa *^ (tt - a)= TT - a
which is ultimately zero. Therefore the three parts of the infinite series in R are all zero ; and thus R =f{z) in the limit ; which establishes Riemann's lemma.
Next, we shall establish another lemma, due to Schwartz *, which may be stated as follows : If a and b are two of the exceptional points, so that between z = a and z = b the series S converges to the sum zero, then F {z) is a linear Junction of z between these values.
For assume that o is less than b, and introduce a function <f> (z), defined by
F(z)-F{a)-l_''^{F{b)-Fia)\
- ^{2-a){b-z),
<f>iz)=0
where ^ = 1 and h is any constant.
Then substituting in the result of Riemann's lemma, we have Limit ■^(^ + «) + <^(^-«)-2<^(^) = h;
a = 0 «'
Therefore (f> (z + a) + (f> {z — a) — 2(f> [z) is positive when a is very small, whatever be the value of z.
Now <f>{a) = 0 and (f>{b) = 0. Also <l)(z) is continuous, since F{z) is uniformly convergent, and consequently continuous. Therefore if <f}{z) can be positive between the values a and b of z, it will have a maximum ; let this occur at the value c of z.
* Quoted by G. Cantor, Journal fiir Math. i,xxii.
156 THE PROCESSES OF ANALYSIS. [CHAP. VII.
Then when a i.s small, we have
(^ (c + a) — (^ (c) < 0, and <^ {c — <x) — <p {c) < 0.
Adding these relations, we see that the condition jii.st found is violated, and so <^ {z) can not be positive at all within the range.
Again, take h small. Choose ^= ± 1, so choosing the sign that the first term 6 \_F {z) — ...] is positive. Then (^ {z) is clearly positive, if this first term is not zero.
But ^ {z) is not positive ; and thus we must have
F{z)-F{a)-l-J'^\F{h)-F{a)\ = 0.
Therefore F {z) is a linear function of z, which establishes Schwartz's lemma.
We see then that the curve y = F{z) represents a series of straight lines, the beginning and end of each line corresponding to an exceptional point ; and as F (z), being uniformly convergent, is a continuous function of z, these lines must form parts of a polygon.
But by Riemann's lemma
Limit li^+^tllM _ Fi^JZ^zli^ = 0. a-o a -a
Now the first of these fractions gives the inclination of the earlier side of the polygon at a vertex and the second of the later ; therefore the two sides are continuous in direction, so the equation y = F (z) represents a single line. If then we write F(z) = cz + c', it follows that c and c' have the same values throughout the range. Thus
and therefore
A,^-A,-...---...=cz + c
A^ Jr-c^-c^=^, + ... + -^ + 2 n-
the right-hand side of this equation being periodic, with period 27r.
The left-hand side of this equation must therefore be periodic, with period 277. Thus we have
^„ = 0, c = 0,
and — c' = yl, -h ... -I- -^' cos nz H — ^ sin ?i2 -|- —
n- n^
Now the right-hand side of this equation converges uniformly, so we can
I
86] FOURIER SERIES. 157
multiply the equation by cos m, sin nz, respectively, and integrate. This gives
TT -" = - c' COS nzdz = 0,
b '■''''
and TT ^ = - c
I -Sin iizdz = I Jo
Therefore the a's and b's vanish, so all the coefficients in S vanish ; which establishes the result that the Fourier expansion is unique.
Miscellaneous Examples.
1. Obtain the expansions
\ — r cos 2 , o ^
l-2rcosj + r^
(^) Qlog(l — 2rcoss+r^)= -rcosz — - r^cos 2z -- »'^co8 3j — ...,
, . . . r sin £ 1 .. . rt 1 1 ■ rt
(c) tan~' , = rsia 2 + -r-sin2z+ = r'sin32 + ... ,
1 — r cos z 2 3 '
/ T^ i 1 2rsin2 . 1 , . „ 1 , . ,
(a) t<an~' — -„ =''sin2 + „ r'sin3z + r r° sin 50+... , 1 — r- o o
and shew that, when |r|<l, they are convergent for all values of z in certain belts parallel to the real axis in the 2-plane.
2. Shew that the series
n- 1 n ^2~~^
r- . 2lTi . (m — l)2jr2 , ,,2^2 . k . 2nZ
sni — sin ^ sin(?i+l) .sin
jj n a n
—I +... + + — ; +... + ; + ...
1 n-\ n+l k
where all the terms for which k is a multiple of n are omitted, represents the gi-eatest integer contained in z, for all real values of z between 0 and n.
3. Shew that the expansions
- log (2 cos „ ) = cos2 - - cos 2z + 5 cos 32... and
- log \ 2 sin s ) = - cos z-- cos 22 - - cos Zz ...
are valid for all real values of z, except multiples of rr.
4. Obtain the expansion
» i—\)™(Xismz , ^ s, ^„ z\ z , . „ . .
2 , - ,V7 rr=(cOS2 + COS22)log 2C0S r ) + -(sm22 + Sin2) — C0S2,
m=o(m + l)(m + i) °\ 2/ 2
and find the range of values of z for which it is applicable.
(Trinity College, 1898.)
158 THK PKOCESSES OF ANALYSIS. [CHAK VII.
5. Let n bo an integer^ 2, and let x\, .v,^, ... .i;„_i be quantities .satisfying the conditions
0<.i\<x.^<.,.<.t\^i<l,
and write .ro = 0, .r„=l.
Let Od, r,, c.,, ... c„_i be real arbitrary constants and let a function (p (.r) be defined by the equalitie.s
<t>{:v) = C0+c^ + ...+c„ {orx,<x<Xi + ^ (« = 0, 1, 2, ... n-1),
<j) (x) = Cq for x = Xfj,
<^(:r) = Co + c, + ... + c,_, + 2', for a- = a:, (.s = l, 2, ... n- 1),
<^(.i-) = Co4-c, + ... + c„-i, for .i- = .r„.
Shew that
</>(^) = "5+ 2 (a„cos 2mrrx + 6OT.sin 27njr.r), for 0<.r<l,
2 in=I
and
^(0) + <>(l)_a,
= ^+ 2 a„,
2 2 „=,
where the coefficients a„ and 6„ are given by
it-i ao = 2 S c,(l-^r),
r=0
1 n-l
am= 2 CrSinimirXr, formal,
WiTT r=o
6^= 2 c,.(l — cos2mnx^) for m > 1.
(Berger.)
6. Shew that between the values - jt and + tt of i the following expansions hold :
/ sin 2 2 sin 2z 3 sin 32 \
Vl2-m2 ~ 22 -m2 "^ 32-TO2 ~ •■■y '
(1 m cos 2 OT COS 22 m cos 32 \
2m"^12-m2~ 22 -m-'' "^ 32^^^" "V '
/I m cos 2 ni cos 22 to cos 32 N
2 .
smTO2=- sin mn- jr
cos m2 = - sin mw
,m« I rt— m*
e^' + e
gmi7 _ g -mir ^
7. Obtain the expansions
S sinfi(2 + TOjr) iB=-« 2 + OTjr
and
cos ^l(2 + TOn-)
' V ^ (2;K^<2« + 2)
^sin2n2Cot2 (;i = 27i)
cos (271 +1)2 /-, ^ ^ , o\"l
- -' (27!</i<n + 2)
2
.00 z + mn I
cos 2n2 cot 2 (/i = 2n)
=_oo 2 + TOjr I I
)j
illSC. EXS.] FOUKIEK SERIES. lot)
If p and q lire positive iiitogoi'M, show that
. Shtt
sin uim +/>) — n
2 i-=-sin^*^ cot^— ,
,„=-- </'"+/' 111
. inn cos (om + p) ,
2 ■^— — ^■^^^^^— = — cos cot — .
m=-« </m+j> q q q
8. Prove that the locus represented by
"=» (-1)"-' . 2 ^- — sin ?i.i' sni »iy = 0
is two systems of lines at right angles, dividing the coordinate plane into squares of area n--.
(Cambridge Mathematical Tripos, Part I., 1895.)
9. If 7(1 is an integer, shew that
* , 1.3.5...(2w-l) fl , m „ , in{m-\)
2. 4. 6. ..2m (2 m + 1 (m + l)(m + 2;
7n(m-l)(OT-2) „ 1
_i ^ LS '- cos fi' 4- V
^ (m + 1) (7)1 + 2) (771 + 3) ^••■J
(a terminating series).
n- 1.3.5. ..(277»- (an infinite series).
Shew also that
1) 12 + 2m + l ''"^ ^"+(2771 + 1) (2m + 3) °°^^ ^ -]
and
4 / cos 3i cos 52 cos "iz cos 9^ \
8 /cos 2 COS 32 cos 52 COS Iz cos 92 \
'^^Vr73 "''1.3.5 ~3r5T7'*' 5.7.9 "■ 7.9. 11 + — j '
10. A point moves in a straight line with a velocity which is initially », and which receives cousUint increments, each equal to u, at equal intervals t. ^Prove that the velocity at any time t after the beginning of the motion is
u ut M °° 1 . 27n7r< _- + -+- 2 - sm ,
Z T 7r ,;, = ! 771 T
and that the distance traversed is
ut , ^ Ut Ut " I Zmnt
--(< + r)+ -- ; 2 -jCOS .
2r 12 2n' „_i m^ r
^4V
11. Shew that
. , „ , .sinxir " (-l)''8in (a + 2nv7r) sin(a + 2i«-7r)= 2 ^ ■ ^^^ ■',
where u is the difl'erence between the real quantity v (supposed not to be an odd multiple of i) and the integer to which v is most nearly equal.
(Cambridge Mathematical Tripos, Part II., 1895.)
I(i0 THE PKOCKSSES OF ANALYSIS. [CUAP. VII.
12. Let II be an integer ^ 3, and let g^, gi, g^, ... be an infinite set of quantities, which satisfy the conditions,
^r + n=£'r. for '•>0-
Let X be a real variable, and let * be the greatest integer contained in nx. Shew that when x > 0,
^ 9r- n "^ " (a,„cos 2mn-.r + 6„. .sin 2m7rx),
r=0 * m = l
if r is not a multiple of - ;
but
' * 0 Gin *
^ ffr-^ = -^+ 2 (o,„ COS 2mnx + 6„ sin 2»iir.v), r=0 -^ -^ m=l
if )• is a multiple of - ; the coefficients a„ and b„ being determined by the formulae
2 "-1
" r=l
1 «-> . 2TOr7r , ^,,
^ T/iTT ,-=1 Ti
1 «— 1 2ftlTTT
6m= — 2 O'rcos (»»^1). (Berger.)
niTT ,-=1 n. \ o /
13. Let X be a real variable between 0 and 1, and let n be an integer ^ 5, of the form im + 1 , where »i is an integer.
Let E (a) denote the greatest integer contained in a.
Shew that
, ,N-s("^), , ^A"^) 2 2 » 1 2m,r „ (-1) V-/ + ( — 1) V -i / = - + - 2 -tan cos2m7ra:,
if .r is not a multiple of - ; n
but
. rnrx cosnnx 2 2*1. 2mir
sm „ + — „ — = — + — 2 — tan cos 2imrx,
2. z n n „,=i m n
if X is a multiple of - . (Berger.)
14. Let X be a real variable Ijetweeu 0 and 1, and let n be an odd number > 3. Shew that
, -.- 1 2 " 1 ^ mir ( — 1)"=- + — 2 —tan — cos 2mjrx, n n ,„=! m n
if X is not a multiple of - , where s i.s the greatest integer contained in nx; but
MISC. EXS.] KOUIUEU SEKIES. 161
/^ 1 , 2 " 1 ^ BUT
0 = -+- 2 -tan cosZmjrX,
n n ,„=i m n
itx is a multiple of - . (Berger.)
I 15. Let X denote a real variable between 0 ami 1, aiiJ let ii be an integer > 3 ; further,
let E(a) be the greatest integer contained in a. Shew tliat
E(nx){Elnx) + \-n} (n-l)(«-2) 1 - 1 ^ ottt
—5^ — '-'^-^ — ^^ '=- ^ '■ + - 2 -cot — cosZmirx,
n 071 rr m=i m n
if X is not a multiple of - ; but
n.v' - juf + „ = - j-^ -' + - 2 - cot cos imirx,
2 6« TT ,„_i m n
j if X is a multiple of — . (Berger.)
16. Assuming the possibility of expanding f{x) in a scries of the form 2Ji. sin/u-, where i: is a root of the equation ^-cosa/' + isin aX- = 0, and the summation is extended to all positive roots of this equation, determine the constants A^.
'• (Cambridge Mathematical Tripos, Part I., 1898.)
17. If
shew that
18. If |
shew that |
If |
shew that |
a«^ _ " g" V„ (x) e«- 1 ~ n=o n\ '
cos in-X cos GnX , , . ,22" ^n-^" „ , ,
cos2,r.r+— ^^,^+-p^ +- = ^-^) 2^rT~ ^2"^-''^'
. , sin 47r.c sin Gtt^ , ,. .,2-''jr2" + ' ,. , .
(Cambridge Mathematical Tripos, Part II., 1896.)
/(j;) = i«Q + a, cos.r + a2Cos2.c+... , u„=— I /(.r;cos «j;tani.i- -.
TT _/ 0 - X
0(x) = 6, sin .r + 62sin2.F+... ,
6„= - j <f>{x) sin 7ix tan ^x — . (Beau.)
n J 0 X
19. Prove that the .series 2.1„sin " ,
1 "■
where -^n= - / sin -^/(t>) rf»,
a J 0 fit
is equal to/(A-) for any value of x lying between 0 and a about which f{x) is continuous.
W. A. " 11
162 THE PHOCESSES OF ANALYSIS. [CHAP. VII.
If /(O), f{a) ai'c the limits of/((), f{a—f), when the positive quantity f diminishes to zero, and if'/(.''') lias sudden increases of value /;, /.•, corresponding to the values a, li, ... of x, the limit for n = cc of nA„ can be written in the form
-i/(0) -(-!)"/(«) + /' cos +/{:cos- '^ + ...L ■
Shew that the series
sin3a."+= sin 9a:+-sin 15.)'+... - 2 ( .sin.j,- + - sin 3j; + -siu 5j' + ... ) tJ 5 \ 6 o I
3n/3 f . 1 . , 1 . „ 1 . , , " 1
H <sina;- rr, sm 5.r + — , .sm Ix— ,-rnsm 11.); + ...^
■n \ 0^ 7" 11- J
has the limit -jn- when j', lying between 0 and tt, approaches indefinitely near to one or other value, and that it has sudden changes of value -in and + i tt corresponding to the values J TT and 5 n of x.
(Cambridge Mathematical Tripos, Part I., 1893.)
20. If, for all real values of x,
F{x) = A^ + A^ cos,f + ^2Cos 2X + A3COS3X + ... ,
then
a) f^ cos (g^) F{x)dx = {U+ V) f "'''^^^
2
r(4n + l) /,.2\ /-,•
ii) J^ cos[j-J F{x)dx = j^ A' F(x)dx,
where
U=Aff + Aj^ cos w + JjCos 4k' + ^13Cos9m'4- ... ,
V = Ai .sin ic + vlj sin 4w + A^ sin 9w+ ... ,
a:2 „ 47r2+.r2 tt.^ „ 4(27r)2+A-2 27r.r
J: = cos - — 1-2 cos — cos hacos ^ — r cos (-...
4w 4^t) w ixo w
4(2«7r)2 + .r2 'innx
+ 2 cos ^ — -^ cos .
Aw w
Prove these formulae, and thence deduce the result
^^''^ ^"^ (|^V = ^i^(0)-(-F(!(;)cos^-'^"'-|-F(2«.)cos^'j+...
+ -^F{0) COS -^^ -F(to) cos (^- + — j + /"(^tO cos l^-+ ^ j - - ,
27r where 'U) = -r , k being a positive integer. When k is even, the last term of each series
involves F(hkv]) and is to be multiplied by i ; when k is uneven, the last term involves
F{i{k-\)w}.
(Cambridge Mathematical Tripos, Part II., 1896.)
CHAPTER VIII.
Asymptotic Expansions.
87. Simple example of an asymptotic expansion. Consider the function
where .r is real and positive, and the path of integration is the real axis in the /-plane.
Integrating by parts, we have and by repeated integration by jJarts, we obtain
hi connexion with the function /(.'), we therefore consider the series
11 2: (-l)"-'(ji-l)!
1 ... -t-^^ ^^ + ....
X X- ar* a'"
We shall denote this series by S, and shall write
1_1 ,2^^_ , (-l)"n!_o X a? of ■■•"^ *■»+'
n-1
The ratio of the »ith term <jf the series S to the (n — l)th term is ;
X
for values of n greater than I +x, this is greater than unity. The sei'ies S is therefore divergent for all values of x. In spite of this, however, the series can under certain circumstances be used for the calculation of /(«); this can be seen in the following way.
11—2
164 THE PROCESSES OF ANALYSIS. [CHAP. Vlir.
Take any definite value for the number n, and calculate the value of »S„. We have
and therefore
< (n + 1)! I — ^„ , since e*~' < 1 and t is positive, I
,n+\ ■
nl
X'
For values of x which are sufficiently large, the right-hand member of this equation is very small. Thus if we take x> In, we have
which for large values of n is very small. It follows therefore that the value- of the function f(x) can be calculated ivith great accuracij fur large values of X, hy taking the sum of a finite number of terms of the series 8.
The series is on this account said to be an asymptotic expansion of the function f{x). The precise definition of an asymptotic expansion will now be given.
88. Definition of an asymptotic expansion. A divergent series
^0 + — +—-+...+ 7p +■...,
in which the sum of the (« + 1) first terms is S„, is said to be an asymptotic expansion of a function f{x), if the e.xpression x'^ \f(x) — <S'„1 tends to zero as *' (supposed for the present to be real and positive) increases indefinitely. When this is the case, if x is sufficiently great, we have
where e is very small ; and the error — committed in taking for f(x) the
(n + l) first terms of the series is very small. This error is in fact infini- tesimal compared with the error committed in taking for f(x) the n first terms of the series : for this latter error is
a," '
and 6 is in general infinitely small compared with A,^ + e.
NS, Ml] ASYMPTOTIC EXPANSIONS. 1G5
The definition which has just been given is due to PoincariS*. Special asymptotic expansions had, however, been discovered and used in the . ighteenth century by Stirling, Maclaurin and Euler. Asymptotic expan- sions are of great importance in tho tiieory of Lijjear Differential Equations, and in Dynamical Astronomy ; these applications are, however, outside the scope of the present work, and for them reference may be made to Schle- singer's Handhuch der Theone der linearen Differentialgleichungen, and the second volume of Poincare's Les Metliodes Nuuvelles de la Micanique CMeste.
The example discussed in the preceding article clearly satisfies the definition just given : for
and the right-hand member of this equation tends to zero as x tends to infinity.
The term " asymptotic expansion " is sometimes used in a somewhat wider sense ; if F, <p, and J are three functions of ,t, and if a series
X il- ls the asymptotic expansion of the function
F ' we can say that the series
, „, FA, FA.
^ ° X X-
is an asymptotic expansion of the function J.
For the sake of simphcity, we shall cmisider asymptotic expansions only in connexion with real positive values of the argument. The theory for complex values of the argument may be di.scussed by an extension of the analysis.
89. Another example of an asymptotic expansion.
As a setond example, consider the function / (x), represented by the series
/(x)=% -5^. (1),
where c is a positive constant less than unity.
The ratio of the ^th term of this series to the (k ~ l)th is less than unity when k is large, except when x is a negative integer, and conse-
• Ada Mallwmaliiii. vni. (188G), pp. 295—344.
166 THE PROCESSES OF ANALYSIS. [CHAP. VIII.
queutly the series converges for all values of x except negative integral values. We shall confine our attention to positive values of x. We have, when X > k;
1 -1_:^ *!_^ ^_
X ■
■k X x'^ a? X* x^
If, therefore, it were allowable to e.xpand each fraction — . in this way,
and to rearrange the series (1) according to descending powers of x, we should obtain the series
-+-. + -+^+ (2),
X x^ x^
where ^,= S c*; A„ = —^ kd', etc.
A-=l ' k = \
But this procedure is not legitimate, and in fact the series (2) diverges. We can, however, shew that the series (2) is an asymptotic expansion of fix), which will enable us to calculate f(x) for large values of x.
Forlet s„ = ^ + 4^+...+4S^\
Sn=^ r + — 7-+... + ^ ^,
Then «„=:£ --~ + '^+...+
i=i\x X- x'
4 = 1 ( \ x) ] x + k'
so f(x) - >S„ = S - - -, ,
•'^ ' A=iV -W. x+k'
and .«{/W-^4 = 4^,s/-:^.
CO lai+lfJs
Now S ris finite, and so when x is infinitely great the right-hand
k = l X ~T~ fC
member is infinitesimal.
Therefore «" {/(a;)— /S„} tends to zero when x tends to infinity; and so the series (2) is an asymptotic expansion of /(«).
Example. lif{x)= j e^-'^dt, where x is supposed to be real and positive and the path of integration is real, prove that the divergent series
_1 ^ 1_^ _ 1.3.5
2x 2-x^'^¥^ 2*x'' ■^■"
is the asymptotic expansion oif{x).
:•()] ASYMFrOTIC EXPANSIONS. 167
90. Multiplication of asymptotic expansions.
We shall now shew that two as}'mptotic expansions can be multiplied together in the same way as ordinary series, the result being a new asymptotic expansion.
For suppose that
A, + — + —■+...+ -" + ...,
X X^ 0-'"
X X- x'-
are asymptotic expansions representing functions J{x) and J'(x) respectively, and let S,, and jS,,' be the sums of their {n + 1) first terms ; so that
Limit A-" (/ - 8n) = 0\
Limit a;»(J'-,Sf„') = 0'
Form the product of the two series in the ordinary way ; let it be
and let S„ be the sum of its n first terms.
As S„ , Sn and 2a are simply polynomials in - , we have clearly
Limit x"(SnSn'- In) =0 (2).
J=QO
Now by (1), we can write
x"
where Limit e = 0, Limit e = 0.
Then a;" (JJ' - Sn Sn') = S„'e + Sne' + ^' .
The terms in the right-hand member tend to zero as x tends to infinity. Hence
Limit X'' {JJ' - S^Sn')= 0 (3).
168 THE PUOCESSES OF ANALYSIS. [CHAP. VIII.
From (2) and (3) we have
Limit a" {J J' - S„) = 0,
.r = X
and therefore the series
„ B, B. X il- ia the asymptotic expansion of the function JJ'.
91. Integration of asymptotic expansions.
We shall now shew that it is permissible to integrate an asymptotic Ji<p,d.l d]% expansion term by term, the resulting series being the asymptotic exj)ansion of the function represented by the original series.
For let the series
— -\ - + ...+-+...
represent the function J{x) asymptotically, and let Sn denote the sum
— ^ H - + ... + —^ .
a-' x^ x"
Then, however small a real positive constant quantity e may be taken, it is possible to choose x so large that
l'^(^)-^"l<^..
and therefore
I J {x) dx - I Sndx ^ j \ J (x) — Sn\ dx
\ J i- J X J X
(n~\)x^-'' and therefore the integrated series
-^ + — ^ + ... H h ...
X 2x- (n — 1) a""'
is the asymptotic expansion of the function
I J{a!) dx.
■I X
On the other hand, it is not in general permissible to differentiate an asymptotic expansion.
92. Uniqueness of an asymptotic expansion.
A question naturally suggests itself, as to whether a given series can be the asymptotic expansion of several distinct functions. The answer to this
91, 92] ASYMITOTIC EXPANSIONS. lOf)
is in the affirmative. To shew this, we first observe that there are functions L{x) which are represented asymptotically by a series all of whose terms are zero, i.e. functions such tliat
Limit *''Z(x) = 0,
whatever n may be, when x (supposed to be real and positive) increases indefinitely. The function e~^ is in fact such a function. The asymptotic expansion of a function J{x) is therefore also the asymptotic expansion of
J(a;) + L(a').
On the other hand, a function cannot be represented by more than one distinct asymptotic expansion for real positive values of x; for if
. A, A.
^o+ — + —■+■■• X X-
and £o + — +—:+.. •
X X-
are two asymptotic expansions of the same function, then
Limits" (A + -'+...+-:-5o-^-...-^)-0,
which can only be if ^0=2^0 ; -4, = 5], etc.
Inipoi-tant examples of asymptotic expansions will be discussed later, in connexion with the Gamma-fimction and the Bessel functions.
Miscellaneous Examples.
1. Shew that the series
1 1 ! 2! 3!
X X- X' X"*
is the asymptotic expansion of the function
'-e
dt Jot
'vihea X is real and positive.
2. Discuss the representation of the function
f(x)=j (f)(t)e'^dt
(where x is supposed real and positive, and (p i» an arbitrary function of its argument) by means of the series
170 THE PROCESSES OF ANALYSIS. [CHAP. VIII.
Shew that in certain cii-ses (e.g. (p {I) = e"') the .series is .absolutely convergent, and represents /(.e) for large positive values of x ; but that in certain other cases the series is the a.syraptotic expansion o(f(.v).
3. Shew that the divergent series
1 , a-1 , (a-l)(a-2),
is the asymptotic expansion of the function
z-" log
a r" l-Jz
e~
for large positive values of z. 4. Shew that the function
/(.,=/; (log., log (^„)}^A'
has the asymptotic expansion
ffv)-— _ A4. A__ A +
where 5,, B^, ... are Bernoulli's numbers.
Shew also that/ (u.-) can be developed as an absolutely convergent series of the form
5. Shew that the function
has the asymptotic expansion
2
1.3...(2?i-3)
:n„2n-l
2»a
PART 11.
THE TRANSCENDENTAL FUNCTIONS.
CHAPTEK IX. The gamma-function.
93. Definition of the Gamma-function : Eider's form. Consider the infinite product
Z «-i z
"-' 1 + -
11
This product clearly diverges if z is a negative integer, for then one of the denominator-factors vanishes. If z is not a negative integer, the product will (§ 23) be absolutely convergent, provided the series
Jj.log(l + lUlog(l+3}
is absolutely convergent ; but since when n is large we have
^log fl-i--)= '- ■^^+... V nj n zn-
and log ( 1 -I- -) = 1 --"-+... ,
\ »/ n 2n-
the terms of this series ultimately bear a finite value to the terms of the series
^ z'-z »=i '2n-
and therefore to the terms of the series 2 -^ , which is absolutely convergent.
The infinite product is therefore absolutely convergent for all values of z, except negative integral values.
174 TKANSCENDENTAL FUNCTIONS. [CIIAP. IX.
This product may be regarded as the definition of a new function of the variable z ; we shall call it the Gamma- function, and denote it by V {£), so that
r(^) = ^n
n
This form of the function was first given by Euler ; but the notation r(^) is due to Legendre, who applied it in 1814 to an integral which will presently be discussed, and which represents the Gamma-function in some cases.
Example. Prove that
, . T • -^ 1.2 ...(n — 1) ,„
94. The Weierstrassian form fur the Gamma-function.
Another form of the Gamma-function can be obtained as follows :
We have
, n+l
r(^) = -^ u-
■2 n=l 1 I i
n
1 ml
= - Limit e^'°Bi™+" IT
^ m = ao n=l ( 1 I ^
= - Lnnit el 2 mj U .
■2^ Mi = 00 ''^~^ T 4.?.
n Now l + -+...-f-_log(,«+l) = ^S^(^-^-log-^
Jo (,n =
0 (n = l»l (« + *•)
dx.
Now the series S — -^ ~ is absolutely and uniformly convergent for
n=\ n {n + x)
real values of x between 0 and 1, as is seen by comparing it with the series
„=i n^'
04] THE CAMiMA-KUN'CTlON. 175
hence as vi increases, the right-Iiaud nionilnT nf this ofpiation tends to the limit
which is finite, since the range of integration is finite and the sum of the
CO ^
series ^ ' , is finite. This limit is known as Euler's constant, and we
shall denote it by 7. Its numerical value is
0-O772157....
Thus Limit ] 1 + ^ + . . . + log (m + l)j- = 7,
1 " e"
and so T (2) = - e-^' 1 1 ,
z „.,-,^-:
n or T^. = ^e'' n \(\+^)e~"
This form (due to Weierstrass) shews that ttt— . is a regular function of z
1 {z)
for all values of ^.
Example 1. Prove that
1^(1)= -7, where y is Euler's constant.
For differentiating logarithmically the equation
r-fe-»'1(('n)"}.
and putting z = \ after the differentiations have been performed, we have
or r'(:)=-y.
Example 2. Shew that
1 1 1 /"'I -(1-2)"
'1 .5 n
and hence that Euler's constiint y is given by
l^^^--^io'^^'^'
1 /■' \-e-'-e~l , y=\ — : dz.
Jo "
Example 3. Shew that the infinite product
S(i— f).
,. = ! \ 2 + «/
17G TRANSCENDENTAL FUNCTIONS. [CHAP. IX.
has the value
r{z-.v+\}'
For nil e" = n
X -
- e™
2
_ " (/t + 2-.r)e~ n
"=' (n + z)e-n
n (H ]e n
n=i V n
n 1 + - e~„
71=1 \ «/
The numerator of this expression is Weierstrass' form of
1
and the denominator is
1
Therefore the given expression has the value
e'"'zV{z)
{z — x)T{z — x)'
95. The difference-equation satisfied by the Gamma-function.
We shall now shew that tlie function F {z) satisfies the difference- equation
r(^ + i) = ^r(^).
We have
1 -^(1 + .
r(^ + i) = -'— n ^
n /, 1,^/1 + 1
: J_ n ililii_!L_
Z-\-l ,(=1 ?t + g + 1
n
1 n('''^)^ n^'^^.
;i + 1 ?i
This is one of the most characteristic properties of the Gamma-function. It follows that if 2^ is a positive integer, we have
r{z) = {z-i)\
95,96] THE GAMMA-FUNCTION. 177
Example. Prove that
1 1 1
/1_1_L+I L.J. J +
r(z)
For consider the quantity
1 I 1
i''"i(«+l)"^2(.'+l)(z + 2) "•■••••
This can be expressed as the sum of a iumil>er of partial fractions, in the form
Z J+1 Z + 71
To tind the coefficients n, multi|ily by (i + n) and put z= - n ; we thus obtain 1 f, 1 1 11 (-)''e
""-(^iT^V +1 + r2 + 172.3 + -I =^T •
Therefore
11, 1 fl 1 1
:(j+l)"^.'(.-+l)(j + 2)^- ^ [z (2+1)1! ^(2 + 2)2! '"j "
But
1 ^r(£'+n + l)
z(z+1)...(z + n) r{z) '
whence the required result follows.
96. Evaluation of a general class of inlinite products.
By means of the Gamma-function, it is possible to evaluate the general class of infinite products of the form
» IT lu,
n = l
where «„ is an}' I'ational function of its index n.
For resolving «„ into its factors with respect to n, we can write the infinite product in the form
YY l(»-«l)(»-«;)-.-("-«t)
«=i I («-6,) ...(n-bi)
In order tliat this product may converge, it is clearly necessary that the number of factors in the numerator may be the same as the number of factors in the denominator; for otherwise the general teiin of the product would not tend to the value unity as n tends to infinity.
W. A. 12
178 TRANSCENDENTAL FUNCTIONS. [CHAP. IX.
We have tlierefoio h- = (. and can write (denoting the product by P)
„ = i l(«-6,) ... (n-b^) For iaige values of )(, this general term can be expanded in the form
i'-^)-('-3('-r-('-r'
Oi + «.,+ .. . + a,. — 6, — ... — 6,. ^ . 1
or 1 i + terms in -^+....
n 71-
In order that the infinite product may be absolutely convergent, it is therefore further necessary that
«!+... + Oj, - &i - . . . - 6,, = 0.
We can therefore introduce a factor
ai + ...+ak-bi-...-h
e "
in the general term of the product, without altering its value ; and we thus have
(7j (t., a/i
1_- U" 1--^ e"... 1-^- e" -^ ~ ^* 1 I,. I,. r ■
'■■
Therefore P =
a,r(-a,)...at,r{-a^)
a formula which expresses the general infinite product P in terms of the Gamma-function.
Example 1. Prove that
»=" s{a + b+s) _ r(a + l)r(6-H) ,=i {a + s){b + s)~ r{a + b + l) ■
Example 2. Shew that
.r(l -p.) (l- !-.)... = {-^(--.^-)^(-a.r«)...^(-a»-l.^■'')}-^ where
2?r . . 27r a = C03 I-ISIU .
n n
97, i'S] THE (iAMMA-KUNCTIOX. 179
97. Connexion between the Ganuna- function and the circular functions.
We now proceed to establish another of the characteristic properties of the Lianiina-fuuction, expressed by the equation
sin TTZ
We have
]^\ Z+l-2
1 - ('n)
r (2) r ( 1 - .-) = ,, -N n — ^^ — ^ ,
fi "^'
(i + g(«-fi-.)
^(1 -2)„ = 1 1 "
' ,, , n
1 "
= i n
?!/ V « + 1
2 n = l /j _^
Sin TTZ which is the result stated.
Corollary. If we assign to z the special value .^, this formula gives
{r(',]'=„o,rQ)„..
98. The multiplication-theorem, of Gauss and Legendre. We shall next obtain the result
r(.)r(..^^)r(.+?)...r(. + ^) = r(„.)(2^)''^«i-n--.
n'^^T {z)r (z ^'^\ ...T iz+''~ ^] For let <^(., = ^ -"^ ^
nV (iiz)
»''-'+"r(^+ i)r(^ + i + M ...rf^+i +'i— I")
Ihen 0(i+I) =
(7i2 + /i - 1 ) (H^ + H - 2) . . . (^(ii;)
</>(-')
1-2—2
ISO
TRANSCENDENTAL FUNCTIONS.
[chap IX.
It follows from this that (f> (z) is a one-valued function of 2, with the period unity ; and (p {z) has no singularities when the real part of z is positive, since
j^ — ^^ is everywhere regular ; it has therefore no singularity for any value
of z, and so by Liouville's theorem (§ 47) it is a constant.
Thus ^(z) is equal to the value which it has when z = -; which gives
Therefore
(by § 97)
Thus
or
'^<^) = ^©^fi
*«*Hr(i)r(i-i)l{r©r(i-?
TT'
n—1
sm - sm — ... sin ?? n
^ (27r)»-'
{n — l)Tr n
4>{z)^
(27r)
«-i
2
V«
T{z)V[z +
1 \
r(5+"^j=r(«^);ii-"-'(27r) ^
Example. If
shew that
B {np, nq) = n - "3
B{p,q):
J{p)r{q)
r(j> + q) '
B(p,q)B(^p + l,qy..B(^p + '^,q) li{'h<i)B[iq,q)...B\{,i-\)q,q\ '
99. Expansions for the logarithmic derivates of the ijainma-f unction. We have
{r(2 + l)j->=ei'^nfl + -)e"«.
Differentiating logarithmically, this gives
rfiogr(^ + i)_
dz
■7 + 5
1
+
1
1(5 + 1) 2(5+2) ' 3(5 + 3)
+ ..
Also
so
l0gr(5 + I) = l0g5 + l0gr(5),
'^iogr(5 + i)=i + ;^iogr(5).
dz
5 dz
99, 100] THE OAMMA-FUNCTION. 1«1
Therefore |
||||||||
d' dz"- |
log |
r(^) |
_ 1 z- _ 1 |
d' , d j |
\z+l) z z |
+ |
||
'^ dz\l{z+l)^2(z + |
2) |
|||||||
_ 1 z- |
1 '^{z + iy |
1 |
These expansions are occasionally used in applications of the theory.
100. Heine's expression of F {z) as a contour integral.
It has long been recojjnised that the Gamma-function is intimately connected with the theory of a large and important group of definite integrals ; and in fact the function has frequently been defined by means of a definite integral. We now proceed to consider various definite integrals in this connexion, the most general of which is due to Heine and can be obtained in the following way.
We have /, \Y
1 +
1 " \ m
T{z) = - n ^
^ "1=1 14.^
in
1 " 7H .
Now if we express - 11 in partial fractions, we obtain
z m = l z + m
1 " 1)1 » <n ! 1
- n —~ = I (-1)"
z,„^iZ + m „,"Io ml {n — m)l z + m'
Consider now the function
This, when a is a complex quantity, may be defined as being equiva- lent to
g(a-nlog(-l)
Now the logarithmic function is many-valued, since the value of the function log (— .%•) is increased or decreased by im when the variable .x describes a simple circuit round the point x = 0. In order that the function (— .i)*"' may have a unique value, we have therefore to select one of the different determinations of log(— .z): and this may be done in the following way.
We first make the stipulation that the variable x is not to cross the real axis at any point on the positive side of the origin ; this prevents x from making circuits round the origin, and so makes each of the determinations of log (— .)■) a single-valued function. Then we select, from these determinations, that one which makes log(— a') real when a; is a real negative (quantity. The value of log (—x) being thus uniquely defined for every value of x, it follows that the value of (— x)"-' is likewise uniquely defined.
182
TRANSCENDENTAL FUNCTIONS.
[chap. IX.
With these presuppo.sitiotis, if C be any simple contour enclosing the origin and cutting the real axis in the point x = I, we have clearly
L
(-xy-^dx =
i-x)'
o
2i sin 7ra
Therefore
1 " I
1 +
1
- n
•2 m=l
1 +
; sin TTZ
>4
1 +
(_ a;)^+'»-i dx
= i^~^ (n + IV f (- xY-' (1 - a-)" dx.
2 sin 772^ ^ J c
Writing i/ = nx in this equality, we obtain
1 + i^' 1 ^ V^ mj
S-- 1 + -
2 sin 7r2 V w,
(-y)^-
1-
vX
dy,
where D denotes any simple contour in the plane of the complex variable y, enclosing the point y = 0, and cutting the real axis in the point y=n. If now we make n increase without limit, we have
r (^) = ^ — { (- iif-' e-y dy, ^ ' 2sm7r2J ^ -" ^
2 sin 7r2j
where the integral is taken along a curve commencing at positive infinity, circulating round the origin in the counter-clockwise direction, and returning to positive infinity again ; and in the integrand we must take (— yy~'^ as equivalent to e<2-i)ioK(-j/)_ where the real value of log {— y) is to be taken when y is negative, and the logarithm is rendered one- valued by the stipulation that the variable is not to cross the real axis at any point on the positive side of the origin.
Since
sin TTZ
this result can be written in the form
This theorem is valid for all values of z — in contrast to that found in the next article, which is true only for restricted values of the variable.
Example 1. Bourgiiefs expressiom for the Gamma- function. By a slight extension of the above proof, it is seen that
1
rW=.
h/'-^enli/,
I
101] THE GAMMA-FUNCTIdX. l.S.'J
where the path of integration is restrictoil only to contain tlio origin and to be oxtomlod indefinitely at both ends in the direction of the negative part of the real axis ; the contour need not be closed.
Take then an contour two lines inclined at an angle a to the axis of .c, passing through the origin, and a small circle round the origin. The integral round the small circle is zero when ; hiia its real part comprised between 0 and 1. The integration along the two lines gives the result
Sni ZTT J 0
which can he written in the form
TU) ^ L ^ f"p--ie'"^°' "sin (p + .-a) (/p.
This formula is true for all values of a which are not less than — . Taking a equal to IT, we have the result
r(z)=[ p'-ie-crfp.
Example 2. By taking for contour of integration a jiarabola with the origin as focus, shew that
r(z) = -— ,~^— I e-4x»(H..i-2)ir-icos[(2i-l)tan-'.i-+.c]f;.t. (Boiu-guet.) % Sin ztt J 0
101. Expression of F (z) as a definite integral, whose path of integration is real.
We have, by the result of the preceding article,
r (z) = ^ 1 e-v+ "-" '"*■' '-■"' dy.
Take a path ABODE, commencing at tlie positive infinitely distant extremity of the real axis (which considered as initial point we denote by A), proceedini,' close to the real axis until it arrives at the neighbourhood of the origin, describing a small circle BCD round the origin, and returning, close to the real axis, to positive infinity again (which, considered as terminal point, we denote by E). With the conventions that have been made, the integral along the part AB of the path becomes
z sin TTZ . in which log y is supposed to have its real determination.
The part of the integral due to the small circle BCD is easily seen to be zero if the real part of z is positive. For the part of the integral due to DC, we have
2 sin TTZ
I g-»+K— Dlogy+tinz-l) j^„
Jo
n
184
TRANSCENDENTAL FUNCTIONS.
[chap. IX.
Thus
ru)
I e'
■f (z-l) _ ->-iir(z-l)) /•»
or
2 sin TTZ
J n
Jo
This integral is called the Eiderian Integral of the Second Kind. It is frequently given as the definition of the Gamma-function : but for this purpose it is unsuited, since the integral exists only when the real part of z is positive.
Example 1. Prove that when z is positive
dx.
Example 2. Prove that
/:
e-''^x'--^dx=^\
Example 3. Prove that
_l 1 J. _ J^ [
(^H
e-'^x'-'^d.v 0 «^-l " ■
102. Extension of the definite-integral expression to the case in which the argument of the Gamma- function is negative.
The formula of the last article is no longer applicable when the argument z is negative. Saalschlitz has shewn however that, for negative arguments, an analogous theorem exists. This can be obtained in the following way.
Consider the function
T, (z) = j\~-^ (e- - 1 + ^ - i:^ + ... + (- 1/-+' |!j dx,
where « is a negative number lying between the negative integers — k and -(Ar + 1).
By partial integration we have, when z< — \,
\\ {z) =
a^ *
+
gJc-¥\ a;*"'"- a^+^
J ^~ ^^^ [{k + l)\ ~ (fc + 2) ! ^ (¥+3)1 ■
1 r"^ ^—1
zJu \ (^'-1)!
102, 103] THE GAMMA-FUNCTION. 185
The terms in the let'i-h;uul inumbor whicli arc not under the integral sign ' vanish, since (z + k) is negative ami (^ + i + 1) is positive : so we have
r,(^) = -r,(s+l).
z
The same proof applies when z lies between 0 and — 1, and leads to the result
riz+l) = zr,{z) (Q>z>-\).
1
j The last equation shews that, between the values 0 and — 1 of 2,
r,(z) = r{z).
The preceding equation then shews that Fj (z) is the same as r {z) for all negative values of z less than — 1. Thus for all negative values of z, we have Saaischiitz's result
r (z) =/ V' [e-^ -l+x-~+...+{- !)*+■ 1^) d^,
where k is the integer next less than — z.
Example. If a function P (^i) be such that for positive values of /i we have
P(u)= I J^-^e-'dx, and if for negative values of y. we define P, {n) by the equation
P,{y)= \\,*-^(e--\+x- ... +{-\f*^^^^cb:,
where k is the integer next less than - fi, shew that
A(.) = />W-^,,(^)-...+(-l)-',-T(^. (Saalschutz.)
103. Gauss' expression of the logarithmic derivate of the Gamma-function as a definite integral.
We shall next express the function T-logr(z) as a definite integral, where 2 is supposed to be a positive real quantity.
1 /■* We have - = / e'^dx.
s
s Jo
r» 1 r
Therefore log s= I - ds = I
e-^ - e-" J dx.
186 TllANSCENDENTAI, FUNCTIONS. [CHAP. IX.
Thn.s we have
Jo Jo Jo «
r(.) = r(.)r^-{e— ,^
0 *• r (i+*yi"
This equation is due to Dirichlet.
Writing 1 + x = e' in the second term of the integral, and x= t in the first term, we have
which is Gauss' expression of j- log T{z) as a definite integral. t^
Example 1. Prove that
^ log r (z) = r i-L- - 'f^} dt. (Gauss.)
'''' •'"llog| '-'/
Example 2. Prove that
rf , ,, , 1 n, ra(l-a) a(l-a)(2-a), "I
104. Binet's expression of log T (z) in terms of a definite integral.
Binet* has given an expression for log V{z), which is of great importance as shewing the way in which log T {z) increases as z becomes very large ; his result will be used later in the derivation of the asymptotic expansion
of r {z).
We have by the last article {z being supposed real and positive) r{z)_rfe-' e^ \
changing « to ^ + 1, we have
^iogr(. + i)=f;(^-l'-;^)d.
dz Now
* Journal cle I'Ec. Polyt. xvi. (1839), pp. 123—343.
r" 1
e-''dt^-
Jo z
104] au(i
THE UAMMA-FI'NCTKIN.
("- (e-' - e-'')=rdt \'e-"Jd,j
Jot J u J 0
187
•■dy II = log z.
Therefore
d^ dz
^,logr(.+ l) = l+log.+/"d«{-
+ —
( e'-l
-^ '^l^-U,^,
Lz ^ Jo \2 t e'-l
= -^ + log z — dte' '2z ° Jo
Integrate for z between the limits 1 and z ; so
iogr(.+ i) = |iog^+.(iog.-i)+/J{^-,^-j-^ + i|^
dt
, /•" ( 1 1 11
dt,
1 1 1) e-"
or
^o^Tiz)={z-l)\osz-z+[yi^-] +
2 t
dt
+ 1-
+ J -dt.
.(1).
Jo (e'-l < 2) « The first of these integrals can be otherwise expressed in the following
way.
We have*
1 1 1 _ ., p sin (tu) du
+
e'-l 2 t Jo e^'"-!
Multiplying both sides of this equation by e~^' dt and integrating with respect to t from zero to infinity, we have
r , / 1 1 1\ ^ r°° , 7 f^ sin (tu)du
= 2
(,du
!„ (w^+ja-Xe^'"-!) Integrating this equation from p = z io p=x , we have
f^T-i^-A-'i)
tail-' (- ] f/< 0 e-"" - 1
.(2).
A proof of this equation can be found by making k infinite in the equation S „ . „ = 2 i: / «--'""" sin ((!() (Ju.
188 TKANSCENDENTAL FUNCTIONS.
Thus equation (1) becomes
l0gr{2) = [z-^]\0gZ-2+2
tail"' I - flu e-"" - 1
Now write ^ = ^ in equation (1): since
we obtain
1 , 1 r=" f 1 11) e-i«
dt
1 1 1] e-' ,,
0 (e«-l ^ 2) i
or
Write ji for < in the last intearral. Thus 2 "
Adding this to equation (3), we obtain
1) e-i'
t] t
dt,
e-j^Tdt t \ t '
log T{z) = {^z-^jlogz-2 + 2
[chap. IX.
— /:{.4i-^-H-' «^
tan"'! - 1 du
\zl 1 , 1
■*"' i ie«-l t e'-l'^ t 2
The last integral is
dt \\ , e-'-g-J'i
or
or'
1 tf— ( _ -,— a:t
/
Jo
.'o Ji <
I log a; da;,
rfa;,
.(4).
* This artifice is due to Pringsheim, Math. Ann. xxxi.
105] THE GAMMA-FUNCTION. 189
1 , 11 or --log---.
Substituting this in equation (4), we obtain
tan-' (-] dn
\ogr(z)=(^z-'l]\og2-3 + ]^\og(27r) + 2J
: This is Binet's formula for log F (z) ; as z increases indefinitely, the last- integral diminishes indefinitely, and so the remaining terms furnish an I appro.Kimate expression for log F (z) when .: is large.
Example. Prove tluit where J (z) is given by the absolutely convergent .series
^^'~*U+l'^2(2+l)(2 + 2)''"3(j+l)(£ + 2)(2+3) "•"•••] ' in which
and generally
Cn= j (.v+\){.v+2)...{x + n-\){±v-l)xdx. (Binet.)
' 105. The Eulerian Integral of the First Kind.
The name Eulerian Integral of the First Kind was given by Binet to the integral
•^ (/>, q) = i .r^' (1 - ■v)i-' dx, Jo
whicii was first studied by Euler aud Legendre. In this integral, the real parts of jD and q are supposed to be positive; and a;?"', (1 —*■)«-' are to be understood to mean tho.se values of e'^^" '"'''* and e'9-"'"8 0-»| which correspond to the real determinations of the logarithms.
With these stipulations, it is easily seen that the integral e-xists, since the infinity of the integrand is of less than the first order at the two extremities of the path of integration.
We have, on writing CI - x) for x,
B(p,q) = B(q,p).
Als..
■ 1.1-^(1 -x)'>~
I .r''-'(l- a;)'? (/./■ =
J II
.n P
P'
or
B(p.q+l)=j^B(p + l,q). ^
190 TRANSCENDENTAL FUNCTIONS. [cHAP. IX.
y Also
B (p, q)=[ .'■■f-' (1 - x)i-' d.i- =1 {1-X + x) a-P-' (1 - a;)?-' dx
= B{p+\,q) + B{p,q+\). Combining these results we obtain the formula
'^ Bip,q+l)=^^B{p,q). Example 1. Prove that if n is a positive integer,
1 1.2... 71
{ill
B{p,n+\)--
p{p-\-\)...{p-\-ny Example 2. Prove that
v/
*/ 106. Expression of the Eulerian Integral of the first kind in terms of the Gamma-function.
We shall now establish the important theorem
B (m, n) = „ ,- \ ■ ^ ' r {m + n)
To prove this, we have
T{m) r (?i) = I e-'^x^'-'dx x I e-J/?/"-> dy
Jo JO
(writing x^ for .c, and if for y)
Jo Jo
= 4 I / e- <^'+!''' «-'"-' (/="-' cZa; c?^
i II J 0
(writing ?-cos 0 for a;, and r sin ^ for i/)
TT
= 4 I f e-'-' ?•=<'"+»'-' cos-'"-i 0 sin="-i (9 «!/• dS
J 0 J 0
= r ()/i + m) 2 f cos='"-' ^ sin--"-' ^ dd / ■'u
(putting cos- 0 = u) = r (7K -t- n) B {m, n).
This result connects the Eulerian Integral of the first kind with the Gamma- function.
106 — 108] THE GAMMA-FUNCTION. IIM
Example. Prove that
(Cambridge Mathematical Triixw, Part 1., 1894.)
i07. Evaluation of triijoiiumetrical integrals in terms of the Gamma- function.
We can now evaluate the integral
I cos'""' X sin"~' a- dx,
Jo
where m and n are not re.^tricted to be integral, but have their real parts positive.
For writing sin-a;= t, we have
/:
2 p tn + ?i
2 J r\ m_ J._j
cos"'-'A-sin"-'.<c?j=5 (1-0- i' dt
/
The well-kno\\Ti elementary formulae for the case in which m and n are integers can be at once derived I'roni this ^
Example. Prove that when /' <1,
(Trinity College Examination, 1898.) 108. Dirichlet's multiple integrals. We shall now shew how the integral
^=//// e"-®-©
xJ'-'yi-'z'-'dxdi/dz
may be reduced to a simple integral, where f is an arbitrary function of its argument, and the integration is extended over all the systems of positive values of the variables x, y, z, which .satisfy the inequality
^^-^i)-^:)
y
192
TRANSCENDENTAL FUNCTIONS.
[chap. IX.
Write
111
Then the integral takes the form
/ = "J^' jjj /(*•. + yi + --.) *i"'-' y.*'-' ^Z'-' da; dy, dz, ,
where the integration is now taken over all the systems of positive values of the variables a;,, y^, 2,, which satisfy the inequality
Now let fi = ^i+yi+ 2i- ^ = 0,
be three^quations defining new variables ^, tj, f. Then ^(^" 3/i. ^O, ^(^. ^?. ?)
9(«i. 2/i. ^i)
1 |
0 |
0 |
'; |
-f |
0 |
'Z? |
-^? |
-^v |
I |
1 |
1 |
0 |
1 |
1 |
0 |
0 |
1 |
-^■%
The field of integration is tlearly such that the new variables ^, r), ^, each vary from 0 to 1.
Thus
iiVh'lr'' I'l /"• I'l a/37 Jo.' 0.0
a/37
et^fe'c'' " \a
IS
r
a^7
r(^^^ + '"
+S-+--1
/(f) r- '^ ^ d^.
^a /3 7/ The multiple integral is reduced to a simple integral.
It is easily seen that this method of evaluation can be applied to multiple integrals of a similar form in any number of variables.
109] THE GAMMA- FUNCTION. 193
Example 1. Shew that the moment of inertia of a homogeneous ellipsoid of unit density, taken about the ajtis of z, is
— ^{a- + b-)irabc,
where «, b, c are the semi-axes.
Example 2. Shew that the area of the epicycloid x* + i/* = P is jirP.
Xote. Diriclllet'^s integrals can also be evaluated by performing on the variables the substitution
.r, = r- sin''' ^, sin- ^2)
i/i = r- sin- 5, cos^ d^,
leading to the same result as above ; in the Ciuse of an integral with » variables the corresponding substitution would be
.i-j = /-sin- $1 sin- 6.,.,.si\n- 1)„_^, etc.
109. The asymptotic expansion of the logarithm uf the Qamma- function {Stirling's series).
We now proceed to find an expansion which asymptotically (§ 88) represents the function logr(2), and is actually u.sed in the calculation of the Gainnia-functioii.
For simplicity, we shall consider only real positive values of the argument 2. For a proof and discussion of the expansion when z has complex values the student is referred to a memoir by Stieltjes*.
From Binet's expression for log F (z) (§ 1 04), we have
log r (z^ = {2-l)\ogz-z + l log (27r) + 4> (z),
tan ' - ax z where (^(^) = 2
.. ., X. X \ a? \ 3?
JN ow tan~' -= o^, + -";"•••
Z Z 6 Z^ 0 Z^
(-!)»-> «=*»-' (- 1)» r' f^dt "*" (2n - 1) 2-"^ "'' z^-^ Jo f + z" ■
Substituting this in the integral; and remembering that
f
r a.--"-' dx _ By.
• Liouville'i Journal (4), v. pp. 425 — 444 (1889). W. A. 13
194 TRANSCENDENTAL FUNCTIONS.
where B^, B^, ... are the Bernoullian numbers, we have
« (- iy~'Br 1 2(-l)" r dx__ f^ f^dt^ "P ^^' ~,Zi 2r (2r - 1 ) z'^-' ^ z^-' J o e'"' - ijot' + z''
[chap. IX.
Let us now find approximately the magnitude of the last term when z is very large.
dz f-^ i-"rft
z'
The quantity is less than
or
or
r dx r-^ i-"i
-f {2n + l)z-'}^
dx
lio
<="(?!;
"-1
i^„
4(w + l)(2m+l)2'''
If now any value of ;( be taken, it is clear that this quantity can be made as small as we please by taking z sufficiently large.
It oUows that the quantity
2«'-' U (z) - i *•" ^T^Uk^ r^ ' rZi 2r{2r-l)z^
can be made as small as we please by taking a sufficiently lai-ge value for z ; and therefore (§ 88) the series
i
A
B,
+ ■
5,
1.2.2 3. 4. 2^ 5.6.2' ■■■
is the asymptotic expansion of the function 0 (z) for large real positive i values of z.
We see therefore that the series
^ - -J log . - . +, log (2.) +^S '^-^^ -
is the asymptotic expansion of the function log F (z) for large real positive values of z. This is generally known as Stirling's series.
110. Asymptotic expansion of the Gamma-fanction.
Forming the exponentials of both members of the equation just foundj we have
-8,
js, I -B.
F(2) = e-^^'-i(27r)iei-* s-*-^ *■«•
10] THK GAMMA-FUNCTION. 195
where C, = - — ^ , 1 2 = etc.
Substituting thu mmit-ricul values of the Bernoulliau numbers, the t'unuuhi becomes
This is the asymptotic expansiun of the Gamvia-function. In conjunction with the formula V {\ + z) = zV {z), it is very useful for the purpose of com- jHiting the numerical value of the function.
Tables of the fuuctiou log r (;), correct to 12 decimal places, for \-alues of z between 1 and 2, were constructed in this way by Legendre, and published in his Exercices de Calcv.l Inte'gnd, Tome 11. p. 85, in 1817.
Miscellaneous Examples.
1. Shew that
"-'(-»(-0(-y-7Hy%)-
(Trinity College Examination, 1897.)
2. If 0-,, be the sum of the n tir-st terms of a divergent .series
shew that the series
1 1 1
-+- + -+..., o, a., a.
1 1 1
+ + + .
a,(ri a.,(T.^ a^(T3
I
is divergent.
If the squares of the terms of the latter series form a convergent series, shew that a function G (l+z) can be defined by the equation
G'(l+-'j=Limit — — Z:L. ^ ,
and shew that
0{l+2) „_i IV a„aj J
where c is a constant. (Cesaro.)
13—2
19(j TRANSCENDENTAL FUNCTIONS. [CHAP. IX.
3. Prove that
d log r {z) _ /•■'e-'-e-"
dz Jo l-e-<« " ^
= f"{(l+a)--(l+a)-';^"-y, J u ('
where y is Euler's constant.
4. Prove that
T{z) = Jjimit n'B{2, n).
5. Prove that
B{t + r,t-r) = ^^j^ ^p4-.
6. Prove that, when q> \,
7. Prove that
•S(i?-a, j)^ a? a!(a + l)g(g + l)
^te?) .? + ? l-2-(/> + ?)(i' + ? + l)
8. Prove that
^iP, q)B{p + q, r) = B{q, r)B{q + r, p). (Euler.)
9. Prove that
1 "^ loff 2?i logr(2) = (l-2)log7r + y(^-2)-^logsin27r + - 2 — ^ — sin 2n27r.
"■ n=l »
(Kummer.)
10. Prove that
I cosf*''-'-ucoii{p-q)udu = -z — ■ r^^r-:, — ,—fi-, r. (Cauchy.)
jo ^ ^ (p + q-\)2P + <'~^ B{p,q) ^ -^ ^
11. Prove that
■ log B („ ,) = log (^^) , jl (1Z1!ULZ|!) ,.. (Kuler.)
12. Prove that
^(i'.i' + s; ^« 1 2(2jo + l) 2.4. (2p + l)(2^ + 3) + •••)■ ^ '^
13. Prove that
r(rt ^ fl I L , 1-3^ U
14. Prove that
(r(,p+i)Y^2p-\( 1 1.3^ 1
I r(jo) J 2 1 ■^2(2^ + l)'^2.4.(2;3 + 3)(2;) + 5)'^ -J ■
MISC. EXS.]
IT). I'rovo thftt
THE GAMMA-FUNCTION.
197
2SP-1 r o«2 c 1 1.3'- }~\i
Vir
16. Prove that
where y is Euler's constant.
17. Prove that
18. If shew that
j^\ogr(z)=-y +
jo ^-1
B(p,p)B(j>+ip + i) = ^^^.
fx+l
I \ogr{z)dz = u,
(Binet.)
(Legendre.) (Binet.)
dti dx
= log.r,
and hence (or otherwise) that
« = .i'log j: — .r + i log 2rr. 19. Prove that, for all value-s of z except negative real values, logr(;) = (2-*)logz-z + ilog(2n-)+ 2 f -^
11 = 1 J l> x -^ z
(Kaabe.)
sin 2«7r.r
(Bourguet.)
20. Prove that
r(a +
l)r(q + 6 + c+l)_ n j;°(l-a-*)l b + \)T{a + c+\)~ U (i_^,)i
)log-
21. Prove that
22. Prove that
nj.»-i-.,-P-i^ V 2 / V2
Jo(l-|-:r)log^- ''^r/^!'\r^'^ + l
^l^jM -2
(Kummer.)
^ r («+?) + i)r(a+c-f-i)r(&+c+i) ^ f ' "r(a+i)r(6+i)r(c+i)r(a+6 + c-i-i) jo
> (l-.!■")(l-.^*)(l-.^^')
(l-j-)log
rfj;.
23. When x is positive, shew that
2n!
T{x+^) „_o22».n! n! ;»■ + ?!
(Cambridge Mathematical Tripos, Part I, 1897.)
24. If a is positive, shew that
r(z)r(a-H)^ ^ (-l)"a(a-l)(a-2)...(a-«) 1 r(2 + a) ~ n=o n! 2 + "'
198 TRANSCENDENTAL FUNCTIONS. [CHAP. IX.
25. Shew that
I "'cos"* cos 7(Arfrf) = 2-P-'7r ^(P + '^)
26. The curve r'" = 2"'~'cosmd is composed of m equal closed loops. Show that the length of the arc of half of one of the loops is
1 f^ i-1
2™ ' . — . / (cos x)" dx, m. Jo
and hence that the total perimeter of the curve is
27. Prove that
log r (z) = (Z - i) log 2 - Z + ^ log (277)
, f 1 " 1 2 " ^^ 3_ " JL \
+ *|2.3,=i (2+r)2 + 3.4.=, (z + r)''^i.brli{z + rY'^-i-
^l0gr(.) = l0g.-j^^j-^yj^{l0g^+l-x}.
logr(.) = /;^
28. |
Prove that |
d dz |
29. |
Prove that |
|
30. |
Prove that |
Iogr(j + a) = logr(2) + alog0--
a I a(l — a)da-l a {I - a) da Jo Jo
22 (.-+1)
ai a(\-a){2 - a)da- i a{l — a){2-a)da
Jo J_o
3a (2+1) (2 + 2)
31. Prove that
32. Prove that
MISC. EXS.] THE OAMMA-FUNCTION. 190
where
and p'=^- + V"+/'<y-
33. Expand
{r(a)}-i as a series of ascending powers of a.
(Various evaluations of the coofficionts in this expansion have been given by Rourgnet, Bull, des Sci. Math. v. (1881), p. 43; Hoiwguot, Acta Math. ii. (1883), p. 261 ; Schlomilch, Zeilschrift fur Math. xxv. (1880), pp. 35, 351.)
34. Shew that
and
where
3.5. If
shew that and
/ j;'"e-'«cos 6.rc;.r = cos {{m + 1) (tt - <!>)] ^^7"i >
[ j™e - ■" sin hxd.v = sin {(to + 1 ) (tt - <^)} ^-"^ti ^ > -a + 6j = r(eos0 + isin <f>).
P{x)= Te-'z^-^dz,
p ^ 1_1 JLj.l_i 1 J_
^^^~i lU-+l"'"2! x+2 3!a;+3"'"""'
P[x+\)=xP{x)--.
36. Prove that
r
d, Viz + x) X , x{x-\) ,.!;(.?:- l)(ji;- 21 dz^'^^—' "' -
37. If a is negative, and if
U, . I \i-^X) _X .X\X— Ij ,X\^X— \.)(X— _. Tz'^'vW ~2~*zirfT)" * 2(^ + 1) (2 + 2)' "^
a = - v + a, where v is an integer and a is positive, shew that
where
(-l)-(a-l)(.-2)...(a-.)^
71 ! \ /'
<'«-(' + .4i)('-.!.) (■ + .4-.).
200 TRANSCENDENTAL FUNCTIONS. [CHAP. IX.
38. When - oX a < 1, show that
r(x)r(a-x)^ I Jln^_ ^ _Rn r(a) „_i x-\-n n-^x-a-n^
where
„ _(-l)"a(a + l)...(a+»-l)
"" ^n •
39. When a > 1, and v and n are respectively the integral and fractional parts of a, shew that
T{x)T{a-x)^ I Gix)p„ ^ G{x)py^„
where and
r(a) „=i X + 71 n=i x—a-n
_(-l)"a(u + l)...(a + K-l)
40. If pi, pj, ... p^ are the roots of the equation
p" + a,p''-' + ...+a^ = 0,
shew that
71=1 l\ ^ + ft ''(z + re)^ (2 + a) / J
/'<^""-^r''(z)
r (2- Pi-^) r (z- p^a;) ... r (z- p,a-) ■
41. If a and 6 are real and positive, prove that
\ I 6''u-"v''-'^u''-''-^dudv^r(,a)r{b).
42. By taking as contour of integration a parabola with its vertex at the origin, derive from the formula
r(«) = s-^^ \(^z'^-^dz
ziamaTTj
the result
I'(«) = S-^ / e''^^"'(H-.!-^)=[3sin{^ + acot-i(-.r)}
z sin dTT J 0
+ sin {.^• + (a - 2) cot"' ( - x)}] dx.
(Bourguet.)
43. Prove that, when 1 < i < 2,
r sin hx , 6'->
and when 0 < a < 1,
) . TTZ'
sm-
/"cos 6.1- , 6"-! TT
0 .t'» 2r(z) TTZ
vz COS 2-
MISC. EXS.] THE GAMMA-KUNCTION. 201
44. Show that
IT
-l.\ If
M
22
niid
'(-I)
02
and if a function /'(.r) be defined by the equation
shew (1) that F{x) satisfies the equation
F(x+\) = xF{x)^■^ ^
(2) that for all positive integral vahie.s of .r,
(3) that F{x) is regular for all finite values of x,
(4) that
4G. Prove that the function G (x), defined by the equation '
G{x+l) = {27ry-e 2 2 n M+-M e2* ,
has the properties expressed by the equations
a{x+i)=r{x)0{x),
C(l) = l,
, 0(l-X) /■"= . T , „
log TTTi X — I '^•^ t^ot 7r.-t-a.r - .r log Ztt,
"■ (!+•'■) jo
r (i;-l)(x-2) "-lr(l+i?-)"l
<?(x)=Ljmit[(«+i)-T-{r(.4-i)}^-'n^pi^J.
(Alexerewsky.) 47. If « is a positive quantity (not neces-sarily integral), and z is a real quantity between — ^ and - , shew that
COS'Z-g,.
1 r(« + l) f, , » , , »(»-2) , ^ 1
and draw graphs of the series and of the function cos'z.
202 TRANSCENDENTAL FUNCTIONS.
48. Obt<ain the expansion
[chap. IX.
cos«.j;=-^jr(.f + l)
+ ■
cos Sax
r(^J^+i)i-r-'.i)'rrV'%i)r(^+i
anil tind the values of .f for which it is applicable. 49. Prove that
"(
^+4^^;
50. If
({s,x)= 2 J,
where | ^ | < 1 and the real part of x is positive, shew that
and
Limit (l-xy-'C{it, x) = r(\-s). 51. If X, w, and s be real, and 0 < w < 1, and s> 1, and if
shew that and
52. If
shew that (1)
(2) (3)
00 ft'inlTix
(h (w, X, s)= 2 , r- ,
,p{w,x,l-s)= ^')
»=1 '^
f«
1 f' x'-^dx
f (s) = (27r)»-i.sin"
"j„ W-1 x)
- ).2;~'(i.j.',
rm^-2!rw=r( .^)^^ai-«)-
1 - s\ '^ .
53. Let the function 0(»)(.r) be defined by the equation
x{t)t'e-''dt
(-l)'0W(.r) =
n (l-e-""')
where z is an integer 5 m, the function ;^ (t) is defined by the equation
tt=0 " •
and the quantities a^ are constants whose real part is positive.
+ ...
(Cauchy.)
(Lerch.)
MISC. EXS.] THE GAMMA-FUNCTION. 203
Shew that <^(') (x) can bo exprossod by the series
0(')(;f) = S/('I(x-|-w) ; where tc = :i\ya„,
and where
(-!)•/"' (■'•) = J"x(0«'«-"rf<.
j Shew also that <^('> (x) satisfies the functional equation
j (f,(-)x-S4,W{x + a,)+ 2 0C)(.r + a^ + a.)- ... +( - IT^O (.r + a, + a„ + ... + a„.)=/C)(x).
Shew further that when ;^(<) = 1, <(,<•) (x) becomes a function ^j/'-'Hx), which has the • multiplication-theorem
«•*• ^ \ n n
where all the quantities X vary from 0 to (« - 1).
* A,,,0,. + ...+
n
(Pincherle.) 4. If
where shew that
f /^ „ ,^-^ fn\x(y + v + 7i-l) fn\ x{x + l)(i/ + v + n-l)(y + v + n)
., riy)riy-x + n)r{x + v)r(v + n)
and that
J- , \ (v - x+n — 1) (x+v+n) . , , ,
I
r (y) r (^ +tO
(y-T-l)r(-n)r(:c+l)r(y+»-l-«-l)'
(Saalschutz.)
I
CHAPTER X. Legendre Functions.
111. Definition of Legendre polynomials. The expression {\ —2zh +h-)~''
can, when | A | is sufficiently small, be expanded by the multinomial theorem as a series of ascending powers of/;, in the form
l + hP,{z) + h'P, (z) + h'P, iz) + ...,
where P, (z) = z,
„ r>z^-Sz
P, (z) = 2 ■ *5*°'
The expressions P, (z), P. (^) ..., which are clearly all polynomials in z, are known as Legendre polynomials. Pn (z) is called the Legendre polynomial of order n.
It will appear later (§ 116) that these polynomials are particular cases of a more
extended class of functions, known as Legendre functions.
«
Example 1. Prove that
P^ (cos 6) = ^ — r- cosec" + ' 6 -,, ^ ' .
"^ ' n\ d (cot ey-
(Cambridge Mathematical Tripos, Part 11, 1893.) Let & be an ansrle such that
I
sin &
(1-2/1008 6'+/!') " =
Then
(l-2Aoos^+A2)-i = *'^. sin S
[sm'^ 6 \
cos d-h sin 5
= cot 6 — h cosec 6.
Ill, 112] LEGENDRE FUNCTIONS. 205
Therefore by Taylor's theorem we have
,(-h cosec d)» d" (8in 6)
8mfl' = 2^
n! d{cotey
^ ' „ n\ d (cot BY
Equating coefficients of /(", we obtivin the required result. Example 2. Shew that
For
/>-©'
Therefore
Thus
2
B=0
1 /"" / J\" /l"
whence the result follows.
Example 3. By equating coefficients of powers of h in the expansion
(l-2Aco8fl + A2)i V - 2.4
1.3
"2.4
A+iAe-« + ^A2e-2'« + .
• shew that
^"(^■°^^)=''2'fi?i^{^'^"^"^+2nk^)'=°«('^-2)^+-"}-
■ 112. SchUifli's inte(jral for Fn{z). r Let /i be any quantity which i.s not greater than the radius of convergence of the series S h"P„(z).
11 = 0
Then (1 —2zh + h-)~^ can bo expanded as the series 1 + hl\ (z) + h'P, {z) + h'P, {z)+.... But {\ — 2zh + h^Y^ is the residue, at the pole
I
_ 1 {\-2zh + A=)i
C — 7"
of the function — 2/t~' \ 1 1
h h
l\- \-1zh + h-
h) h'
206 TRANSCENDENTAL FUNCTIONS. [CHAP. X.
Now the last expression has two poles, namely at the points
_1 (1 - Izh + /t^)i *~h h
and t = r + i .
h h
When h is very small, the former of these poles is close to the point t = z, while the second pole is in the infinitely distant part of the plane. Therefore, if G be a contour in the ^-plane, including the point z, the former , pole only is contained within G when /) is not large, and so we have
|o^"^"(-) = 2L.L..-2;+2.-/.^^
(^^-^^" dt.
■ (t - zY
Equating coefficients of //", we have the result
1 r 1
which is called Hclddjiis integral-formula for the Legendre polynomials*.
113. Rodrigues formula for the Legendre polynomials. From Schlafli's integral
^"^^'~27riJc2"{t-zr+''^^ we immediately deduce, b}- the theorem of § 38, the result
which is called Rodrigues' formula.
114. Legendres differential equation.
We shall now prove that the function y = P„ [z) is a solution of the differential equation.
(1-2=) J-2^^ + «(«+l)y = 0, which is called Legendre' s differential equation of order n.
* Schlafii, Ueher die bciden Hcine'schen Kugclfunctionen ; Bern, 1*381.
113—115] LEGENDKE FUNCTIONS.
For on substituting Schiivtli's integral, we have
207
(1 - z"')
dz'
_2«— j'A--' + „(„ + !) P„ (2) az
(h + 1) r {t^-iydt
•Ziri
Ic 2^{t -Ir^ '" ^" + -^ ^'^ -l) + -2^n+l)tit- z)]
_(n + l) r d ((<'-l)''^M ~27ri.2"Jcdt\(t-zY*'r'
and this integral is zero, since the function (t- - 1)"+' (t — 2)~''~- resumes its original value after describing the contour C. The Legendre polynomial therefore satisfies the differential e([uation.
The differential equation can clearly be written in the alternative form
dz
(1 _ ,.) <il'i(^[ + ,^ „ + 1 ) p„ (^) = 0.
dz )
115. The integral-properties of the Legendre polynomials.
\Vu shall now shew that
)' 1
l\.Az)Pn{z)dz = 0, 1
and that | ' {P„ (z)Y dz = -^-—- ,
. — 1 — /( "7" 1
if m and n are positive integers and m is not equal to n For since
we have
iJ(i-^=)¥^K"(« + i)A=o,
dPr,
dz
dz
dz \^'' " ' dz
-\- {m- n){m + n-\)\ P,nPndz = 0. ■ —1
Integrating by parts, this equation gives - {m - n) (m + « - 1) f ' P,„ (z) P„ (z) dz
= 0.
'(l-.nlp.„(.)'^^-p„(.)'^^-(^>
dz
dz
which shews that the integral
I ' P,„ (z) P„ (z) dz has the value zero when m is not equal to ?!.
208 TRANSCENDENTAL FUNCTIONS. [CHAP. X.
To establish the second part of the theorem, let the equation (l-2/(2 + /r)-i= i /t"P„(«)
n = 0
be squared, and the resulting equation integrated between the limits — 1 and + 1 ; using the result already proved in the first part of the theorem, we thus obtain
n dz ' " p
-- — -'^^
n=o J -I n i.
Equating coefficients of A-" in this equality, we have
which is the desired result.
Example 1. Prove that, if m is not equal to n,
X {l+C-l)"*-"}. (Cambridge Mathematical Tripos, Part I, 1897.) Example 2. Prove that
/:.-
dz'- dzr (^ '■' "'"-" "'^ (2» + l)(»-r)!'
according as m, n are unequal or equal.
(Cambridge Mathematical Tripos, Part I, 1893.)
116. Legendre functions.
Hitherto we have supposed that the inde.x n of P„ {z) is a positive . integer ; in fact, P„ {z) has not been defined e.xcept when n is a positive integer. We shall now see how the definition can be extended so as to furnish a definition of P„ {z), even when n is not integral.
An analogy can bo drawn from the theory of the Gamma-function. The expression z\ as ordinarily defined (viz. as z (z— 1) (z-2)...2 . 1) has a meaning only for positive integral values of z; but when the Gamma- function has been introduced, 2! can be defined to be r(2-M), and so a function z\ will exist for all values of z.
Referring to § 114, we see the differential equation • (l-^=)g-2.| + n(« + l), = 0
]I(j] LEGENDRE FUNCTIONS. 209
is satisfied by tho expression
even when n is not a positive integer, provided that C is a contour such that the function
(t - z)''+»
resumes its original value after describing C.
Suppose then that n is no longer taken to be a positive integer.
Now the function (<'— 1)"+' (t — z)~^~- has three singularities, namely the points t=\,t=—'\,t=z; and it is clear that after describing a small closed contour enclosing the point t = \, the function resumes its original value multiplied by e^irMn+i) . w'hile after describing a small closed contour enclosing the point t = z, the function resumes its original value multiplied by
If therefore C be a simple contour enclosing the points t = \ and t = z, but not enclosing the point t = — \, then the function
(^--1)"+'
It - zY+^
will after describing C resume its original value multiplied by e~"", i.e. it will resume its original value. Hence whatever n he, the Legendrian differential equation oj' order n,
is satisfied by the expression
'j=^iUnit"-int-^)--dt.
where C is a simple contour in the t-plane enclosing the points t = \ and t = z, but not enclosing the point t = — \.
This expression will be denoted by P„ (z), and will he termed the Legendre function of the first kind and of order n.
We have thus obtained a definition of P„ (z) which is valid even when n is not integral.
The Legendre function is a mere polynomial when n is integral, but is a new transcendental function when n is not integral ; just as F (z) is the polynomial (z— 1)! when z is integral, but is a transcendental function when z is not integral.
w. A. 14.
210 TRANSCENDENTAL FUNCTIONS. [CHAP. X.
We shall suppose the many-valued function z", which occurs in the defining integral, to have the value 1 when z is equal to 1, and when z is not equal to 1 to have that value which would be obtained by con- tinuation along a rectilinear path from the point 1 to the point s.
117. The Recurrence-formulae.
We proceed to establish a group of formulae which connect Legendre functions of different orders.
We have by § 116, for all real or complex values of «,
2-7riJc n.2" '*|(«_2)»|- Integrating by parts, we have
1 ft(t^-ir-
27riJc2"~'{t-z)" and hence we have
p„(.)-.p,._.(.)=A.f jj;-i)- ,, (A).
, , c ^- ' {t - zy
Differentiating this equality, we obtain
SO
'-T^*-%^''-»^-.« m-
This is the first of the required formulae. Next, from the identity
we deduce 0- f (^'-l)""'rfn [ 2^Hn-l)(^'-l)"- f tit^-l)"-^
or
_ f (<^-l)"-' ,, , [ 2 {«^-l) + li(n-l)(<'-l)»-^(J<
r (n-l){(<-^)-H ^1(^^-1)"-^
117] LEGENDRE FITNCTIOXS. 211
or
or
f._ n r «'-iy'-' n-i r {f-irjdt
~ 27ri .' c 2"-' (< - 2)"-' 2771 j c 2"-= (< - z)"-'
_("-iur (t'-i)"-' , •iwi Jc2"-'{t-zr '
or, by formula (A) above and Schlafli's formula,
0 = n[PAz)-zP„_,(z)]+(n-\)P„.,(z)-{n-l)zPn-d^),
or
nP„iz)-(-2n-l)2P„_,(z) + (H-l)P„_,(z)^0 (II),
a relation connecting three Legendre functions of consecutive orders. This is the second of the required formulae.
Other formulae can be deduced from (I) and (II) in the following way :
Differentiating (II), we have
,^^^^)_(2.-l).'^^^V(„-l)^-^^=(2.-l)P,.^^ Substituting for
dPn(z)
dz from (I), we have
„l/Jv^)+„p,._.^_(2„_l)/-^)+(„-l)'^^
= (2»-l)P„_,(2), or
-(» -\)Z j-^ — +(»-!) — j-^ — =-(«-l)-P,._l(2),
or
^%^^-'%^^^ = (.-l)P,.-.(.).
Changing (h — 1) to n in this equality, we have
,^^)_^^^^) = „p,.(,) (III).
Ne.\t, changing n to (« + 1) in (I), we have
^2^^->-/4^ = («+1)P„(.).
14—2
212 TRANSCENDENTAL FUNCTIONS. [CHAP. X.
Addiug this to (III), we have
^+lSlK^gi^=(^2n + \)Pn(z) (IV).
Lastly, combining (I) and (III), we obtain the result
{z^-l)^^^'''^nzP,iz)-nP„^,(z) (V).
The formulae (I) to (V) are called the recurrence-formulae.
The above proof holds whether n is an integer or not, i.e. it is applicable to the general Legendre functions. Another proof which, however, only ajjplies to the case when m is a positive integer (i.e. is only applicable to the Legendre polynomials) is as follows :
Write
V={l~2hz + h-)-^.
Then equating coefl&cients of powers of h in the equality
(l-2A^+^2)U=(j-A)r,
we have
nPA^)-{2a-l)zP„_,{z) + {,i-l)F„_,{z) = 0,
which is the formula (II).
Similarly, equating coefficients of powers of A in the equality
we have
_dP^{z) _dP„.,(z)
" dz ~~dz "'^"^^■''
which is the formula (III). The others can be deduced from these.
Eicample. Shew that, for all values of n,
{2n + Z) P\,,-{2n + \) P,? = ~{z{P„^ + P\,,)-2P„P„,,].
(Hargreaves.) For
j; (^ {Pn + -^"n + l) — 2P„P„ + ,} _/)2i P2 a.O-P '^"n.a.p dPn+\ .7 p ""n+1 pp """
\
= iV + P\,i + 2/'„{/'j^-_'i^j+2i>„,,{.
dP^^^ dP^^
dz dz J
= i'„2 + P^„,, + 2P„(-)i-l)/'„ + 2P„ + i(« + l)/'„,, (as is seen by using formulae (I) and (III))
= {2n + Z)P^„^^-{2n + \)P„\ which is the required result.
118] LEOENDRE FUNCTIONS. 213
118. Evaluation of the integral-expression for Pn{2), as a power-seines.
When n is a positive integer, we have seen that P„ (z) is a polynomial in z. When n is not a positive integer, however, P„ {z) is not a polynomial ; and as P„ (s) is not a regular function of s for all finite values of z unless n is integral, it follows that no power-series exists which represents P„ (2) for all finite values of z, when n is not integral. In order to find a power-series capable of representing P„ (z), we must tiierefore make some supposition regarding the part of the z-plane on which the point z lies. We shall suppose that z lie.s within a circle of radius 2, whose centre is the point 1 ; so that
|l-z|<2.
As the contour C of § 116 was subject oiil}- to the condition of enclosing t—l and t = z without enclosing < = - 1, it is clear that we can choose it so as to lie entirely within the circle of centre 1 and radius 2 in the t-plane, i.e. to be such that the inequality 1 1 — <| < 2 is satisfied for all points t on C.
Now write t—l = (z — l)j(. When t describes the contour C, the point representing the variable w will describe a contour 7 on the «-plane ; since C encloses the points t = z and t = l,y will enclose the points m = 1 and m = 0 ;
2 and since |1 — <l < 2, we shall have |m| < . ^"1 f"^^' ^'^ points u on 7.
\z—l\
Then changing the variable of integration from < to m in the integral which represents P„ (2), we have
■^" ^'^ = 2^-/ ^' "" {(^ - 1) « + -1" (" - 1)~"~' ^"
1 -I- ^-—^ u ] i(" (« - 1)-"-' du.
2Tn]y
2
Since \u\< ■. —\ ^^ ^^^ expand this in the form
Now on integrating by parts, we have the i-esult
r ~ ("m — 1^~"1 r-^nf
Jy \_y 11 J n Jy
The first expression on the right-hand side is zero, and so we have I !('•+" (« - 1)-"-' (in = -— [ «"■+"-' (« - 1 )-"-' (» - 1) du
. y n J y
= ^-^ ( H"-^" (u - 1)-"-' du - ^^^ I" «'■+''-' (u - 1)-"-' du, n Jy n Jy
or
/.
ii'-+" (u - !)-»-' du = '"-"'"— 1 «'■+»-' (u - 1)-"-' du.
y
214 TRANSCENDENTAL FUNCTIONS. [CHAP. X.
Therefore
V
Now transform the integral on the right-hand side, by writing u = - -. ■ The integral I m"(m — 1)~"~'(Z« becomes — I , where the integration
J y J ^ V 1
has now to be taken in the positive sense round a contour S enclosing the
points V = 0 and «> = go , but not enclosing the point v=\. This integral can
/' v'^clv be replaced by + I , where the integration has to be taken in the
positive sense round a contour 8' enclosing the point v=l, but not enclosing the points v = 0, or y = co (since the integrand has no singularities in the region between the contours B and S'). The contour B' can now be diminished until it becomes an infinitesimal circle surrounding the point v=l. The
value of the integral is then 1" ( , where the integration is taken round
this contour ; or 27n', since the many-valued function v" has been taken to have the meaning 1 at the point v=l. We thus have
If r
^r— . «'■+" ( K - 1 )"-' dll = -
J,TnJy
r + n r —1 + II 1 + n
r ' r —\
and on substituting this in the expression already found for P„ (z), we obtain
P / \_i _L. ? T^Jn-l) ...{n -r +\) jr + n) {r -I -\-n) ...{I +n) fz-iy
an expansion of P„ (s) as a series of powers of (2— 1).
If now, as in § 14, a series of the form
, a.b a(a + l)b(b+l) „ 1 . c 1 . 2 c(c+ 1)
(a hyper geometric series) be denoted by
F(a, h, c, z), then the expansion can be written in the form
P^{z) = F{-n, 71 + 1, 1. ^^
This is the required expression for P„ {z) as an infinite series. It is valid at all points z within the circle whose equation is |1 — z\<2.
119] LEOENDRE FUNCTIONS. 215
Corollary. Since this series is clearly unaffected when n is changed to
— n — I, we have
P„(z) = P_„_,(ir).
Xote. When n is a positive integer, the above series terminates and gives the expression
of /"„ {:) as a polynomial in —^ .
119. Laplace's integral-expression for P„ (z).
We shall next shew that, for all values of n and for certain values of z, the Legendre function P„ (2) can be represented by the integral (called Laplace's integral),
~r{z + cos <l)(z--l)i]"d<f>.
When n is not an integer it is necessary to state which of the branches of the many-valued function in the integrand is to be taken : we shall take that branch of the function {z + cos (/> {z- — 1)*]" which reduces to unity when taken by the process of continuation along a straight path to the point z=l. It will appear later that it is immaterial which branch of the two-valued function {z' — 1)' is taken.
(A) Proof applicable onhj to the Legendre polynomials.
When n is a positive integer, the result can easily be obtained in the following way. We have
i /i-P,. {z) = (1 - -Ihz + h-)-K
n = 0
But ( 1 - -Ihz + h-)-i = - -j T-^ — j-^ — ^xT T ,
IT J 0 {I — nz) — h{z- — iy cos <p
as is seen by applying the ordinary formula for the integration of
J a + b cos (f) ' Expanding the integrand of the integral in ascending powers of h, we have
(1 - 2hz + /r)-i = - i A" ('{z + cos <p (z^- l)il"#, "■ «=o Jo
and on equating coefficients of /;" on the two sides of this equation, the
required result is obtained.
As however the theorem is true whether n is an integer or not (i.e. as it
is equally true for the Legendre functions and the Legendre polynomials),
it is necessary to have a general proof independent of the character of n;
this will now be given.
(B) General proof.
First, we shall shew that Laplace's integral satisfies Legendre's equation.
216 TRANSCENDENTAL FUNCTIONS.
For if we write
1 f y=- {z + cos<f){z'-l)i]''d<ji,
we liave
[chap. X.
dz^
= - I (2 + cos (/> (z^ - 1)1)"-= [n sm^(f> - 1 - 2 cos 0 (2= - l)-i) d0.
But
I {z + cos (f) {z'' - l)ij"-= sm-(f)d<j)
JO
~n ~
= - {2 + cos<^(22-l)ij''-='sin(^coS(^ _o
+ j cos (^ -TT [sin <f>[z + cos (^ (2^ - 1)*}"-=] d<f)
= 1 (2 + C0S^(2=-l)i}"-=iC0S°-(^d^
•'0
- (n - 2) / {2 + cos 4, (z- - 1)1)"-' COS <f> (z- - 1)^ sin-^dcf) .'0
= f {2 + cos (/) (2= - l)^Y'--d<f> - (»i - 1) f " {2 + cos 0 (z- - 1)1)"-= sin=(/)rf0 JO Jo
+ {n - 2) 2 r3m-<pd(f> [z + cos ^ (2- - 1)*)"-^ Jo
Therefore
?j I (2 + cos (/) (2= - l)il»-2 sin=^ci^
JO
= ( [2 + cos^(2"--l)iJ''--d</> + Cn-2)2 |''{2 + cos<^(22-l)4)»-'sin=(^df Jo 7o
Thus we have
= - (n - 2) 2 f ' (2 + cos ^ (2' - 1)*)"-' sin=<^d^
- - 2 (2^ - l)-l f " [2 + cos </) (2= - l)ij"-= cos (^d(/) TT Jo
= _ ^ 2 (2= _ 1)-J f A [U + COS ^ (2^ - l)i)"-= sin 0] c?^
TT Jo <*<»
^0,
119] LEOENDRE FUNCTIONS. 217
•which shews that Laplace's integral satisfies Legendre's equation, whatever « and z may be.
\—z
Now suppose that z is nearly unity, and put = u. Then the integral
becomes
1 T"
- (1 - 2» + cos ^ (- 4» + w=)l}" di>,
•which for small values of it can be expanded in the form
1 + L f 'd^ I »(»-!).. (.-r + l) ,_ 2^^ ^ ^^^ ^ ^_ ^^, ^ ^,,^j,
TrJo r=l » •
This is a series of powers of «*; the first terms (neglecting u^) are
If IT'
1 + linii^ - cos ^d4> - 2nu - Ml + (« - 1) cos'rf)) d(^,
or 1 — Inu —^ — ,
or 1 - n (n + 1) u.
It is clear that odd powers of v^ can arise only in conjunction with odd powers of cos <f> in the integrand, and so here vanish when integrated. Laplace's integi-al can therefore, when u is small, be expanded in ascending powers of u in the form
1 — « (?i + 1) ?( + (IM- + (Is't" + OiU* + ... .
But the coefficients a^, a^, ... can be found by substituting this expression in Legendre's equation, and equating to zero the coefficients of each power of u. We thus find that
_ , , . ^ ?i(>i -!)...(» -?•+ 1).(1 +»)■■■ (r-1 +n)(r+n) ■
""■"^ ?•!»•:
and thus Laplace's integral is equal to
F{-n, n + l, 1, ^^),
or (§118) to P„(z).
We thus have, for all real or complex values of ?;, the result
P„iz) = - r[z + cos <f>iz'-l)iy^d<}>.
IT J 0
It must be observed that as the power-series Fy — n, n + \, 1, — ^^^
was used in the proof, this proof is valid only for values of z which satisfy
218 TRANSCENDENTAL FUNCTIONS. [CHAP. X.
I 1 —2 I the inequality - — ^— < 1. As however P„(i) is an analytic function of z,
the result will be true for a more extended region including this, provided the integral
-I'
TT Jo
is an analytic function of z within this more extended region : since if these two expressions are equal for any region however small in which they are analytic functions, they must be always equal so long as they remain analytic functions. But it is easily seen that for the integral
-r
TT Jo
z + COS (f> (z- - lf}"d<f).
every point on the imaginary axis in the 2-plane is a singularity : and therefore the region in the ^-plane for which the equality
P„ (z) = i ['[Z+ cos <f> (z"- - 1)*)" d<l>
TT Jo
is established is the region for which the real part of z is positive. Corollary. Since
we have for all values of n, real or complex, the result
Pn (Z) = - r{z + cos <}> {Z"- - l)i}-»-' CZ</,, 71" Jo
so long as the real part of z is positive. Example. If
{i-uLe+h^y = Jo ^' ''"' '^' ''^°'' ' < ''
shew that
, 2 . , . /•! h'x'
[l-xhy
(Binet.)
120. The Mehler-Dirichlet definite integral for Pn {z).
Another expression for the Legendre function as a definite integral may be obtained in the following way.
For all values of n, we have by the preceding theorem
Pn (2) = - Tl^ + COS <h {Z^ - l)i|" dcf>. 71" Jo
In this integral, replace the variable ^ by a new variable h, defined by
the equation
h = z + {z- — 1)* cos (f>
1
120] LEOENDRE FUNCTIONS. 219
SO that
and
We thus have
aud therefore
rfA == •- ( :■ — 1 )* sin <f)d<p, ■: (1 - ifi: + /r)4 = (z- - l)i sin (f>. idh _ .
, -i dh.
Xow write s = cos ^. Thus
P„ (cos 6) = * P' ., A" (1 - 2/i cos (9 + /r)-* f^/*. Writing A = e'*, this becomes
P„ (cos 0) = -^- j _^ ^2 cos <^- 2 cos g)^ ■
or
D / /3^ 2 r" COs(»l + ^)<f) ,
P„ (cos ^) = - rj,7 4 ^j,,. rtl/>
^ TTJo 2(cos<^-cos ^) 4 ^
This is known as Mehler's simplified form of Dirichlet's integral. The result is valid for all values of n.
Example 1. Prove that, when n is a positive integer,
cos <^)} i '
For we have
_ , ^> 2 r' sin »+i P„ (cos fl) = - ,, ; -J- ^ ' "■ j 9 {2 (cos e - <
jo a4-6-(a-6)cos«)~2a*6* Put
2| = (y-l) + Cy+l)cos!(.'. The equation becomes
(1 Writing ^ = co8 (ft, y = coa 6, this gives
TT^ _ /■" (1+A)rf|
-2Ay+A»)i- j _, "(1-2^|+A2) {y+| (»/-l)-^}5 ' y=cos 6, this gives
(l-2Aco3i9 + A-)-i = - /''(l+/0sin(^(l-2/»cos<^ + /r)(l+cos(^)-i(cos5-cos<^)-i(;(>.
Equating coefficients of /(" on both sides, we have
_ , ^, 1 T" sin(?i + l)d>sin 0f/(i)
i*, (costf) = - ^ ,
^ J sin ^ cos ^ {2 (cos 5 - cos <^)Ji
S20
or
TRANSCENDENTAL FUNCTIONS.
[chap. X.
P„{coae) = - I {2(oos5-co8(/))}-»-^in(»+i)(^fl?<^. n J g
Example 2. Prove tliat ■' •.
A"
.(oose) = 2^./
(A2-2Acosd+l)»
^^''*'
the integral being taken along a closed path which encloses the two points /i = e*'*, and the conventional meaning being assigned to the radical.
Hence (or otherwise) prove that, if 6 lie between J n and f w,
^(003^) =
4 2.4,..2n , cos (nd + (j>) 1' cos{nS + Z<t>)
n 3.5...(2»-fr) I (2 sin e)i 2 (2m+3) " (2 sin (9)3
12. 3* co3(nfl + 5(^)
■ 2.4.(2» + 3)(2n + 5) ^(2"s[n fl)« "
where (j) denotes id- Jjt.
Shew also that the first few terms of the series give an approximate value of P„ (cos 6) for all values of 6 between 0 and n which are not nearly equal to either 0 or rr. And explain how this theorem may be used to approximate to the roots of the equation P„ (cos d) = 0.
(Cambridge Mathematical Tripos, Part II, 1895.)
121. Expansion ofPn{s) as a series of powers of-.
We now proceed to find an expansion of the Legendre function which is valid for large values of z.
If the real part of z he positive, we have for all values of n (from Laplace's integral)
P„ (z) = 1 r [s + (2= - l)i cos 0j" d<}). Now suppose that \z\ is very large: then this can be written in the form
Expanding the integrand in ascending powers of - , this gives Pn (^) = 5 /J {(1 + -^-^^ </>) - '-£r + • • •}"#
= ^'' [" 1(1 + cos </,)" - ''-^ (1 + COS </,)»-' + . . . I d(/).
We can evaluate
I (1 + COS 0)" c?<^ and I cos (^ (1 + cos 0)"-' c?<^ Jo JO
121, 122] LEOENDRE FUNCTIONS. 221
by putting <f> — --^ and using the result
IT
and thus we find that i'„ (z) ciin be expressed by a series of powers of - , the first two terms of the expansion being given by the equation
"^''~v/;rr(« + l)l (2n-l)2z^^^-r
The general law of the coefficients in the series can without difficulty be found by substituting in Legendre's differential equation (§ 114) ; and in this way we find that P„ (z) can he expressed by the hypergeonietric series
2-z-V{n+\) [\-n nil
I
)
T(H + l).7r» \ -1 ' 1' 2 ' z in the notation of § 14.
This series has only been proved to hold when z is large and the real part of z is positive : but by § 14 it converges, and so represents an analytical function, over all the area outside the circle of centre 0 and radius 1. The series therefore represents P„ {z) over this region.
122. The Legendre functions of the second kind.
Hitherto we have considered only one solution of the Legendre differential equation, namely P„ (z). We can now proceed to find a second solution.
It appears from § 114, that the differential equation
is satisfied by the integral
j{t^-lY{t-z)-"-'dt,
talven round any contour such that the integrand resumes its initial value after making the circuit of it. Let i) be a figure-of-eight contour in the <-plane, enclosing the point t = + I in one loop and the point t = — l in the other, and not enclosing the point t= z. Then after describing this contour, the above integrand clearly resumes its initial value, since it acquires the factor g»in after describing the first loop, and this is destroyed by the factor e''^'^'" acquired during the description of the second loop. D is therefore a possible contour.
222 TRANSCENDENTAL FUNCTIONS. [CHAP. X.
A solution of Legendre's equation is therefore furnished by tlie function Qn (2), if Qn (•j) be defined by the equation
Q,. (z) = .^ I 4 (<— 1 )" (z - t)—' dt ;
4t sin n-jT J d"
it is supposed that, in describing D, the point t makes a positive, i.e. counter- clockwise, turn round the point t = —\, and then a negative, i.e. clockwise, turn round the point t = +\. The significance of the many- valued functions {t- — 1)" and {t — «)"""' will be supposed to be fixed in the same way as before.
Another form of the integral may be obtained in the following way.
Let the contour become so attenuated as to consist simply of a line joining the points — 1 and -h 1, described twice, and two small circles round the points — 1 and + 1 : when the real part of ()i -t- 1) is positive, the parts of the integral arising from these two loops are at once seen to be infinitesi- mal; and thus we have
I (r- - 1)" {t - 2)-"-i dt = (e""' - e-'"'') X I (1 - t-y (t - 2)-"-' dt
2i sin nir ( 1 - f^ (t - 0)-"-' dt,
-1
so Q„ (2) = gil / \ ( 1 - t"-T i^ - i)-"-' dt.
This last result is valid when the real part of (« + 1) is positive. When n is a positive integer, the original definition of Qn{z) becomes undeterminate: in this case we can use the formulae just found.
Qn{z) is called the Legendre function of the second kind and of order n.
r ■. 123. Expansion of Qn (2) as a power-series.
We now proceed to express the Legendre function of the second kind as
. . 1 a power-series m - .
We have, when the real part of (;? 4- 1) is jjositive,
Qn (^) = 2^ /\(1 - *')" (^ - *)"""' f^^-
Suppose that j^j > 1. Then the integral can be expanded in the form
< V (« + !)(« + 2)... (n + r), ^^
r!
,\ el
dt
123] LEGENURE FUNCTIONS. 223
as is seen on writing r for 2is, since the integrals arising from odd values of ?• obviously vanish.
Writing <•= u, we can evaluate the coefficients of powers of- as follows:
I ( 1 - (■)" r-'ilt = I ( 1 - ")" u'-idu
I
f
I
= £(«+l,A-+i)
^r(n + i)r(s+^)
and thus the formula for Q„ (s) becomes
O /.^_ ^* r(«+l) 1 pfn+l «^2 3 1\
V» \^) 2''+' r (n + f ) 2"+' V 2 ' 2 ' 2 ' W ■
This is the expansion of the Legendre function of the second kind as a power-series in - , corresponding to the expansion obtained for P„ (z) in § 121.
The proof given above applies only when the real part of {n + 1) is positive ; but a similar process can be applied to the integral
the coefficients being evaluated in the same way as those which occurred in
1—2
the expansion of the Legendre function P„ (z) in ascending powers of — ^ ; the same result is reached, which shews that the fonnida
ttJ r(n + l) 1 /n + 1 n + 2 3 1
is true for all values of n, real or complex, and for all values of z represented by points outside the circle of centre 0 and radius unity.
Example 1. Shew that, when ii is a positive integer,
'We CAD write Legendre's differential equation in tlie form
(1-.^ J-2.-J + «(n + l)« = 0. It is easily verified that this equation can Ije derived fmni the equ.ition (1 -.'•-) J+2 (« - 1) .- ^^+2nx=0,
d".v by difierentiating « times and writing u= -j^ .
224 TRANSCENDENTAL FUNCTIONS. [CHAP. X.
Now one solution of the latter equation is x={z^- 1)" ; and a second solution can be derived by the ordinary process for finding a second solution of a linear differential equation of the second order, of which one solution is known. Thus two independent solutions of this equation are found to lie
{z^-\Y and (C--1)" / (i)2- l)-n-irfj,.
It follows that
is a solution of Legendre's equation. As this expression, when expanded in ascending
powers of -, commences with a term in 5"""', it must be a constant multiple of Q„ (2) ; and
on comparing the coefficient of j-»-i in this expression with the coefficient of j-"-' in the expansion of <?„ {z), as found above, we obtain the required result.
Example 2. Shew that, when n is a positive integer, the Legendre function of the second kind can be expressed by the formula
/oo rx (-00 r xi
I / ... / (t)2-l)-»-i(rfr)'' + i.
For on expanding the integrand («-- I)"""' in ascending powers of -, the right-hand side of the equation takes the form
jjoj.-jv^' Vn^2 + „2,..4+ 2!i;2''*8 +-J'
and on performing the integrations this becomes
(2?i + l)(2re-l)...3.1 V + i"*" 2(2?i-l-3) 2» + 3+— /' or§„(2).
Example 3. Shew that, when n is a positive integer.
This result can be obtained by applying the general integration-theorem
to the preceding result.
124. The recurrence-formulae for the Legendre function of the second f kind.
i The functions Pn{z) and Qn{z) have been defined by integrals of pre- «
cisely the same form, namely \
{t--iy'{t-3)-"-'dt.
It follows therefore that the general proof of the recurrence-formulae for P,i(2), given in §117, is equally applicable to the function Qn{z)', and hencel that the Legendre function of the second kind satisfies the recurrence-formulae
12-i, 125] LEGENDllK FUNCTIONS. 225
-dz- -'—dr- = "^-'-'^'^'
nQ„ (z) - ('2„ - 1 ) zQ„_, {z) + (H-\) Q„_, (z) = 0, ^ dQ., (z) dQ,._, (z)
(Iz dz
= nQ„{z),
—Tz ^— =(2»+l)Q„U),
(z= - l)'^^2z^ = nzQ. {z) - nQ,^, (z).
125. Laplace's integral for the Legendre function of the second kind. Consider the expression
ij=\ [z + cosh.e{z"-\)i]-''-'de, Jo
hi which z is supposed not to be a real negative number between — 1 and — », and the real part of (?i + l) is supposed to be positive; under these conditions the integral certainly exists.
If now we form the quantity
(which occurs in Legendre's differential equation), we find for it the value
-()(+!)=[ [z + (z- - l)i cosh d]-"-' sinh- 0dd Jo
+ (n + \) I [z + {z^ - 1)* cosh e}-"-' d0 Jo
+ {n + l)z (z' -!>-*[ {z+ {z- - 1)4 cosh d]-'"' cosh ddd. Jo
This expression can be transformed, by integration by parts, in exactly the same manner as the corresponding expression found in the discussion of Laplace's integral for P„ (2), in § 11!); and thus it is found to be zero. The quantity y therefore satisfies Legendre's equation.
In order to compare y with the solutions P„ (2) and Q„{z) which have already been found, we suppose that | ^ | is large, and write y in the form
2-"-' f {1 + cosh d(l-^^ + . ••)}'""' d0,
W. A. 15
226 TRANSCENDENTAL FUNCTIONS. [CHAP. X.
which when expanded as a power-series becomes
a, a., Ui ,, .
where a„= f {I + co^h 0)~''-' dO
J 0
= fjT.-"il-^)-^du, where . = 3-^-A__,
= ^^xjjB (»i + 1, ^), where B is the Eulerian
TTJ r(?t+i)
■2«+i r(n. + f)"
integral of the first kind,
Now any expression of the form (1) which satisfies Legendre's differential equation must be a multiple of Quiz) (since, by substituting the expansion in the differential equation, we can determine the coefficients a^, «,, as, . . . uniquely in terms of a,,, which shews that all expressions of the kind are multiples of any one of them); and as the value found for «„ is equal to the coefficient of the initial term in the expansion of Qn (z), we have
y=Qn (z)-
Thus we have the result
Q„ (z) =j{z + (z' - 1)5 cosh ^j-"-' d9, Jo
which may be regarded as the analogue of the Laplace's integral already found (§119) for P„ (2).
The theorem is valid only when the real part of (;i-|-l) is positive; and the proof has assumed that | 2 | > 1 ; but the equivalence of Q^ (z) and the integral, having been proved to subsist for this range of values of z, must continue to subsist for all values of z, continuous with this range, for which the integral continues to represent an analytic function of z ; and hence the theorem holds for all values of z except those which are real and less than
— 1, which are singularities of the integral.
126. Relation between Pn (z) und Qn (z), when n is integral.
When n is a positive integer, and z is not a real number between 1 and
— 1, the functions Qn{z) and P„{z) are connected by the relation
Qn(z) = ir Pniy)^,
which we shall now establish.
126] LEGENDKE FUXCTIOXS. 227
When j| > 1, wo havo
Now if ()( + A-) is an odd integer, we liave
f 1\ (:/) fdy = f P„ (y) y'chj - f ' i^. (y) 7/% = 0.
J -\ .'11 .' (I
It' n is loss than A', aiul {11 +k) is an men integer, we have
Jf PAy)!i'd>j=\'Pn('j)!My - . -1 (I
1 r^ rf"
(by Rodrigues' theorem) = ^rr^,^^^ f ^^„ (i/^ - 1)" d<j
(integrating by parts) = .,„^ , A- (A- - 1) . . . (A - » + 1 ) f /-" (1 - ff dy
-II- J 0
A(,A-- IH^'- -)••• (A-- )i + 1)
(i + n+ l)(A;+n-l)...(A;-n+l)'
If on the otlier hand k is less than n, and (/! + A') is an even integer, the same process shews that the integral vanishes.
Therefore
\ r p mJI- = S A-(A--l)...(A-«+l) J_
2 j _j^nu;^_y ^ (^. ^^ _^_ ^y (k + n-l)... (k-n+l) ^*+i '
where the summation is taken for the values A' = n, n + 2, n + i, n + 6, ... 00 . But this expansion, by § 123, represents Q,, (s). The theorem is thus established for the case in which |^j > 1. Since each side of the equation
represents an analytic function even when | ^^ | is not greater than unity, provided z is not a real number between — 1 and + 1, it follows that, with this exception, the result is true universally.
Example. Shew that (?„ (z), where n i.s a positive integer, is the coefficient of /t" in the expansion of(l -•2/(£ + A-)-icosh~i \— A .
For
i AǤ.(.)= 2 i" r ^-(y->-^^
-2J_, » (z-y)
= (l-2zA+A2)"*cosh-'/- ''~^ \.
15—2
228 TRANSCENDENTAL FUNCTIONS. [CHAP. X.
127. Development of the function (t — a:)~' as a seines of Legendre polynomials in x.
We shall now obtain an expansion which will serve as the basis of a general class of expansions involving Legendre functions.
We have, by the recurrence-formulae,
( 2n + 1 ) a;P„ (a;) - ( « + 1 ) P„+, («) - mP„_i («) = 0,
(2n + 1) zP, {z) - (n + 1) P„+, {z) - nP,,-, (z) = 0.
Multiply the first of these equations by P„(2), the second by Pn (■«), and subtract : we thus obtain
(2n + l){z-x)P„(z)Pn(x)
= (n + 1) (P„+, (z) Pn {x) - P„ (z) P„+, (x)]
- n {P„ (Z) Pn-, (X) - P„ (x) Pn-, (Z)}.
Write 71 = 0, 1, 2, 3, ... «. successively, and add the resulting equations. This gives
\Po (x) P„ {z) + 3P, (x) P,(z)+...+{2n + l) P„ (x) Pn (z)] (z - x)
= {n + 1) {Pn+, (Z) Pn {X) - Pn^, (x) P„ (^)j.
Divide throughout by (s — x) (z — t), and integrate from z = —l to ^ = + 1.
Thus
" '•+1 Pr(z)
" r+1 P (z)
2 {2r+l)Pr{x)^^dz oJ-1 z-i
^\(^z-x)lz-t) f^"+^ ^"^ ^» ^""^ ~ ^"+' ^*'^ ^^ ^^^5 ^^
(by partial fractions) =
^' ^3^ {^''+' (^) ^" (*"> - -P«+' (*■) ^" (^)l '^^ ^'^ (P„+: (^) P„ (,r) - P,„+, (^) P„ (^)) ci^
— \Z — 5
Now by the result of the last article, the left-hand side .of this equation can be written
-2i(2r+l)P,(«)a(0-
0
In the first integral on the right-hand side, replace the integrand by its
n
value 2(2r-|- Y)Pr(x) Priz), and integrate ; only the first term survives, since
0
^^^P,{z)dz=0, when r is an integer greater than zero ; so the integral has the value 2.
127] LEGENDRE FUXCTIONS. 229
We thus have ^ ('2r +l)I\ (.<•) Q, (0 = -— + J^ IP„ {.>■) Q«+. (0 -i"«+. (^O Qn (01-
r = 0
t — X t — x
This oiiuatiou is valid for all values of n. Let us now see if x and t can be so chosen as to make the last part of the right-hand side tend to zero as n tends to infinity. We have, from Laplace's formulae for the functions
P. (.) Q... it) - iV. (.) Q. it) = I fJl ^^^^l\ A d,pa^,
where A denotes a quantity which is finite and independent of?;.
It is clear that this double-integral tends to zero only when, for all values of <f> between zero and tt, and all values of -v^ between zero and infinity, the inequality
|a;-»-(a,--l)icos<^
is satisfied.
< + (^'-l)*coshi|f
<1
Wntmg ^. = ^^« + _J, t = -^^: + -
the inequality becomes
\u+- -\-[u cos 0 < v+ +[v cosh ySr .
I i( \ 11/ i r \ vj ^ \
The left-hand side of this relation has its maximum value when cos (/> = !, the value being 2 | w j.
The right-hand side similarly has a minimum value equal to 2|v|.
The condition thus becomes
I «. I < j D I
or \x+ix''-l)'>\<\t + (t--l)i\.
This inequality shews that the point x must be in the interior of an ellipse, which passes through the point t, and which has the points -I- 1 , — 1 for its foci : for if a be the major axis of this ellipse, then
t = a cos 6 + i (a- — l)i sin 0,
where 0 is the eccentric angle of t in the ellipse ; and thus
(t^ - 1 )i = (,t= - l)i cos ^ -1- ia sin 0,
and t + if--l)i={a + (a"- - 1 )*} e''«,
so that \t + (f^-l)i'^ = u + {a--l)K
and hence the above inequality shews that the semi-axis of the ellipse which passes through x is less than the semi-axis of the ellipse which passes through t, i.e. that x is within the ellipse which passes through t.
230 TRANSCENDENTAL FUNCTIONS. [CHAP. X.
Hence if the point x is in the interior of the ellipse which passes thruuyh the point t and has the points + 1, — 1, for its foci, then the expansion
^ = i (2« + l)P„(^)Q„(0
is valid.
128. Neumann's theorem on the expansion of an arhitrarj function in a series of Legendre polynomials.
We proceed now to discuss the expansion of any arbitrarily given function in terms of the polynomials of Legendre. The expansion is of special interest, inasmuch as it represents the case which stands next in simplicity to Taylor's series, among expansions in series of polynomials.
Let f(z) be any function, which is regular at all points in the interior of an ellipse C, whose foci are at the points z = —l and z — + l. We shall shew that it is possible to expand f{z) in a series of the form
ttoPo (z) + OiP, {z) + a,P, (z) + a,P, {z) + ...,
where Wo, rt-i, a.^... are independent of z: and that this expansion is valid for all points z in the interior of the ellipse G.
For \et z = t be any point on the circumference of the ellipse.
Then we have
/(^) = ^i /5-? = ^il/^'^ * lo^'" "- ^^ ^" ^^> «" ^'^' or f{z)= t anPn{z),
M = 0
2h, 4- 1 /" where a„ = ^r-^ f{t) Q,, it) dt.
iTTl J c
This is the required expansion.
Another form for a„ can be obtained in the following way. We have
2u+l fi 1 f f{t)dt
= ^jj\y)Pniy)dy.
The latter is the more usual form for «„.
1:28,129] LEOENDRE FUNCTIONS. 231
Example 1. Shew that the semi-axes of the ellipse, within which the series
converges, are
where p is the radius of convergence of the series
Example 2. If
\y+\J ' {x+\){y-\y
prove that
f ' * .={(-1^+1) (y- 1))- 2 P^ {x) Q„ii/).
129. Tlie associated functions Pn"^{z) and Qn'"(z).
We shall now introduce a more extended class of Legendre functions.
If Hi be any positive integer, the quantities
(1— 2-) - , J ■ and (1—Z-) , ,,
will be called the associated Lecjendre functions of the nth degree and nith order, and will be denoted by P,,'" {z) and Qn'" {z) respectively.
We shall first shew that tlie associated Legendre functions satisfy a differential equation analogous to the Legendre differential equation.
For let the Legendre differential e(|uation
be dififerentiated m times, and let v be written for ^r-^ .
dz™
■ W^e thus have for > the equation
(1 - ^■) Xs -^z{m + \)--=- + (n - m) {n + ni + l)v = 0.
m
Write iu = V (1 - z'-y^ ;
the equation becomes
This is the diflferential equation satisfied by the functions
P,r(z) and Q,r(z).
232 TRANSCENDENTAL FUNCTIONS. [CHAP. X.
Several expressions for the associated Legendro functions can be obtained easily from the above definitions.
Thus from Schliifli's foninila, we have
1)1 = (n-^lHn^^...in + ..) (1 ^p^ f ^^, _ ^^„ ^^ _ ^^_„_„_, ^^_
where C is a simple contour enclosing the points t = l and t = 2, but not enclosing the point t = — 1.
From this result, or from Rodrigues' formula, we have, when n is a positive integer,
130. The definite integrals of the associated Legendre functions.
The theorems already given in § 115, relating to the definite integrals of the Legendre functions, can be genei-alised so as to be stated in the following form : When in and n are positive integers,
I
+1
P,,'" (2) Pr™ (z) dz = 0, when r ^,n,
and r'A"'(^)}'^^^ = ^-Tr^-
J _i ' Zn-\-\ {n—my.
To establish these results, we use the identity
(z"--\f
dzu-m^ > (?l + m)!' ' rf2"+"'
which gives
jjP^r.^z)r-dz=j_^^^a-zT{^^^\ dz
^1 f (n-^m)\
22»(7i!)^ j_i(n-m)!^ '
(integrating by parts) = ^^ ^S-^l^^^'"^ ''
(n + m)! n ,p / s,„ ,^
2 (w + m)!
2ft + 1 (?i — m)!"
130, 131] LEOENDRE FUNCTIONS. 233
We can prove in the saiin" way tl>e otlier result stated, namely that I P,,'" {z) Pr'" (2) dz = 0, when r 4= n.
For this integral in the same manner reduces to a multiple of
\''' P„{z)Pr{z)dz, which is zero when n and r arc different.
131. Expression of P„"'{2) as a definite integral of Laplace's type.
The associated Legendre functions can be expressed by means of definite integrals of the same t\-pe as those found in § 119 and § 125, as will appear from the following investigation.
O CI
We have
I [z + cos </) (s- - 1 )-*]•"-"' sin-'" 4,(l(f>
[z + cos 0 (5- - 1 )i;"-"' sin="'-' ^ cos </>
d . .
]
+ 1 cos (^ |- [sin-"'-' </) [z + cos </> {z- - lf-\ "-'"] d<^ Jo dcp
= (2m - 1) I "cos^ <f) sin-'"-- 4, {z + cos <^ (2= - l)Jj''-'» d<p J 0
- (h - Hi) I [0 + cos 4> (z- - l)i}»-'«-i cos <j) (z- -l)i sin-"' 0f7(^
. II
= (2m - 1) f ''sin='"-= (f){z + cos <f) (z'' - l)4}"-"» d^ .'0
- (2m - 1) [ sin-'" 0 [z + cos <^ (z- - 1 )ijn-'n rf^
- ()i - m) I 'z+ cos <t> (z- - l)ij"-"' sin^'" (/)C?<^
+ (n -m)z j {z + cos (f> (z- - l)J)"-"'-> sin-'" (^c?(/>. We thus have
(n + m) lz + cos <f) (z- - l)*}"-™ sin*'" 4,dd>
Jo
= (2m - 1) f sin-'"-= <^ (2 + cos <^ (2= - 1 )*)"-'" d<j> Jo
- . ^ . (2 + cos (^ (2' - 1)J|»-'" sin«"-' (^ (2 — 1)» |_o
„ ^ ■ . . f {2 + cos (^ (2= - 1)4)"-"' (2 /H - 1) sin-'"--<^ cos 0c?(^,
(2^
234 TRANSCENDENTAL FUNCTIONS. [CHAP. X.
or -— [2 + cos</>(^''-l)4i"-"'sin-'"<^f/0
-HI — 1 J u
= /""siu^"'-^ <P\Z + COS 0 (3-- - 1)1]"-'" {1+2 («= - l)-i COS </)) cZ(/>
1 ci f"
= TT J- (^ + COS d>(2--l )»}"-"■+> sin-"'-^ (ifZrf).
?i — m + 1 a^ j 0 Thus if we write
/,„ = {z + coii(f,(s--l )J)"-'« sin^'" (^(Z()^,
0
we have /.„ = ^— ^- ^^—^ ^"f=-^ ,
{n + 7n){n— m + 1) dz
and therefore /,„ = , (2;» - D (2»^- 3) - 1 ^ _
{n + »i) (;i + //I - 1 ) . . . (/I - ?/i + 1) ^2"'
But /„ = {"[z + cos ^ (^= - 1 )!)" fZ<^ = ttP,, (2),
Jo
when the real part of z is positive.
Therefore /». = (-.,,,)(, ^ ^, _ ^^ ,,_ ^,^_,„^ ^ ^^ ^.P«(.)
^ (2m-l)(2».-3)...l.^ _ -f ^
(« + m)(ft + m-l) ... (n-m + 1)'^ ^ ^nW.
or p,r (z) = ^" +;"^ ^" +;;r ^ > .- ■; ^" - "^ + ^ )( i - .0^"
(2m— l)(2m — 3) ... 1 .TT ^
X ('{z + cos ^ (z- - l)ij"-'"' sin-'"- 0fZ(/). .'0
This result expresses P^{z) as a definite integral of Laplace's type, valid for all values of n whea the real part of z is positive.
132. Alternative expression of P„"' (z) as a definite integral of Laplace's type.
The formula last found can be replaced by another result, found in the following way.
If in Jacobi's well-known theorem *
JV (cos 4>) COS m<i>dj> = 13 5.,\2»t-l) C"^'"" *-*'°^ "^^ ^"'"" '^^'^' we take /(cos ^) = [z -{- {z- - l)i cos ^}",
* Crelle's Journal, xv.
132, 133] LEGENDHE FUNCTIONS. 235
SO that
m
/""'(cos<^)=/i(» -1)... (n-iii + I) {2-- 1)- {z + (2-- l)icos<^l"-"', we obtain
I [z + {2-— l)^ con <f}]" cos III (f) dtp Jo
_ n (n -1) ■■• (>i- >n +_1) /,» _ -i^f 1.3.5...(2m-l) ^"^ ''
X f " [^ + (2' - 1)* COS (^)"-™ siir'" ^d(f)
m
(n + m)(H + /H- 1) ... (/i + 1) " ^ ''
(ft + »i) (71 + m — 1) ... (ft + 1} ^ ^ ,f
Therefore
TT
X [s + {z^ — 1)4 COS (/){" COS m(j)d<p. ■ 0
This formula is valid for all values of », and for all values of z whose real part is positive; hi being a positive integer.
133. The fumtion Cn' {z)-
A function connected with the associated Legendre functions P,,"' {z) is the function C^" {:), which for integral vakies of v is defined to be the coefficient of A" in the expansion, in ascending powers of k, of the quantity
(l-2A2 + /(2)--.
It is easily seen that €„" {£) satisfies the differential equation
I
(Py (2iH^ dy _ n{n + 2v) d^^ 22-1 dz z'--l ^
.dt.
For all values of n and v, it may he shewn tliat C^ (z) can be defined by a contour- integral of the form
Constant xa-.=)J-^j^L_ii__
When n is integral, we have
C-f~w (-2)'i.(.< + l)...(./ + ?t-l)
" ^ '' Ji! (2?i + 2i.-l)(2re-t-2.'-2)...(M + 2v)
which corresponds to Rodrigues' formula for /'„ (z) ; in fact, since
PAz) = C:^{z\ Rodrigues' formula is a particular case of this formula.
236 TUANSCENDEN'TAL FUNCTIONS. [CHAP. X.
Wlien >• is an integer, wo have
C'-"* (z) = I ^ p U)
"-n-A^I (2r-l)(2r-3)...3.1 d^ "^'^
whence we have
(f^^ (,) = {\^ff± J, , , .
S;-rW (2r-l)(2r-3),..3.1 " ^''
The last equation gives the connexion between the functions (',,''(2) and P^ {z).
This function C^" {z) has the following furtlier properties, analogous to the I'eourrence- foriuulae,
j<-(2)=(«-i+2,o^c;_j(z)-2.(i-32)c;:'^(4
Miscellaneous Examples.
1. Shew that when n is a positive integer,
where ti? is to be replaced by (1 -2-) after the differentiation has been performed.
2. Prove that when n is a positive integer,
(Cambridge Mathematical Tripos, Part I, 1898.)
3. Shew that
. _, TV t, fl.3.5...(2«-l)12
(Catalan.)
4. Prove that
P ,, „.dP„dP„,, jJ^^-'-^^-dz'^'
is zero unless m- ?i= + 1, and determine its value in these cases.
(Cambridge Mathematical Tripos, Part 1, 1896.)
5. Shew (by induction or otherwise) that when 11 is a positive integer,
(2n + l)JV„2(2)& = i_,P„2-2.-(P,H/'2- + ... + /^„-i) + 2(AA + A^3 + -+^n-A).
(Cambridge Mathematical Tripos, Part I, 1899.)
6. Shew that, if k is au odd numl ler,
^- ^=2a„/'„(2),
{l-2zh + h"-f
MISC. EXS.] LKOENDRE FUNCTIONS. 237
where
» il-h"-)"-'- 1.3.5. ..H--2)\ d.i'di/J •'
where .t and^ are to be rojilacecl liy unity after the difterentiations have been perfoniied.
(Routh.)
7. If
n-o
shew that
2(» + l)«„ + ,-3(2«+l)/f„ + (2«-l)^„_, = 0 and
nH„+li\..,-zR,; = 0 and
4 (4.-3 - 1) /f„"' + 96.-2/f„" _ . (1 2n2 + 24;< - <)1 ) A',,' - >i (2« + 3) (2« + 9) /?„ = 0,
where
^"' = -— , etc. (Pincherle.)
8. If Hi and n be positive integers, and m ^n, shew that
M p (^W 1 ^n.-r^r^n-r /2« + 2m-4r+l\
where
. 1.3.5... (2»i-l)
(Adams.;
9. Shew that P„ [z) can be e.xpressed a-s a determinant in which all elements parallel to the auxiliary diagonal are equal (i.e. all elements are equal for which the sum of the row-index and column-index is the same) ; the determinant containing (2»-l) rows, and its first row being
I
' 3' 3 • 5' 5''-2»-l 10. Shew that
^' -S' S''-d;rrT^- (Heun.)
'■<')=i-/:*^^isf*-
11. Shew that
/' , ^ ^^-^ ^^^ /'n- , (2) - P„- J {x) P„ (j)} clx=-l,
12. Shew that, when n is a positive integer,
o^6) _ (-1)- o;i jl, fr-z\\ *i n! d!''\2r^\r+zj)'
(Catalan.)
P g.(cosg)_(-l)" o» (1
where 3 = r cos 6.
238 TRANSCENDENTAL FUNCTIONS. [CHAP. X
13. Shew that the complete scihitiun of the Lcgoiidre differential equation is
y = Al\{z) + BP,
^''llo^.
'){Pnwr
14. Shew that
■where
{2 4.(^2_l)i.a= I B^Q,,„.a-l(^),
„ _ a(a + 2m.+J) r(m-|)r(OT-a-i) "'~ 2^r m\ r(m-« + l) '
15. Shew that, when the real part of (« + l) is positive,
and
dh.
'?»(') =
/:
dh.
16. Prove that
2m _L 1
n{n-vV) (Cambridge Mathematical Tripos, Part II, 1894.)
17. Shew that, if )i be a iMsitive integer,
<2n (^) = \ log ( J^') . P„ (.-) - \ {/'n-i (.-) P, (-') + I /'„_, {Z) P, {Z) + 1 Pn-S (') A (^) + • ■ •} ■
18. Shew that
and
^ / N 1 r. / SI 2 + 1 r2?i- 1_ ,, 2re-5_ ,, 271 -9_ ,.
g„(2)=-i>„(2)l0g_ _ |-^ P„_, (3) + ^^^— P„_3(.) + ^^^__^^i>„.J.)+..,
where n is a positive integer, and 2 > 1, and where log " is to be changed into log if 2 is numerically less than unity.
Prove also that
k+ a-1) ''(" + ^) (izS) + fx. _ 1 . A » {n-l){n + l)(7i + 2) (z-X
1-
12 2*
^/, ^ 1 l\w(»i-l)(ji-2)(ra + l)(m + 2){« + 3)/2-l\' \ +>v 2"3; P2532 V"^/ "^
7,11 1
where ^- = 1 + - + - + ...+-. 3 ;^ «.
(Cambridge Mathematical Trijjos, Part II, 1898.)
I
MISC. EXS.]
LEGEXDUE FUN'CTIONS.
239
19. Shew that
1 pm (j) = »(n + l)...(« + »t-l)
2 " '' m\
20. Prove that, if
(2» +l)(2» + 3)... (2« + 2.< - 1 )
then
y-P _-li^+i)^ +?!H:3^, ,
Vi-^n + i 2?i-l "^2h -!"■-'
z"'F{n, n + m, m+1, s^).
3(2n + 3)„ 3(2« + 5) (2n + 3) (2» + 5) „
^3-^n*3 2n-l ^» + »"^ 2»j-3 "-1 (2«-l)(27i-3) "-"
and find the general formula. :.'l. If
(2k-1)(2?i-3) (Cambridge Mathematical Tripos, Part II, 1896.^
shew that
(T^^+A.-r=.f„^»'(^)*"'
C„''{.r.ri-(a:=-l)»(.V-l)*cos<^}
_ n(2K-2) >^=" _
^ n(2y-2) ''=" _ 4An(»-x){n(i/+X-i)p {n(.'-i)P^=o'' ^ n(«+2v+x-i)
2. If
(Gegenbauer.)
<rA^)=j\t'-S'- + l]-it"dt,
where e, is the least root of t^ — 3lz + l =0, shew that
(2ft + l)<r„.^,-3(2«-l).-o-„., + 2(H-l)<r„.2 = 0,
and
4 (4-2-1) (r,."'+ 144^.,o-„" - J (12«- - 24)i - 291) < - (n - 3) (2?f - 7) (2» + 5) o-„ =0,
I where
„,_cPa-„(z)
(Pincherle.)
23. Shew that
(Hobson.)
r(«-»» + l)y„ {. + (22_l)icoshM}''*l
where the real part of (re + 1) is greater than m.
24. The equation of a nearly spherical surface of revolution is
r = l+a{P,(cos5) + P3(cose) + ... + /\„.-i(cos5)},
where a i.s small ; shew that to the first order of a the radius of cui-vature of the meridian is
n-l
1 + a 2 {ii {4m + 3) - (?» + 1 ) (8»i + 3)} 1%^ + , (cos 6).
m=0
(Cambridge Mathematical Tripo.s, Part I, 1894.)
CHAPTER XL
Hypergeometric Functions.
134. The H y per geometric Series.
We have already in § 14 considered the hypergeometric series*
a.b a{a+l)b(b + l) „ a (a + 1 ) {a + '2)b{b+ l){b + 2) ^ "^l.c^"^ 1.2.e(c+l) ^"^ 1.2.3.c(c+l)(c + 2) ^ "^ "•■
from the point of view of its convergence. It was there shewn that the series is absolutely convergent for all values of z represented by points in the interior of the circle whose centre is at the origin and whose radius is unity. It follows from § 22 that all the series which can be derived from the hypergeometric series by differentiation and integration are likewise absolutely convergent within the same region : and by § 5.5, the convergence is not only absolute but uniform over the interior of the circle, and the sums of the series obtained by differentiation and integration of the series term by term are the derivates and integrals respectively of the sum of the series. The hypergeometric series, together with the series which can be * derived from it by the process of continuation (§ 41), will therefore represent an analytic function of the variable z ; this function will be denoted by F(a, b, c, z).
Many of the most important functions of Analysis can be expressed by means of the hypergeometric series. Thus it is easily seen that
{l+z)" = F(-n,/3,^,-z), \ogil+z)=^zF(l,l,2,-z),
e' = Umit F[ 1,0,1,^,
and we have shewn in the preceding chapter that the Legendre functions may be represented by the series
• Tlie name was given by Wallis in 1655.
134, 135] HYPERGEOMETRIC FUNCTIONS. 241
7rir(/( + l) 1 p/n + 1 H + 2 3 1
^ , ,_ 7rU'(» + l) 1 j,/n+J. n_+'^ ,3
These examples are sufficient to shew that the functions represented by the hypergeometric series are in some cases one-valued and in other cases many-valued.
Example. Shew that
^^F(a, b, c, 2) = ^F(a + \, 6+1, c+\, z).
135. Value of the sei'ies F{a, b, c, 1).
We have shewn in §14 that the series F{a,b,c, 1) converges absolutely so long as the real part of c — <( — 6 is positive. Suppose this condition to be satisfied. Then we have
F(abci)=i ri«_4-»)r(6 + »)r(c)
T{c) " 1 r(n + a)r(n + b)r{c-b)
r(a)r(6)r(c-6)„ron! T{n + c)
r(c) ^ ]_r(n + a)B{n + b,c-b)
r(«)r(6)r(c-6)„=on! r(c)
T{a)r{b)r(c-b)-on\)o' '' io^ "^ ' '^^
^ ^^^ —r, ( Cx'-'dx P ^' (1 - ^y-^'2*-> -0)Jo Jo
r(a)r(6)r(c
Writing z = I — t, this becomes
'dz.
^<^' '• '' 1) = r(a)r[/;r(c-6) />/o -'^-"-'^^-'-' (^ " ^^^"'^^
r(a)r(6)r(c
(writing xt = s)
— r-, I ds I e-'s''-H'-''-'>-'(l -tf-' '■-b)Jo Jo
r(a)r(6)r(c
r(c) ■r(a)r(6)r(c-6)
r(c)r(c-a-6)
r(«)B(c-a-6,6)
~r(c-6)r(c-a)' •
W. A. 16
242 TRANSCENDENTAL FUNCTIONS. [CHAP. XI.
The hypergeometric scries with arguineut unity can thus be expressed in terms of Gamma-functions.
136. The differential equation satisfied by the hypergeometric series.
The function represented by the hypergeometric series y = F{a, b, c, z) satisfies the differential equation
for if the series be substituted for y in the left-hand side of the equation, the coefficient of z^ is
a{a + \)...{a + r-\)b{b+l)...{b + r-\) 1.2 ...r.c{c + l)...(c + r)
{r{r-\){c + r)-r{a + r){b + r)-c{a + r){b + r) + r{c + r){a + b + \) + ab{c + r)] or zero ; which establishes the result.
Example. Shew that one integral of the equation
is
z^"F(m-fi, m-v, m~)i + l, z), where
a-l=-i^ + p),
b + 1=7)1 + n, f= — mn.
137. The differential equation of the general hypergeometric function.
The differential equation found in the preceding article is a case of a- more general differential equation, which may be written
d'y ^ fl-g-q' ^ 1-^-/3' ^ I-7-7') dy dz^ \ z — a z — b z — c ] dz
^ fga'(a-6)(a-c) ^ /3/3'(b-a)ib-c) ^ yy (c ~ a) (c-b)] y _^
\ z—a z — b z — c ] (z—a){z—b)(z—c)
...(A),
in which a, b, c, a, jB, y, a, /3', y' are any constants such that the equation
a + ^ + y + a + 13' + y' = l
is satisfied. This will be called the differential equation of the general hyper- geometric function. The forin here given is due to Papperitz*.
* Math. Annalen, xxv.
136, 137] HYI'EUGEOMETIUU FUNCTIONS. 243
Wo shall now shew that the differential donation satisfied by the hypor- peoinotric series is a particular case of this equation.
For ill the equation (A), write
a = 0, b= x , c=l. The equation becomes
rfj;- ( z z — l)dz { z z — \ ) z{z—\)
In this equation, let a aud 7 be replaced by zero. We thus have
dz- \ z z—ljdz z{z—l)
and in this equation the constants a', 7', /3, /3', are to be such as to satisfy the relation
^ + a' + /3' + 7' = 1.
This differential equation can be identified with the equation
^(^-l)g+{-c + (« + 6 + l).-jg + «6y = 0,
■which is the differential equation satisfied by the hypergeometric series, by writing
/9 = a, /3' = ?^ o' = 1 — c ;
which in virtue of the above relation gives y' = c — a — b. The differential equation of the hypergeometric series is therefore a special case of equation (A).
We shall denote any solution of the general differential equation (A) by
the symbol
I a b c "j
P\a 13 y z\.
[a' 13' 7' J
This notation is due to Riemauu*; it enables us to express our result thus:
The hypergeometi~ic series
F{a, b, c, z)
is a solution of the differential equation of the class of functions
0 00 1
0 o 0 z
\ —c b c—a—b )
* Abhandlungen d. K. Gesell. d. W ittentchaften zu Gottingen, vii. (1857).
16—2
244 TUANSCENDENTAL FUNCTIONS.
Although the hypergeometric series itself satisfies only a particular form of the differential etjuation (A), it is nevertheless possible to satisfy the general equation (A) by means of a function derived from the hyper- geometric function. For by the transformation
x{2 - b){c — a) = (z — a)(c — b),
the differential equation (A) is reduced to the form
ax- [ X x—\]dx [ X x — \ ] x{x—\)
In this we take a new dependent variable, defined by the equation
y = .r° (1 — x)y u. The equation becomes d-u , /l-a' + a , \-<y' + 'y\du ,^^, , , ^_, , ,., u .
Now the equation
a + /S + 7 + a' + /3' + 7'=l
will be satisfied if /3, a', ff, y', are expressible in terms of three new constants, a, b, c, defined by the formulae
I /3 = a - a - 7, a' = 1 — c + o, j3' = b — a — y, y' = c — a — b + y.
The differential equation for lo can now be written
x(x—l)-, \- 1(1 + a + b)x — c} -,- + abu= 0.
' dx^ ' ' dx
But this is the differential equation satisfied by the hypergeometric series, a solution of it being
F (a, b, c, x).
Hence we have, as one solution of the equation,
u = F (a -h 0 + y, a + /3' + 7, 1 + a- a', x),
or y = x" (1 - x)y F{(x + /B + y, a + /3' + 7, 1 + at - a', x),
or, disregarding a constant factor,
fz — c\y „ f ^ ^, , , , , z — a.c-b
y={^^\^tfF[.^^ + y,a + ^' + y',l-,u-.'
:-b.
Pa B 7 -
138] HVrKllGlCOMETlUC FUiNCTIONS. 245
This is therefore a sohitioii, expressed by a hypergeomotric scries, of the differential eiiuatioii whicli defines the class of functions
((, b
a' 13' i
The advantage of the differential equation (A) over the equation found in §13(3, which is satisfied by the hypergeonietric series, lies in its greater symmetry and generality. The points z = «, z = h, and z = c, are called the sincitdarlties of the differential equation (A); the quantities a and a' are called the exponents at the singularity a; and similarly /3 and /3' are the exponents at b, and 7 and 7' are the exponents at c.
Example. Shew that
/ 0 00 1 ^ ( -\ a: \
P 0 /3 y ---[=/'-! V -23 y
(Riemann.)
Tliis relation follows from the foct that the difterential equation corresponding to either of the /'-functions is
5?^ + z^-l dz + 1*^'^ + .--^1/ ?^ -°-
138. r/(e Legendve functions as a particular case of the hypergeonietric function.
The expressions \vhich have been found for P„(2) and Quiz) as hyper- geometric series naturally lead us to suppose that Legendre's differential equation is a special case of the differential equation which defines the general hypergeometric function. That this is the case appears from the following investigation.
If in equation (A) of the last article we take
« = — 1, i = X , c = 1, we obtain the differential equation
dz'^X z + l + z-1 \dz^\ z+l^'^'^^z-l\(z-lXz + l)
If now in this equation we take a = 0, a' = 0, 7 = 0, 7' = 0, /3 = n + 1, yS' = — n, we obtain
rf-w 2z dy / . IX y n
which is the Legendre differential equation.
24<6
TRANSCENDENTAL FUNCTIONS.
[chap. XI. 1
It follows from this that any solution of Lcgendrc's equation is a hyper- geometric function of the type
! 0 « + 1 0 2 - .
0 - 7( 0 J
In the same way it can be shewn that the associated Legendre functions P„"'(^) and Q,^{z) are hypergeometric functions of the type
-1
CO
2 " + 1 ^
m m
-2 -" -2
Example 1. Shew that
/ -1 00
|^^P„ (.-) = />] -r n + r + \ az'
\ 0 -» + >•
Exq-mple 2. If 2^=17, shew that the Legendre differential equation takes the form
^-y ^ [ 1 L 1 "^ + w(w + l).y^Q
dri^ (27; \-tl)di) in {I -It)
Shew that this is a hypergeometric differential equation.
139. Transformations of the general hypergeometric f auction.
We shall next consider the effect of performing certain transformations in connexion with the general hypergeometric function
I a b c a /3' 7' The diflferential equation satisfied by this function is d'y fl-a-a' 1-/3-/3' l-y-y'\dy f«a'(a-6)(a-c)
dz^+\
2 — a
+
-b
+
dz
^ 00'(b-a){b-c) ^ yy' (c-a)(c-b)]
z — b z — c ) (z — a)(z — b){z — c)
y
0.
In this equation, let the dependent variable be changed by the trans- formation
(z - bV
y--
by , -a)y-
139] HYPEIIGEOMETIUC FUNCTIONS. 247
Tlic differential ecuiation for y is found after a slight reduction to be
dhj ^ fl - a - g' - 28 1 - ;3 - ;8' 4- 28 1 - 7 - 7") d£ dz" \ z — a z — b z — c \ dz
^ f(a + 8Ha'4-8)(a-/>)(«-c) _^ ( z — a
(0 - B)(^' - B)(b - c)ib - a)
-b
.(c-a)(c-b)
I-. yL
) {z — a){z — t
"^'^'^ z-c ] {z-a){z-b){z-c) ^"
This is the differential equation of a hypergeometric function which has exponents a + 8, a' + 8, at the singularity a, ami exponents /S— 8, /3' — 8', at the singularity 6 ; and so we have
a ?) c "1
a + 8 /3-8 7 s\; a' + 8 /3'-8 7' J
, [abc ,
and hence in general we shall have
Ia b c \ f f' ^ '^
« /3 7 4=p «+8 ^-B-e y + e z a /3 7 J (a+o p—b — e y+e
It will be observed that by this transformation the exponent-differences a — a', /3 — /3', 7 — 7' are unaltered.
Consider now the effect of transformations of the independent variable z.
If we introduce in place of ^ a new variable /, defined by the equation
_ Oiz' + bi ^~ dz'+di'
where aj, b^, c,, d^ are constants, so that
djZ + b,
z =-
we have
and
C-.Z — a.
dy _ a,rf, — 6,c, dy _ {c^z + d^- dy dz {CiZ — a,)" dz' a^d^ — b^Ci dz'
d'y ^ _ 2ci (a,d, - h^c,) dy («.(/, - b.c,)- d'y dz^ (CiZ — Oiy dz (CiZ — OtY dz'^
_ 2ci (c,z' + d,y dy {ciz' + d,)* d^y ~ (a^di-b,c,y dz' (a^d^ - b^ dz' '
i
I
248 TRANSCENDENTAL FUNCTIONS.
Hence if we define quantities a', b', c by the relations
[chap. XI.
so that
0.= , ,, 0 = -,, ,, c = - , -^r ,
z — a =
(Old, - 6,Ci) (s' - a')
(Cis' + cZi)(c,a' + di)'
the general liypergeometric differential equation becomes
d'y . 1 dy ( ^ (1 - g - g') (c,a' + dQ , (1 - ^ - ;3') (c.6' + d,) dz''^c,z' + d,dz'\^''^ z-a! ^ z'-b'
I (l-7-7')(cic' + t/.)) I (ag'(a'-6')(a'-c') , /3;8'(6'- a')(6'- c')
^ — c
.2 — a
^'-6'
7y'(c'-a')(c'-fc')l y ^
0'-c' ]"(3'-a')(/-6')('2'-c')
The coefficient of -^4 in this equation can be written in the form
1 - « - a' 1-/3-/3' 1-7-7' 1 j 2ci - (1 - a - a') c,
z'-a '^ z'-b' "*" /-c'"'^c,2' + d,|-(l-/3-/3')c,-(l-7-7')cJ '
which, in virtue of the relation
a + a' + /3 + /3' + 7 + 7' = l,
reduces to
1 - g^ 1-/3- ;8' 1-7- 7'
/ _ a' ^ / - 6'
^--c'
Hence the differential equation reduces to the differential equation of the function
(a' V c
P]a /3 7 z
U' /S' 7
and thus we have the relation
\a b c P\a /3 7 2
«' /3' 7'
= P
a' b' c
a /3 7 ^'I g' /3' 7'
This shews that the general hypergeometric function is unaltered if the quantities a, b, c, z are replaced by quantities a', b', c, z' , which are derived from them by the same homographic transformation.
HO] HYPERGEOMETRIC FUNCTIONS. 249
140. The twenty-four particular solutions oj the hypergeovietric differential equation.
We have seen iti § 137 tliat a particular solution of the general hyper- geometric differential equation is
We shall suppose that no one of the exponent-differences a — a', ^ — /3', 7 — 7' is zero: it is shewn in treatises on Linear Differential Equations that when this exceptional case occurs, the general solution of the differential equation involves logarithmic terms : the formulae will be found in a memoir* by Lindelijf, to which the reader is referred.
Now if a be interchanged with a', or 7 with 7', in this expression, it must still satisfy the differential equation, since the latter would be unaffected by this change. We thus obtain altogether four expressions for which
(c-h)(z-a) (c — a){z — b) is the argument of the hypergeometric series, namely
fz - fA" fz-cy ^( ^, ^, , ., / (c - h) (z - a)
fz - ay' iz - cy „ (■ , _ , „, , -, . f (c-b)(z-a)
(z - ay fz - cy' { ^ , „, , , (c - 6) (^ - o))
rririHf^f' + ^ + 'y' «' + /?' + % i + «'-'
\z-bj \z-b) { . r- w - ■ r- ■ /. . ■" ^c-a){z-h))
these are all solutions of the differential equation.
Moreover, the differential equation is unaltered if the quantities a, a, a are interchanged respectively with j3, /3', b, or with 7,7', c. If therefore we make such changes in the above solutions, they will still be solutions of the differential equation.
Let a change in which (a, a, a) are interchanged with (/3, /3', h) be denoted for example by
/a, b, c\ [b, a, c) '
each singularity in the bracket being interchanged with the singularity above or below it. Then there are five such changes possible, namely, /o b c\ /a b c /a b c\ /a b c\ fa b c\ \b c a) ' \c a b ' \a c b) ' \c b a) ' \b a c) '
* Acta Soc. Scient. Fennicae, iii. (1893).
250 TIUNSCENDENTAL FUNCTIONS. [CHAP. XI.
To each such change correspond (by interchanging a with a', etc. as ah-eady explained) four new solutions of the differential eipiatiDn. We thus obtain twenty new solutions, which with the original four make altogether twenty- four particular solutions of the hypergeometric differential equation, in the form of hypergeometric series.
The twenty new solutions may be written down as follows :
/z — bY' fz — aY rr { ^, ^, , , . ^, ^ (a — c)(z — b)
(a — b){z — c)\
-bY' (S—aY'{^, , n: I , nr r, (ci - c) (z - b)
(^!)
f /z~c\y (z-bY „{ n • r.. -, ' {b-a){z-c)
2'»=t-J \T=-o) ^|'y+«+/^- 7+«+^, 1+7-7- ;,_,)7,_,;
Iz-cY' (z-bY -r,\ , ^ , , n> -, ' (b-a)(z-c)]
^'«=fcj [jz^j ^{7+«+^,7'+«+/3.i+7-7- ;^_,;^_,;j
^" = (^ U— J ^]7 + « + ^> 7 + «+^, 1 + 7-7. ;^,-i:^)7^|
2/.^=f^T'f'7^T^{7' + « + /9', 7' + «' + /3, 1+7-7 ^^-"^(^-''^'
2 - a\ V'2' - ^V r. f o ,0/-, , {b-c){z-a)\
y-=U^c; VJ^c; -f + 7 + P. a + 7^P.^ + «--. (ft_^)(^_,)j y-=K^c) (^) ^(«+7 + ^- «' + 7' + /3',l + a-«. [^TToy^
3/.-(— :y(^T^{«+7+/3', «+7'+/3, !+«-«', ^f^}'-"^
Cj \Z-cl [ I '^ ' It- (])_^0(Z-C)
(z-aV Iz-bY' r,[ , r^, , , -, , , (})-c)(z-a)
K^ \z—ci\z—ci \ ' ' {b — a)(z — c)
( (z- c\y fz-aY Ti{ r. r., r . , U'- - b) (z — c)
^■'=(^J [z^b) ^i7+/3+a. 7+/3'+«. 1+7-7'- ;-,_,;;,_,;
^"=(.3^) (^6) ^{7 + /3 + a.7 + ^' + «. 1+7-7- ^3^F)}
i^=«=u— J t^i) i^{7+^+«',7'+/3'+«, 1+7-7. (,_,;;,_^)}
141] HYPERGEOMETRIC FUNCTIONS. 251
y-'={^-7.) U_J ^'W + ct + y', /3 + u+y, 1+13-13',' '^ '-
\z-al \z — aj { {c — b){z — a))
-h\^' (z-c\>'{r>, ■ a' ■ -, a' o (c-a){z-h)
,2/«=(f^:) (H)Vk+.+y./3'+«'+7, 1+^
/S,
(c-6)(^-a)
The existence of these twenty-four values was tirst shewn by Kunimer*.
Example. Find the twenty-four solutions of tlie Legendre differential equation, lon'csponding to the above set of solutions of the hyiiergconietric differential equation ; and oxi>ress each of tliem in terms of the two independent solutions P„ (z) and Q„ (z).
141. Relations between the particular solutions of the hypergeometric differential equation.
Since the twenty-four expressions found in the last article are solutions of the same linear difiorential equation of the second order, any three of them must be connected by a linear relation with constant coefficients.
We proceed to find the relations which thus connect them.
First, consider the set of four solutions
Vi, y-i, yv6, Vxr/,
it is clear that, in the neighbourhood of the point z = a, each of them can be expanded in a power-series of the form
A{z- ay 11 -f -B (^ - a) + C(z - af +...].
But there is only one series of the form
{z-a)'[l+B(z-a) + C(z-ar- + ...]
which .satisfies the differential equation; for the coefficients B, C, ... can be uniquely determined by actual substitution in the differential equation. Let this solution be denoted by P"'.
Thus the solutions
yi. 3/3. y.3, yi5
must be mere multiples of P'°'. Moreover, for y, the factor A is (a - c)y {a - 6)-'»+i" ;
for y, it is (a - c)y'(a - 6)-|''+t'i ;
for y,3 it is (a — by (a — c)"'"^ ;
and for y,, it is (a - by'(a - c)~""^'.
* Crelle't Journal, xv.
252 TKANSCENDKNTAL FUNCTIONS. [CHAP. XI.
Thus we have
\a — cl \a — of
, (c — h){z — a)\
' {c—a){z — h)
, . fz-c\>' /z-b\-''-y'
n ( r, ' r.f -. , (C — h)(z — a)
= (^-a)»
2 — c\-'^-^ ( s —by
a — c' \a — b
, (b — c)(z — a)'
y F ■la + l3 + y, a + l3' + y', 1 + a-a
(b - a) {z - c) z — cN~""^' fz — b\^'
r< \ -,/ o /I , (^ — c) (^ — a)
xi?'^a + /3+7, a + ,S + 7, 1 + a - a' ^ ^^ ''
(b-a){z-c)\ ■
Similarly solutions P'"'', P^^\ P'^'i, P^y\ P'f'^ exist, each of which is equi- valent to four of the above hypergeometric series.
Having thus classified the twenty-four solutions into six distinct solutions, namely
pia.) pia') piS) pi^') p(y) p(y')
we proceed to find the relations between these latter six solutions. We know
that P'"' must be expressible linearly in terms of P^t^ and P'^'*. Let the
relation between them be
pw =a^pir. +a^,pi/i.
We have then to find the coefficients o^ and Oy'.
Now this equation can be written in the form
<^-«)-(~)'fe')"""'
, f z - aY / z - bY'^-y
xFia+0 + y, a'+/S'-l-7, 1 + 7-7. !"^ rrr: — rl
+ ay,{z-cy'
+ ^ + 7, «+^+7. 1 + 7-7, ;,_,)(,_^-;
- aj \c — bj
n f ^ - , ^, . . ■ {a-b'){z-c)\
xP|a + /3+7, «' + /3'+7. 1+7-7, (,_,)^,_^)}
142] HYPERQEOMETRIC FUNCTIONS. 253
Dividing throughout by the common factor {z — a)', and writing z = a and z = c successively in the resulting C(|uation, we obtain two equations, from which 'jy and a.^- can be found : tlie iiyi)ergeometric functions reduce to the
I type
F{u, V, w, 1),
I which in §135 was shewn to be expressible in terms of Gamma-functions, and i the type F {u, v, w, 0), which clearly has the value unity.
As already explained, in certain cases (e.g. when one o{ the exponent-differences is an integer) the above theory of the sohitioiis requires niodilioation. For a discussion of these cases the student is referred to Lindelof's paper ah'cady mentioned, and Klein's Lectures " Ueber die h\iiergeoiiietrische Function."
142.' Solution of the general hypergeometric differential equation by a definite integral.
Wc ne.xt proceed to establish a result of great importance, relating to the expression of the hypergeometric function by means of definite integrals.
Let the dependent variable y in the differential equation of the general hypergeometric function ((A) of § 137) be replaced by a new dependent variable /, defined by the relation
y = (z- ay (z - by (z - c^ I.
The differential equation satisfied by I is easily found to be
dz^ \ z—a z — b z— c \ dz
(a + 13 + y){(a + ^ + y + 1) z + la(a + 0' + y' - I)} (z-a)(z-b){z-c)
which can be written in the form
^ = Q(')dz'-l^^--^Q'(')+^(')^'!fz
+ f-^^-— ^ * <2" (^) + (? -.1) ii' (^)} /,
where [ ^ = 1 — <z — j3 — y = a' + f3' + y',
Q(z) = {z-a)iz-b)(z-c), R (z) = ^(a+l3+y){z-b)(z- c).
It must be observed that the function / is not regular at i = ac, and consequently the above differential equation in /is not a case of the generalised hj-pergeometric equation.
We shall now shew that this differential equation can be satisfied by an integral of the form
254 TRANSCENDENTAL FUNCTIONS. [CHAP. XI.
1= I (t- aY'+^+y-' {t - b^+^'+y-' {t - c)«+^+V-' {z - i)-''-s-7 dt, J c
provided the path G of integration i.s suitably chosen.
For on substituting this value of / in the differential equation, the condition that the equation should be satisfied becomes
0 = [ (i - aY'+^+y-' {t - by+^'+y-' (t - cY+^-^y'-' {z - ty'^-^-y-- Kdt, J c
where
K={^-2)^^Q(z) + {t-z)Q'(z) + ^^^^Q"iz)^
+ {t-z){R{z) + {t-z)R'{2)} = i^-2}{Q(t)-(t-zy} + it-z){R{t)-(t-zy^(a+^ + j)} = {^--2)Q(t) + {t-z)R{t) = -{l+a + l3 + r^){t-a){t-b)(t-c)
+ 1, (a + 0 + y){t - b){t- c){t - z), or K = {t- ay-'^'-i^-y (t - i)>-— P'-c (t - c)i-»-^-r' {z - i)»+''+v+=
-^ \{t - aY'+^+y {t - by+^'+y (t - cY+i'+y' (t - ^)-fi+a+^+v)}.
It follows that the condition to be satisfied reduces to
where F= {t - aY'+^+y (t - 6)»+^'+r (t - cY+^+y' (t - ^)-n+«+3+v).
The integral / will therefore be a solution of the differential equation, provided the path of integration C is such that the quantity V resumes its initial value after describing- the arc C.
Now V=it- ay'+^+y-' (t - by+^'+y~' (t - cy+^+y'-' (z - <)-«-^-y u,
where U = {t- a){t-h){t- c) (z - <)"' ;
and the quantity U resumes its original value after describing any contour : hence if 0 be a closed contour, it must be such that the integi-and in the integral I resumes its original value after describing the contour.
Hence finally any integral of the type
(z-aY {z- by (z - cy [ (t- a)3+r+»-i {t - 6)v+«+^'-i (t - cY+^+y'-' (z- ty-^'^-y dt,
J c
142] HYPERGEOMETRIC FUNCTIONS. 255
where C is either a closed contour in the t-plane such that the integrand resumes its i)ntial value after describing it, or else is an arc such that the quantity V has the same value at its termini, is a solution of the differential equation of the general hypergeometric function.
Example 1. As an cx;imiilo, wo shall now clriluce a real definite integral which (for a certain range of values of the quantities involved) represents the hypergeometric series.
The hvporgoonietric series F{a, h, c, z) is, as already shown, a solution of the difi'erential equation of the function
0 X 1
0 a 0
1-f b c—a—b
The integral
thus becomes in this case
f f'-'{t-\Y-'>-'^{t-z)-''dt. J c
Xow the quantity V is in this case
and this tends to zero at i = l and i=a:>, provided c > b > 0.
Hence if these conditions are fulfilled, we can take as the contour C an arc in the ^plane joining the points t = \ and ; = x ; so that a solution of the differential equation is
/.
t"-' {t-l)'-''-^ {t-z)-''dl.
In this integral, write C=- ; the integral becomes
['«''-» (1 -m)'-'-! (l-vz)-" du ; ' 0
this integral is therefore a solution of the differential equation for the hypergeometric series.
It is easily seen that this integral i.s in fact a mere multiple of the hypergeometric series
F{a, b, c, z) ;
for supposing \z\< 1, and expanding the quantity (1 -?(:)"" in ascending powers of c by the Binomial Theorem, the integral takes the form
./ 0 r=I r\ Jo
r=l rl
du.
256
TRANSCENDENTAL FUNCTIONS.
[chap. XI.
or or
[ r=i r!c(c + l)...(c + r-l) J'
B(b,c-b)F(a, b,c,z), which establishes the result stated.
Example 2. Deduce SchliiflCs integral for the Legendre functions, as a case of the general hypergeometric integral.
Since the Legendre equation corresponds to the hypergeometric function
-1 X 1
0 n+1 0 :
\ 0 -n 0 J the corresponding integral is
or
j^(t'-l)n(t-z)-"-^dt,
taken round a contour Csuch that the integrand resumes its initial value after describing it ; and this is Schliifli's integral.
Example 3. Deduee Laplacc^s integral for the Legendre functions, as a case of the general hypergeometric integral.
If we write the Legendre difterential equation becomes
6?2y /I I \dy n{n + \)y_
de \u^-\)d^ 4 e
This corresponds to the hypergeometric function
^0 00 1
n n-\-\
2 2 i+l n
0 I
0
2 2
and so the hypergeometric integral becomes in this case
|~5 L» ( 1 _ y)-h (^ _ „)-i dii.
taken round a contour enclosing the points !( = 1 and i( = |.
Write | = f-,
so f=z + (z2-l)i.
Then the integral becomes
f-"-i I (l-!/)-i i\-C~-u)-iu''du, taken round a contour enclosing the points u = \ and M = f^.
14."]] HYPERGEOMETRIC FUNCTIONS. 257
^K Write ii = h( in this integral ; we thus obtain
/'(l-2.-A + /t2)-J/i"rf/,, the integral being now taken round a contom' in tlic /(-[ilane enclosing the points /i = f and
Suiipose now that the rojil part of z is positive ; and let the conto\ir Ixrcome so attenuated as to reduce to a small circle surrounding the point A = f, another small circle surrounding the point /i = f-i, and the line joining the points f and f "', described twice. The small circles contribute only intinitesimally to the integral, which thus becomes a multiple of
P Jl-23A + A2)-iA"rf/i.
Writing /i = z + {z' - 1 )4 cos (f>
in this integral, we obtain
/ {z + {z--l)i cos cf)]'' d<p, J 0
which is one of Laplace's integrals (§ 119).
143. Determination of the integral which represents P'°'.
We shall now shew how the integral which represents the particular solution P'"' (I 141) of the hypergeometric differential equation can be found.
Wc have seen (§ 142) that the integral
I={z-aY{z-bf(2-c)yl {t-af+y+'''-'(t-b)y+''^^'-'{t-cy+^+y'-\2-t)-''-^-ydt
J c
satisfies the differential equation of the hypergeonietric function, provided C is a closed contour such that the integrand resumes its initial value after describing C. Now the singularities of this intes^rand in the (-plane are the points a, b, c, z; and on describing a simple closed contour enclosing the singularity b alone, the integrand resumes its initial value multiplied by
8 >
as is seen by writing it in the form
(S+y+a'-l)log«-a)+(-y + o+/3'-l)log((-A) + (a + |3+y'-l)IoK(<-c)-(a -(-3 + ^)10^(7-0
Take then a point 0 in the (-plane, and draw a loop in the (-plane passing through 0 and encircling the point b, but not encircling any of the points a, c, z. Let an integral taken in the positive or counter-clockwise direction of circulation round the perimeter of this loop be denoted by the sign
Jo
'0 W. A 17
258 TRANSCENDENTAL FUNCTIONS. [CHAP. XI.
and let an integral taken in tlie negative direction of circulation round the perimeter of the loop be denoted by
i« ' so that we have the equation
nb + ) r(b^)
Jo Jo
where it is understood that the initial value of the integrand in the second integral is taken equal to the final value of the integrand in the first integral.
Let now a contour C be drawn in the following way. Take first a loop starting from 0, encircling the point b in the positive direction, and returning to 0 ; then a loop starting from 0, encircling the point c in the positive direction, and returning to 0 ; then a loop encircling the point b in the negative direction ; and lastly a loop encircling the point c in the negative or clockwise direction.
Conformably to the notation already explained, an integral taken round this contour will be denoted by
r(b + , c+, !)-, C-)
Now after description of this contour, the integrand of the integral / ; already considered resumes its initial value multiplied by
g2wi (y+a+^'-l+a+34 y'-l-y-a-^' + l-a-^-y'+l)
or 1, i.e. the integrand resumes its initial value*.
Hence if C be taken as the contour, the integral / will satisfy the differential equation.
Thus
I = (z-ay{2-by(2-c)T [ ' '' ' ' ' {t- af +i'+«'-i (t - b)y+''+^'-'
J 0
(^t - cy+P+y-^ (2 - <)-«-3-r dt
satisfies the differential equation of the hypergeometric function.
Now suppose that the jjoint z is taken near to the point «., so that \z — a\ is less than either \b — a\ or |c — al. We can clearly draw the contour just
* These double-circuit integral were introduced by Jordan iu 1887. Clearly any number of contours can be formed in this way, it beiug necessary only to ensure that each singular point is ' encircled as often in the negative or clockwise direction of circulation as in the positive or counter- clockwise direction.
14-I-] HYPERGEOMETRIC FUNCTIONS. 259
describctl in such away that, for all points < on it, \t — a\ is greater than 1^ — a|. Thus we can write
/■(4 + ,f + ,6-.c-) X I {t- «)»+>+"■-' (t - 6)Y+«+S-l (^ _ ^Y+fi+y-i
(.,-,r-.(i-i^«)--'*.
Under the conditions already stated, each of the expressions
(•-Hr.(>-^)'.-('-sr"'.
can be expanded by the Binomial Theorem in ascending powers of (z — a). We thus obtain for / an expansion of the form
I = {2- ay [A + B(z- n) + C(2- af + ...},
and as / satisfies the differential equation it must therefore be a multiple of the particular solution P'"' of § 141.
Thus
f(b+,c + , b-, C-)
P'"' = Constant x (z - a)" (z -hf{z- c)> (t - rt)^+r+« ->
{t - 6)>+''+^'-i (t - c)«+3+i'-i {z - ty'-^-y dt. Similarly
/•(!) + , r + , b-, C-)
P'"' = Constant y.{z- a)"' {z -by{z- c)y {t - a)''+v+»-'
Jo
(t - 6)^+»+'''-> {t - c)» +3+r'-i (z - <)--'-S-r dt.
In the same way the particular solutions P"^', P^'>, Pw, P'r\ can be expressed as contour-integrals.
144. Evaluation of a double-contour integral. ^Ve may note that an integral
/(a + , I + , a - , !>-)
can Ijc expressed in terms of the integral.-}
jo ""'i„ '
in the following way.
Let the initial value of the integi-and at the point 0 be denoted by T. After describing the loop round a, the integrand will have at U the value c-'"('»'+^+y-i) y^ and the part
17—2
260 TRANSCENDENTAL FUNCTIONS. [CHAP. XI.
/•(a+) /■(a + . t>*, a-, b-)
I of tlic integral I will have been obtained. Describing next the loop
J » Jo
/(n+, b + , a-, b-) will therefore be 9
and the integrand will return to 0 with the value
g2n-t (a' + ^ + y-l + i + P' + Y-l) yr
Describing next the loop round a in the negative direction, we observe that the corre- sponding part of the integral would have been
/:'
if the integrand had had for initial value
g2Tri{a' + p + y-l) j,
which is its final value when the loop is described with the initial value T : it is therefore actually
.' 0
■"/:"
■/:
and lastly, describing the loop round b in the negative direction, we obtain the part
/■('' + )
.'0
of the integral.
Collecting these results, we have
a formula which furnishes the value of the doublc-ooutour integral in terms of two simple- contour integrals.
145. Relations between contiguous hypergeonietric functions.
Let P {z) be a hypergeoraetric function with the argument z, the singu- larities a, h, c, and the exponents a, a', /3, /3', 7, 7'. Let P;+i „,_i(2^) denote the function which is obtained by replacing two of the exponents, I and m, in P {z)hy l + \ and m — 1 respectively. Such functions P;^.,,„i_i(^') are said to be contiguous to P {z). There are clearly 6 x 5 or 30 contiguous functions, since I and m may be any two of the six exponents.
It was first shewn by Rieraann* that the function P (z) and any two of its contiguous functions are connected by a linear relation, the coefficients in which are polynomials in z.
* Ahhanilmujen der Kon. Ges. der Wiss. zu Gottingen, 1857.
145] HYPKROEOMETRIC FUNCTIONS. 261
; . 30 29
There will clearly be '^~— or 435 of these relations. In order to obtain
them, we shall take P (z) in the form
P (z) = (z- a)' (z - by (z - c)y I (t- a)^+v+.'-i (,; _ 6)r+''+?-i
■ c
(t - c)«+3+y'-i (t - z)-'-^-y dt,
where C may be any closed contour in the <-plane such that the integrand resumes its initial value after describing C.
First, since the integral round C of the differential of any function which resumes its initial value after describing C is zero, we have
0 = f ^A(i- aY'+^+y {t - t)"+^'+Y-' (t - c)-«+8+>-' (t - z)---^-y] dt,
J cat or
0 = (,a' + /3 + 7) I (t~ a)"'+''+i'-i (t - 6)<«+3'+r-i {t - c)»+^+r'-> {t - z)---^-y dt J c
+ (a + ^' + 7 - 1) f {t- aY'+^+y {t - hy+^'+y-- {t - cy+^+r-i (t - z)—-^-y dt J c
+ (a + /3 + 7 - 1) f (t- aY'+^+y (t - bY+^'+y-' (t - cY+^+y'-' {t - zy^-^-y dt
J c
- (a + /3 + 7) f (t- aY'^^+y (t - 6)«+^'+''-i (i _ c)»+^+r'-i (t - ^)-«-3-r-i dt, J c
or
(a + ^ + 7) P + (a + /3' + 7 - 1) P..+,, 3_, + (a + /3 + y'-l) Pa'+,,y-i
_(a + ^ + 7)p z-b ^^+'->'—
Considerations of symmetry shew that the right-hand side of this equation can be replaced by
(« + /3 + 7) p
These, together with the analogous formulae obtained by cyclical inter- change of (a, a, a') with (b, /3, /3') and (c, 7, 7'), are six linear relations connecting the hypergeometric function P with the twelve contiguous functions
P«+l,3'-l, i p+l,y_l, Py+io-li Pa+1,Y-1. -' 3 + l,a-l. -I Y+1. 3'-l ' ■'a'+l.e'-li Po'+I.y-l, i ^■+i,y'_i, P^'.t.i,a'-1. Py+l,o'-li -^ r'+'. ^'-1-
262 TRANSCENDENTAL FUNCTIONS. [CHAP. XI.
Next, writing t — a = (t — b} + {b — a), and lusing P^'-i to denote the result of writing a' — 1 for a' in P, we have
Similarly P = P.._,,y+, + (c - a) Pa_i.
Eliminating P„'_i from these equations, we have
(c-b)P + (a - c) P.._,, p.+i + {b-a) Pa'-,,y+, = 0.
This and the analogous formulae are three more linear relations con- necting P with the last six of the twelve contiguous functions written above.
Next, writing {t —z) = {t — a) — {z — a) we readily find the relation
X ( {t- ay+y+'''-' {z - a)i'+«+3'-' (z - 6)«+^+y'-i (t - ^)-«-^-y-i rf^,
which gives the equations
(^ _ „,)-! jP - (^ - 6)-^ P^+,,y_,} = (^ - 6r {P - (^ - c)-' P.+,,»--,l
= (^-c)-'{P-(^-a)-'P,+,,,_,).
These are two more linear equations between P and the above twelve contiguous functions.
We have therefore now altogether found eleven linear relations between P and these twelve functions, the coefficients in these relations being rational functions of z. Hence each of thes'e functions can be expressed linearly in terms of P and some selected one of them ; that is, bettueen P and any hvo of the above functions there exists a linear relation. The coefficients in this relation will be rational functions of ^, and therefore will become polynomials in z when the relation is multiplied throughout by the least common multiple of their denominators.
The theorem is therefore proved, so far as the above twelve contiguous functions are concerned. It can in the same way be extended so as to be established for the rest of the thirty contiguous functions.
Corollary. If functions be derived from P by replacing the exponents a, a', /3, /3', % 7', by a + p, ol +q, ^ -{-r, 13' + s, y + t, y + u, where p, q, r, s, t, u, are integers satisfying the relation
p + q-\-r+s + t + u = 0,
then between P and these functions there exists a linear relation, the co- efficients in which are polynomials in z.
MISC. EXS.] HYPERGEOMETRIC FUNCTIONS. 2()3
This result can be obtained by connecting P with the two functions by a chain of intermediate contiguous functions, writing down the linear relations which connect them with P and the two functions, and from these relations eliminating the intermediate contiguous functions.
It will be noticed that many of the theorems found elsewhere in this book, e.g. the recuri-ence-fornuilue for the Legendre functions (§ 117), are really Civses of the theorem of this article.
Miscellaneous Examples.
1. Shew that
F{a,h-{-\,c,z)-F{a,h,c, z) = ^ F{a + 1, b + \,e+l,i).
2. Shew that
F {a + \, b + -[, c, z)~ F{a, b, c, z) = ^ F{a+-l, b + l, c + 1, j).
dP d^P
3. If P{:) be a h\ijergeometric function, express its derivates -j- and -y^ linearly in
dP terms of J' and contiguous functions, and hence find the linear relation between P, -j- ,
and -j .,- , i.e. verify that P satisfies the hypergeometric differential equation.
If
W(a,b,x) denote F (—^ , 1, 2, -bx\
shew that the equation y= W{a, b, x)
is equivalent to x= W{b, a, y).
5. Shew that a second solution of the differential equation for
F (a, b, c, x) is x^-<'F{a-c-\-\, 6-c + l, 2-c, x).
6. Shew that the equation
(Oj + h^) £, + («! + 6,.r) ^. + (rt„ + V) y = 0 can, by change of variables, be brought to the form
and that this latter equ.ation can be derived from the hypergeometric equation
x" by the substitution 6 = »», x=— , where m is infinitely large.
m
264
7. Shew that
TRANSCENDENTAL FUNCTIONS.
[chap. XI.
(
•1 00
c:,{z)=p\ i-- H+2r h-v z \,
-n 0 )
where C" (z) is the coefficient of A" in the expansion of (1 - ihz+lfi)-' in ascending powers of A.
8. Shew that, for values oi x between 0 and 1, the solution of the equation
dhi 1
\ -. •'J (
dy
is AF{\a, iA i, (1 - 2^)2} + Z? (1 - 2^-) ii'{i (a+ 1), i 0 + 1), I, (1 - 2.r)2},
where ^, 5, are arbitrary constants and F{a, /3, y, «) represents the hypergeometric series.
(Cambridge Mathematical Tripos, Part I, 1896.)
9. Shew that the differential equation for the associated Legendre function P^ {z) of order n and degree m is satisfied by the three functions
0 00
1
p] ^m -n -TO
1 1-0
2"^ T"
y,
1 , 1
-rTO ?!+l -s™
/ 0 00 1
P-^ 2 ™ 2 2- (22-1)}}-,
71 + 1 n + l
1-2" -"^ ^
, 0 00 1 ,
n TO 1
PJ "i 2 " V-?
n + l TO 1 ~2 " ~ 2 2
(Olbricht.)
10. Shew that the hypergeometric equation
''i^-'^)'^-^- {7-i^+^+l)^}-£+<^!/=0
is satisfied by the two integrals
f z^-'^{l-zy-^-\\-xz)-'dz
J 0 I /-Vl-2)°->{l-(l-^)^}"'«^^-
and
I
i
MISC. EXS.] HYPERGEOMETRIC FUNCTIONS. 265
11. If
(1 - x)'-^^-^ F{-2a, 2/3, 2y, :r) = 1 + I}x + Cx- + Dx^ + ... , ahow that
^K 3, V+i, x)F(y-n,y-li, y + A, .r)
= l + -+i^-'+(y + i)(y + S)^-^ +(y + i)(y + S)(y + ^)''-^ +
(Cayley.)
12. Prove that
where /*„(;) and (?„(2) are the Legendre function.s of the first and second kind of order «.
13. If a function F(a, fi, 0, y, x, y) be defined by the equation
F.{a, A 3', y ; -r, !/) = r^a)T%-a) /„'"""' ^^ " "''"'"' ^^ ~ "*^"^ ^^ -uyr^'du,
then shew that between F and any three of its eight contiguous functions
F{a±l), F(ff±l), F{?±1), F{y±l),
there cxi.sts a homogeneous linear equation, whose coefficients are polynomials in x and y.
(Levavasseur.)
14. If y - n - 3 < 0, shew that, for values of x nearly equal to unity, Ft„ R , ,.N r(y)r(a + g-y), -,Ny-.-3
and that if y- a — 3 = 0, the corresponding approximate formula is
^(a,Ay,x) = j,^^y^^yog -^.
(Cambridge Mathematical Tripos, Part II, 1893.) 16. Shew that when \x\ < 1,
/•(r.O.i-.O-)
= - ie'"^ sin QTT sin (p - q) 77 . I'(p-°)^W /-(„, „„ p, .r),
"(P)
where c denotes a point on the finite line joining the points 0, x, the initial arguments of v — x and of v are the same as that of .r, and that of (1 - 1-) reduces to zero at the origin.
(Pochhammer.)
CHAPTER XII.
Bessel Functions.
146. The Bessel coefficients.
%
In this chapter we shall consider a class of functions known as Bessel functions, which present many analogies with the Legendre functions con- sidered in Chapter X. As in the case of the Legendre functions, we shall first introduce the functions, or rather a certain set of them, as coeflficients in an expansion.
For all finite values of z, and all finite values of t except t = 0, the function
can be expanded by Laurent's theorem (§ ■l.S) in a series of ascending and descending powers of t. If the coefficient of i", where n is any positive or negative integer, be denoted by J,, {z), we have (by § 43)
J„{z)=^-Jn—^e-'^"-^^du,
the integral being taken round any simple contour in the w-plane enclosing the point u = 0.
To express this quantity J^ (z) as a power-series in z, write
2t
u= - . z
Thus Jn{z) = ^. (^] ' jt-"-'e-ftdt,
the integral being taken round any simple contour in the (-plane enclosing the point i = 0. This can be written
^w=^®"j.^'(r/--'-'
'e'dt.
r = 0
UGJ BESSEt, FUNCTIONS. ^ 2fi7
Now (§ 56) we have
—.1 <""""■"' e'(/^= the residue of the function i~""''~'e' at its pole, the origin.
2in.'
If n is a positive integer, this residue is
1
if ;i is a negative integer, say = — s, the residue is zero when r = 0, 1, 2, ... s — 1,
anil when r > s it is
1 (r - s) ! ■
In any case, the residue is
r(n+r+l)' Thus if n is a positive integer, we have
and if n is a negative integer, equal to — s, we have
0! tl '
J (,._ % f^Y'-' (-1)' _ K pY^=' (-i)'^_l
•^"^^>-,-l2J H7.--S)!- ,roV2J (s +
or j-„(^)=(-iyj,(4
Whether n be a positive or negative integer, the expansion can clearly be written in the form
"^ '' ,ro2"+»-r!r(w + r+l)"
The function J„ (z) thus defined for integral values of n is called the Bessel coefficient of the nth oidcr.
We .shall see subsequently (§ 149) that the Bessel coefficients <are a particular case of a more extended cla.ss of function.s known a.s Bessel functions.
Bessel coefficients were introduced by Bessel in 1824 in his " Untersuchung des Theils der jilanetarischen Storungen, welcher aiis der Bewegung der Sonne entsteht."
In reading some of the airlier jjapers on the .subject, it is to be remembered that the notation has changed, what wiis formerly denoted by J,, {2) being now denoted by J„{2z).
Example 1. Prove that if
2^(1+9')
-^ = A, + A.fi-\-A^e--\-...,
then will e"sin6j = .l,./i {z) + A.iJ2 {z) + AfJ3{z) + ... .
(Cambridge Mathematical Tripos, Part I, 1896.)
268 TRANSCENDENTAL FUNCTIONS. [cHAP. XII.
For replacing the Bo.ssel functions in tho given scries by their values as definite integrals, we have
A , J, {z) + A,J„{z) + A 3-/3 {z) + ...
1 1
\ u u-J u' the integrals being taken round any simple contour in the ?(-plane enclosing the origin.
Taking a new variable t, defined Ijy the equation
1 / 1
2 V It
we thus have
where the integration is now to bo taken in the clockwise direction round any large simple contour in the <-plane. This expression is (§ 56) equal to minus the sum of the residues of the function
'fi + W at its poles t — ib and t= -ib ; that is, it is equal to
l(^(o + i6)_i^((.-i4)
28 ii
or e^sini^,
which is the required result.
Example 2. Shew that, when n is an integer,
J„(z+y)= 2 J,n{!')Jn-,»{y)- m = — «
We have e ^ 'i=e'^ '' .c'^ '',
or i «V„(.^+,y)= 2 r./„.(2) 2 fJriy).
n= — 00 m= — « r=:— oo
Equating coefficients of Z" on both sides of this equation, we have the required result.
147. Bessel's differential equation.
We have seen that, for all integer values of n, the Bessel coefficient of order n is expressed by the formula
where (7 is a simple contour in the ^plane enclosing the point < = 0.
U7. 148]
BESSEL FUNCTIONS.
269
We shall now shew that the function Jni^) is a solution of a certain liiH'ur differential eciuation of the second order, namely,
For we find in performing the differentiations that dz^ z as \ z-j
. I /-"-If' 4/ J 1 — H
= _ J^f^)"/" ^ (e'-Sr»-')d«
dt
0,
since the function e *' t~"~' resumes its original value after the point t has described the contour in question.
Thus Jn(z) satisfies the differential equation
dz- z dz V z-J ^ '
This is called Bessel's equation of order n. Its properties in many respects resemble those of Legendre's differential equation, which is also a linear differential equation of the second order.
, 148. Bessel's equation as a case of the hi/perf/eonietric equation.
Iff be any finite quantity, the differential equation of the hypergeomctric function
/Ox c \
IC T, + ^C z
is (§ 137)
— » — w
— IC
d'u 1 dii ( n-
fz +
c^
= 0.
dz'^ z dz ' \z ' z — c J z(z — c) If in this ecpiation we make c tend to an infinitely large value, we obtain
270
TRANSCENDENTAL FUNCTIONS.
[CHAI'. XII.
which is Bessol's equation of order n. Thus Bessel's equation can he rer/arded
as a liiniting case of the hi/jjergeometric equation, corresponding to the
function
0 X c \
Limit P
ic ^i + ic z
Another representation of Bessel's equation as a limiting case of the hypergeometric equation is the following.
If we change the dependent variable in Bessel's equation, by writing y = e^'ii, the differential equation for w is easily found to be
d-u /„. \\ du fi n-\ ^
Now if c be any quantity, the differential equation of the hypergeometric
function
0 oc c
p.
0
— )( '-. — 2ic 2ic — 1
4
IS
d^ dz^ \z
1 2 — 2ic\ da /n-c 3
z — c J dz \ z 8 J z(z — c)
= 0.
If in this equation we make c tend to infinity, we obtain
d-u dz-
1 -.\ du ( n- i\ _
which is the above equation. Hence Bessel's equation is a limiting case of the hypergeometric equation, being the equation for the function
e'^ Limit P .
0
— n ~ — lie
4
c 0
lie - 1
Bessel's equation is connected not merely with the general hypergeometric equation, but with that special form of it which we have considered in con- nexion with tiie Legendre functions.
148] BESSEL FUNCTIONS. 271
For the differential equation of the associated Legeudre function (§ 129)
is (§ 13S) the equation of the function
- 1 X 1
•-£
or (§ 139)
P^
III 2 " |
+ 1 |
m 2 |
m |
m |
|
2" |
— n |
~ 2 |
■in" |
00 |
0 |
m 2 |
?i + 1 |
111 5" |
m |
m |
|
~¥ |
— ?( |
~ •) |
= 0.
The differential equation of this function is
d'll I J , 1\ J^y , (_ m- _ n + l 7n=\ n'-ij
d{2-f W- 4n= zV d (^») [ z"- 4?i= n z") z" {z^ - 4>ii')
If in this equation we make n tend to infinity, it becomes
Ay +lliL^(-i + ']l\y =0
d {z^y z'd {z"-) \ z'-J iz"- '
which is Bessel's equation. Thus Bessel's equation of order m is the same as the equation for the function
Limit P,/" I 1
2n-J
By considering Bessel's equation as a limiting case of the hj'pergeometric equation, we can deduce certain nolution.s in the form of definite integrals.
. For the differential equation of tlie function
0 00 e
P -, n - i'c i + >c z
, — 71 + ic i — ic is satisfied by the integral
272 TRANSCENDENTAL FUNCTIONS. [CHAP. XII.
if C is a contour such that after describing C the integrand returns to its initial value. When c bocomea infinite, this expression reduces to
-ti I t-n-
J c
h{t-z)-^-he^^dt,
which accordingly satisfies Bessel's eqiiation if C be a contour of the kind described ; C can for instance be a tigiu'e-of-eight contour encircling the points t = 0 and t=z.
In fact, if we write
we have
(P'y 1 dy /, n-\
-I «+- ^J"-!-'"
[j{t!'-"e^"U-s)-"-^dt
2 )
= 0.
Other solutions can be found by changing the signs of n and i.
Example. Shew that Bessel's differential equation is the limiting case of the equation of the hyijergeometric function
0 X c2
hn \{c-n) 0
-hi -h{c + n) m + 1 when c tends to infinity.
149. The general solution of Bessel's equation by Bessel functions whose order is not necessarily an integer.
We now proceed, in the same way as in § 116, to extend our definition of the function Jn{z) to the general case in which n is not an integer.
It appears from the proof given in § 147 that, whatever n may be, the differential equation
d-y 1 dy f "°^ _ a dz' z dz \ z-j^~
is satisfied by an integral of the form
y=z^\ r»-' e'~^' fZ<,
provided the path of integration Q is a contour on the <-pIane, so chosen that
the function
t-- e " <-"-'
resumes its initial value after describing C.
149] BESSEI. FUNCTIONS. 273
Now when the real jiart of t is a very large negative number, the function
is infinitesimal. Hence y will be a solution of the differential equation, provided the contour G begins and ends with values of t whose real part is infinitely large and negative.
Let therefore a contour C be taken which begins at the negative end of the real axis, and after proceeding close to the real axis to the neighbour- hood of the origin makes a circuit of the origin and returns, close to the real axis, to the negative end of the real axis again. The integral y taken round this contour satisfies Bessel's differential equation.
We shall now shew that this solution y can be expressed in the form of a series of powers of z.
Suppose as usual that by i~"~' is understood that branch of the function t~"^^ wliich when continued (§ 4.1) to the point t—1 by a straight path, arrives at the point t = \ with the value unity.
Then we have
-'e'. e'it dt
y=z-\ t-"- J c
,.=0 2-.r! Jc'^ ''*■
But (§ 100) we have
r {k) 2iri
^ ' t->=etdt.
■ c
But when n is an integer, we have (§ HG)
r (~\- V V '■> ^
"^ ^ ,ro2»+«-r! r(n + r-i-l)'
Comparing these results, we have, when n is an integer,
where C is the contour already described.
Now we have seen that the right-hand side of this e(| nation has a meaning and satisfies Bessel's differential equation for all values of 2 and all values of ?i ; whereas, up to the present, Jni^) has been defined only for integral values w. A. 18
274 TRANSCENDENTAL FUNCTIONS. [CHAP. XII.
of n. We shall take this opportunity of extending the definition of J„ (z), in the following way.
or
For all values of n and of z, the f auction
Zo 2»+«-r! V{n + r + \)
will he denoted by J,, {z). In the integral, i^""' is to have the value which becomes unity when the variable t travels in a straight line to the point t — \: and (7 is a contour which encircles the point t = Q and begins and ends at the negative end of the real axis in the i-plane. The function Jn{z) thus defined is called the Bessel function of z, of the first kind and of the order n ; it satisfies Bessel's differential equation of order n.
Since Bessel's differential equation is unaltered by the change of n into — n, we see that J_n{z) is also a solution of the equation; and therefore the general solution of Bessel's equation is of the form
aJniz) + bJ_n(z),
where a and h are arbitrary constants, except in the case in v^^hich J,, {z) and J-n {^) are not independent functions ; this exceptional case happens when n is an integer, for then, as we have already seen, we have the relation
J„ {z) = (- 1)" J_„ {z).
A second solution of Bessel's equation in the case when n is an integer will be given later.
150. Tlie recurrence-forvfiulae for the Bessel functions.
As the Bessel functions, like the Legendre functions, are members of the general class of hypergeometric functions, it is to be expected that recurrence-formulae will exist between them, corresponding to the relations between contiguous hypergeometric functions (§ 145).
We shall now establish these recurrence-relations: the proof given does not assume the order n to be an integer, and consequently the formulae are valid for all values of n, real or complex.
Let G be the contour described in the last article, which begins and ends at the negative end of the real axis in the <-plane, and encircles the point t = 0.
Then since the function
e* *H-
150, 1.'>1] BKSSKI, FUNCTIONS. 275
is iiiHiiitfsim.il at the extremities of this contour, we have the equation
or J„_,(-') + J,.+,(^)=y/„(2) (A),
which is the first of the recurrence-formulae. Ne.xt we have, by differentiation,
or
d f / -' 1 r f-€
~r t-"-'e *tdt = -^2 r"-=e *tdt, dzJc ~ J c
^-^^-^Jn{^)-Jn^.(^) (B).
From (A) and (B) it is easy to derive other recurrence-formulae, e.g.
'^^^=^(^n-.(^)-^«+.(^)l (C),
and ^^:^)=J„_^(^)_«J„(^) (D).
az z
Example 1. Shew that
,M/ (','1
Example 2. Shew that
151. Relation between two Bessel Junctions whose orders differ by an integer.
The various recurrence-formulae found in the last article can however be easily deduced from a .'iingle equation, which connects any two Bessel functions whose orders differ by an integer, namely
{^r {Z) = / 1 y Jl_ K"^
^»+'- ^ ' {zdzfX 2» J'
where n is any number (real or complex) and r is any positive integer.
18—2
276 TRANSCENDENTAL FUNCTIONS. [cHAP. Xll.
To establish this result, wc have, by ^ 149, d(z"-y\ z" \~d(z"-Y27ri.2"]c
(-2y ^"+>- "+'■ '-^'''
which is the equation recjuired.
The recui-rence-fonmilae can be derived without difficult}' from this result. Thus, equation (B) of the last ai'ticle is obtained by taking ?•=! in this equation : and equation (A) of the last article may be derived in the following way.
Taking ;■= 1 and ?• = 2 successively in the formula just proved, we can express the first and second derivates of J„ (a) in terms of J„ (z), Jn+i {z) and Jn+2 (•2), in the form
/^ =--0--n)Jn iz) Jn+^ {Z) + Jn+1 {z)-
dz- z- z
Substituting these values in Bessel's equation
d:-Jn{z)^ldJj^^/^_n^ dz- z dz \ z-j
=) (.,1 ^ I ) we have Jn+'^iz) ~ — ^ •A.+i (z) + J„ (z) = 0.
Changing n to (n — 1) in this result, we have
Jn^dz)-^JAZ)+Jn-Az) = 0,
z which is the formula (A) of the last article.
The other recurrence-formulae can be derived in a similar way.
152, 153] BESSfX FUNCTIONS. 277
152. The roots of Bessel functions.
The rolatinn ostalili.slifd in the preceding .irticle enables us to deduce the interesting theorem that between anij two consecutive real roots of Jn{^) there lies one and onli/ one root of J„+,(3)*.
For since J„(z) satisfies Bessei's eiiuatioii, it follows that the function i/ = 2~" Jn(z) satisfies the dift'ernitial cijuatinn
or 2 ,^ + (2n + 1 ) -/ + ^(/ = 0.
dz^ dz
From this equation it is evident that if ^ be a value of z (real and not
zero) for which ,- is zero, then the signs of -, ' and u must be unlike at the dz ° dz-
point z = ^. Now let 2 = f , and 2 = f ... be two consecutive roots of the
rlii d^V
function j- . It is clear from the differential equation that neither 1/ nor ~j-^
d 1/ d^v can be zero at either of these points. Then the function -7- -p^ has a
different sign just before reaching z — ^., to that which it has just after
leaving z = ^i; and hence it follows that the function >/ -^ has a different
.■^ign just before reaching z= ^^to that which it has just after leaving 2 = ^,. The function y must therefore have an odd number of roots between the points z = ^, and z = ^.j.
But from Rolle's Theorem it follows that >j cannot be zero more than once in this interval : so y must have one and only one zero between the
points 2 = f , and z — ^.,: and therefore the zeros of y and of -j- occur
alternately.
Thus, between any two consecutive roots of the function z~"Jn(z) there
lies one and only one root of the function j- {z'" J„ (z)} or - «"" J„+i (z): which
establishes the theorem.
153. Expression of the Bessel coefficients as trigonometric integrals.
We shall next obtain a form for the Bessel coefficients (i.e. the Bessel functions for which the order n is an integer), which in some respects corresponds to the Laplacian integrals obtained in §§ 119 and 132 for the Legendre functions.
* The proof here given is due to Gegenbauer, Moiiatthefte fur Math. vni. (1897).
I
278 TRANSCENDENTAL FUNCTIONS.
If in the equation
we write t = e''^, we have
[chap. XII.
Changing i to — i in this equation, we have
7J = - OO
Adding and subtracting these results, we have
■» cos {z sin <f>)= S J„ (z) cos w(^,
n= - tw
CO
sin (^^ sin (/)) = S Jn(2)sinn(^.
71= - 00
Since J„ (z) = (— 1 )" /_„ (a'), these equations give
cos (z sin (j)) = /„ (2) + 2 Jj (2") cos 2</) + 2/4 (2) cos 4(^ + . . . , sin {z sin ^) = 2 J, (z) sin </> + 2./3 (z) sin 3</) + . . . .
As these are Fourier series, we have (§ 82)
e/„ (2) = - cos?;^cos(2; sin ^)rf^, (« even),
TT j 0 1 f''
0 = - I cos n0 cos {z sin ^) d^, (» odd),
•"■Jo
1 T'' J^(z) = ~l sin »^sin(2 sin^)rf^, (n odd),
0 = — I sin «^ sin (2: sin 6) dO, (n even).
Since
cos {nd — z sin 0) = cos n0 cos (z sin 0) + sin nd sin (^r sin d),
we have in all cases when n is an integer
1 f" J„ (2) = — cos (?t^ — 2 sin 6) dd, T Jo
the formula required.
Example. To shew that for all values of n, real or complex, the integral
1 f" y=- / COB {nd - z ain d) dd "■ .' 0
154] BESSEL FUNCTIONS. 279
satisfies the difibrential equation
d-y 1 dy /, n'\ sin tctt /I n\
which reduces to Bessel's equation when « is an integer.
1 /'" For if y = - j cos {n6 - z sin 6) d6,
we have :r^ = -| sin ^siu (w^-^sin d)cW,
dz n J 0
yi=-- f''sin2(9cos(«d-;sinfl)d^,
dz' TT J 0
d-v 1 /' "■ so y + j2=^/ cos-5cos(M.5-0sin ^)c?^,
dz TT _/ 0
and - -f — 5y = - I .sin (?i5 - 2 sin d) rf^ - - I ^ cos(?jd-2 8m5) aS.
z dz z^^ n J 0 z ^ ■^ J 0 ^~
Xow integrating by parts, we have
-I - sin(7i5-jsm5)Vfl = — sin n7r + - / {n-zcos6)coa{n6-zs\n6)d6,
T .( 0 ' jr^ n J 0 z
and therefore
j^ + - J' + f 1-T )w = — sinKn- + -y- I (m2 cos 5 - ?^2) cos (m^ - J sin 5) (^5
1 jj /■»=«■
= — sinKTT--^ I cos(n5-isin5).f/(w^-2sin5) n-2 z^'r;e=o
1 . n .
= — sin»)7r - ., sin 7I1T irz z'n
_.sin im /I ?j\ which is the required result.
154. Extension of the integral-formula to the case in which n is not an integer.
We shall now shew how the result
J„ («) = - I cos (n0 — z sin &) dO
must be generalised in order to meet the case in which n is not an integer, i.e. the case of the Bessel functions, as opposed to the Bessel coefficients.
I
280 TRANSCENDENTAL FUNCTIONS. [cHAP. XII.
Suppose that tho real part of 2 is positive. Write t = „zu in the formula
we thus have
where m~"~' has that value which becomes unity when the variable u travels by a rectilinear path to the point u= 1. Since values of t whose real part is large and negative correspond to values of u whose real part is large and negative, we see that the path in the «-plane, along which this integral is to be taken, is still a path leading from u = — cc round the point it=0 and returning to m = — 00 .
Let this contour be chosen so as to consist of
(a) a straight line parallel to, and below, but indefinitely close to, the real axis from u = — <x> to m = — 1 ;
(/8) a circle I of radius unity described round the origin ;
(7) a straight line parallel to, and above, but indefinitely close to, the real axis from u= — \ to u= — <x) .
Thus
dit
du,
Jn (2) = ~ f ' «-"-' e^ ^"""^ du + J'.\ w-"-' e' ^" ' ^
2TnJ -I
where w~"~' has in the first integral the value e'""^''*" at m = — 1, and in the third integral has the value e~""+"'''at u= — l. Hence, writing u = — t in • the first and third integrals, and u = e'* in the second integral, we have
Jn (2) = :^ e -»'»+«■ »i" » d(9 4- %^^-r- *-"" ' e^ ^ >' dt
'^TTj-w f ZTTl J I
-^— ^ t-"-'e^^ '' dt,
where, in the last two integrals, f^-^ has the value 1 at the point i = 1. Writing t = e', we have
_^sJnOL+l)Zf%-n.-.s,nh«^0 TT .'0
= — I fgi(i.<ina-ii9> I ^-ifisiiifl-.,ell ^g I Sin(?i + i)7r / g-nS-isinhS^g SttJo TT Jo
I
154] BESSEL KI'NCTIONS. 281
or J„U) = - ['cosizsme-ueydd-^-^"'' re-"0-""''^'de (1).
This formula is valid when tho real part of s is positive. When the real part of z is negative, a similar pi-ocedure leads to the result
J„ (z) = ^""' \ r cos [3 sin 8 + 1,0), 10- sin nir ( ('-««+"!"'•« cWl (2).
TT [J 0 Jo J
When n is an integer, the t'orniula (1) gives
J„ (z) = i f " cos (/,(9 - z sin 6) rW, when the real part of z is positive ; and the formula (2) gives
or, since J„ ( z) = ( - 1 )" ./_„ {z),
1 ;■" J,, (z) = - ms{nO - z sin 0)<I0,
when the real ])art of z is negative.
Thus in either case when n is an integer, we have again the result of the last article, namely the formula
Jn{2)=- j''cos(n0-zsm0)(W (3).
The equation (3) was known to Bossel. Equation (1) is due to Sohlafli, Math. Ann. in. (1871) ; equation (2) was first given by Sonine, Math. Ann. xvi. (1880).
The trigonometric integral-formula for ./„ {z) may be regarded as corresponding to the Laplacian definite integrals for the Legendre functions. For we have seen that the Bessel function J^ {z) satisfies the difierential equation of the function
But the Laplacian integral shews that this quantity is a multiple of
^i=i' /," [' " 2«'^ + {(^ - &)' - 4* ""' "^T ™' '"''* '^'^
or Limit I (l+-*cos(^l con m<f) d(f>,
or I e"">'^ cos m(l)d(p,
Jo
the similarity of which to the above result (3) will bo observed.
282 TIIANSCENDENTAL FUNCTIONS. [CHAP. XII.
155. A second expression of J,, (z) as a definite integral whose path of integration is real.
Another definite-integral fcjnmila, which is vaUd for all values of z and a certain range of values of n, can be obtained in the following way.
The function Jn{z) is expressed for all values of n and z by the series
r=o2''+"-r!r(w-i-r + l)" Since (§ 95) we have
this can be written in the form
"71 \Z) = « /I \ .
Now by § 107 we have
j^cos=-./,sin^».^#^ \^^:\,Xl) ' provided the real parts of (r + ^j and in + ^ J are positive. Thus if the real part of ( « + - j be positive, we have
2" r Q j r («. + ^) '•=0 2H .'o
■ ^ (-l)'-.g"cos=^0
But cos (^ cos d)) = Z -^r—; -.
r=o 2r!
Thus we have
z'" /""
/„ (z) = TT^; -. rr COS (z COS ^) sin=" ^ rf^.
This formula is true for all values of z, and for all values of 7i whose real part is greater than - ^ .
Example 1. Shew that
1 f
F^{cosd)= — I e-^'=<»«J„(.rsin5).r"rfx.
I
lo5, 156] BESSEL FUNCTIONS. 283
For we have
1 /■"
^ J 0
so f e ""»»^„(j;8infl)^''rfx=i ("./(^ I e"^''"''*. e-"'"'«"'»*^"c£^
^1 /■",, ri«+i)
n J 0 ''' (cos ^ + isme cos 0)» * I
= r(ft+l)/^„(eos^), which establishes the result.
Example 2. Shew that
1 ["
r(n-OT + l)_/o
(Cambridge Mathematical Tripos, Part II, 1893.)
156. llankel's definite-integral solution of Bessel's differential equation. If in the result of the last article we write
t = cos (p, we obtain the result
It will now be shewn that this integral is a member of a very general class of definite integi-als which satisfy Bessel's differential equation, namely, integrals of the form
y= z" I e"Ht-- ly-^-dt,
• c
where C may be any one of a number of contours in the ^-plane. The importance of solutions of this type was first shewn by Hankel*.
To shew that integrals of this class satisfy Bessel's equation, we form the first and second derivates of the expression //, and find that
J c
.' c at
• Math. Ann. i.
284 TKANSCENDKNTAL KUNCTIONS. [ClIAP. XII.
FniMi this it is clear that Bcssel's equatimi will be satisfieil by the integral
J c provided C is a closed contour sueh that the integrand resumes its initial value after making a circuit of G.
The similarity of this result to the general theorem of § 142 is very apparent.
157. Expression of J^ (z), fur all values of n and z, by an integral of Hankel's type.
We shall now shew how the particular solution Jn{z) of Be-ssel's equation can be expressed by an integral of Hankel's type. Consider the contour formed by a figure-of-eight in the ^plane, enclo.sing the point t = + 1 in one loop and the point t = — \ in the other, so that a description of the contour in the positive sense involves a turn in the positive direction round the point ^ = + 1 and a turn in the negative direction round the point i = — 1. After turning round the point ^ = -I- 1 in the positive sense, the integrand resumes its original value multii)lied by e<""i''^"'', as can be .seen by writing it in the form
z(t+(ii-J)li)t<((-l)+(n-4)liig((+l).
p,nd after turning round ^ = - 1 in the negative sense, it is further multi- plied by
-(»-J)27ri
Hence after describina' the whole contour, the inte>;rand resumes its original value.
Thus y=^" e'^'{t--l)'''idt
is a solution of the differential equation, valid for all values of z and of n ; the .symbol (1 -h, — 1 — ) placed at the upper limit of the integral indicating that the path of integration consists of a positive revolution round 1 and a negative revolution round — 1
111 this equation we shall suppose as usual that z" has the value which reduces to 1 when z travels by a straight path to the point z = 1, and we shall suppose (<^ — l)""i to have initially the value which reduces to g-fn-S)"' when t travels by a straight path to the point t=0.
To find the relation between this quantity y and the particular solution Jn (z) of Bessel's equation, we expand y in the form
y= 2 ^, I r(«»-l)''-Mi.
1 \r)7] HKSSKI, KUSCTIOXS. 2H5
i
I To evaluate the iiitct^^rals wliirli oi^cnr in tliis series, write
F{r, n) = I r (<•■■- l)"-if/<.
/•(i+.-i-)
Then /'(/■, H+ 0 = (r+= - r ) ( r- - i)" - * </?
-I'
Thus wo have F (r, n) = - ""/ '" t " F (r, 7i + 1 ).
2?i + 1
This result enables us to reduce the evaluation of F(r, n) to the evalua- tion of F (r, n + 1), anil thus to the evaluation of F (r, n + k), where k is a
positive integer so chosen that the real part of (» + k) is greater than — y^ .
We have therefore to evaluate the integral
r(i + ,-i-) '
where we may now suppose that the real part of n is greater than — ^ . The
contour can be supposed to start at the point t = 0, where (f— 1)"~* has the value e-(''-i)'", then to proceed to the neighbourhood of the point t = 1 along the real axis, then to make a positive turn in a small circle round t = I, then to return along the real axis to the point t = 0, where (<-— !)""* has now the value e*""^''", then to proceed along the real axis to the neighbourhood of the point t = —l, then to make a negative turn in a small circle round ^ = — 1, and lastly to return along the real axis to the point t = 0, where (<--!)""* has now the value e"'""*'". Since the real part of n is greater
than — ^ , the integrals round the small circles at t — l and t = — ] are
infinitesimal, and we therefore have
Jo J n
4.e(»-J)«[ <r(l _^3)n-J(^^_g-(n-i)-ri I f (\ - t'^-^ (it,
.' 0 .' 0
where in each of these integrals the quantity (1 — <-)""■* is now supposed to
286 TRANSCENDENTAL FUNCTIONS. [cHAP. XII.
have the vahic unity at t = 0. Writing — t fni' t in tlie two last integiaLs, we have
F{r, m)= - [et"-!)"' - e-C"-*)"') {1 - (- !)'■+'} | fH- f')"-'^ dt.
= — 2i cos- -^ sin [n — ^ j tt ) i/i''-" (1 — i/)"-J dv, where v = t", (by § Wo) = - 2r cos^ ~ sin fn - 1) ttS (^ , n + J)
(by §106) =-2tcos''— sin rn-^JTr
r(. + ^ + i
This result has now been proved to hold .so long as the real part of n is greater than — = : and in virtue of the formula
n / N 2?! + ?• + 2 r, . , .
F(r, «)=-— ^^^-j^^ F(r, n + l),
we see that it holds universally.
Thus we have F(r, n) = 0, when r is odd; and it is therefore sufficient to take ?• even. Let r= '2s. Then the formula becomes
. .. r (.5 + .^) r (r* + J
F(-2s, n)=2ism(n+\]7r—l 11 1 1
^ V ay r(/, +S + 1)
But (§ 97) we have
^ ' ^ ' sm I ?n- ., ) TT
Therefore '' /V F f s + i
F{2s, «)=_^ "
rg-«jr(/. + 5 + 1)
(-1)"^''+=' liirvi^s^y
and so y = S „ , — -p. r ,
.=0 2s! rQ-?i)r(7i + s + i)
, " (-l)»2"+"-» ■ ^^'^^(•i)
or y = S
But Jn{z)= S
=0 2-s! rQ-„)r(« + s+i)-
=„2"+'"s!r(n + s + !)■
I I
158] BESSKI, FUNCTIONS. 287
ThtTcfore J„ {z) = rrr y,
or .. , ,
2"r
This formula gives the rcimired expression of J„ {z). It is valid for all
values of n and of z\ but when n is of the form {k-\--\, where A' is a
positive integer, the factor T (., - ii\ becomes infinite and the integral
/■(1+, -1-)
I e'"(^'-l)"-Jrf<
becomes zero (since the integiand is now rey;ular at all points within the contour), so that for this exceptional case the formula is indeterminate.
Example. Deduce the formula
from the result of this article.
158. Bessel functions an a limitin'j case of Legendre functions.
We have already (§148) shewn that Bessel's differential equation of order //) is the same as the dififerential equation of the associated Legendre functions
Limit Pn- (l - £^ and Liniit Q,.'" (^ " |^) •
We shall now express this connexion more precisely, by establishing the
formula
z-
J„.(2)=Limit«-'"P„"' 1- ,.
For taking the expression of the associated Legendre function by a definite integral (^ KU), we have
-m pml] _ ^' \ = (n + m-){n + m-\) (n-TO+l)^"' / _ ^y™
" " \ 2nV (2m- 1) (2m -3) 1.7r.»i^" V 4W
f f z^ I z- z^W """*
and as n becomes infinitely great, the right-hand side of this equation tends to the limiting value
(2m - 1) (
5^ — I I 1 + - cos<^ s\n^^d4>,
2m — 3)...I .TrJo \ n ^/ ^ ^
288 TKANSCENDENTAI, FUNtJTlONS. [cHAP. XII.
(2ni — 1 ) (2w — 3) . . . 1 . TT .' 0 or (§155) -fmiz),
which establishes the result stated ; it is due to Heine*.
159. Bessel functions whose order is half an odd integer.
The result of § 157 suggests that when the order w of a Bessel function
Jniz) is a number of the form k + -^, where k is a positive integer, certain
exceptional circumstances arisef in connexion with the function. In this case it is in fact possible to express the Bessel function
2*+i \ Z- "* ''
2*+4r (/.- + ^) i -2 (2i- + 3) ' 2 . 4 . (2A; + 3) (2A; + 5)
in terms of well-known elementary functions.
For by § 151 we have, if k be a positive integer,
But the series-expansion of the function /j {z) is
/, (2) = — - 1 _ + — _- — ; — _ _ . . I = ( __ Sin 2.
-^ ' TT* [ 2.32.3.4.5 I \7rzJ
Therefore J,,, (.) = -^ ^ ^^,^-, ^-
which is the required expression of the function Jj.^.j {z) in terms of more elementary functions.
■ The .student will without difficulty be able to prove that a second solution of Bessel's differential equation in this case is
* Heine's definition of the aRSociated Legendre function is somewhat different from that which has since become Reneral and which is adopted in this book : this leads to differences of statement in many other formulae, such as that of this article.
+ The student who is familiar with the theory of linear diiJerential equations will observe that in this ease, and also in the other exceptional case of n an integer, the difference of the roots of the " indicial equation " of Bessel's equation is an integer.
159, 160] BESSKL FUNCTIONS. 289
Example. Shew that tho sDliitioii of the equation
whore the quantities c^ are arbitrary constants, and oq, a,, ... uv,,,, are the roots of the equation
Q-im + i — ;- (Lommel.)
160. Expression of Jn{z) in a form which furnishes an approximate value to Jn {i)for large real positive values oj z.
We now proceed to form an integral which will be found to play the same part in the theory of the function /„ {£) as the integral of § 104 plays in the theory of the function V {z). We shall suppose z to be real and positive. Then, by § 15.5, we have, for all positive values of ■/!.,
\
Jn {z) = -^ — TT / cos {z cos d)) sin*"d) dS.
2''.rhi. +.-,j.7ri-'o
Writing cos <f>= x, this becomes
-.11 r+i
Jn (z) = r— — ; I (1 ~ ar)'^~^ cos zxdx,
or J„ {z) = Real part of ■ 1(1- «=)""* e"" dx.
In order to transform this integral, we take in the plane of a complex variable t a contour OPQBCO, formed in the following way. 0 is the origin (( = 0) ; P is the point t = I — p, where p is a small quantity, and OP is the part of the real axis between 0 and P. Q is the point t=l + ip, and PQ is a quadrant of a circle which has its centre at the point t=l. B is the point t=l +ik, where k is a large positive quantity, and QB is the line (parallel to the imaginary axis in the <-plane) joining Q and B. C is the point t = ik, and BC is the line (parallel to the real axis) joining B and C. Lastly, CO is the part of the imaginary axis between C and 0. Then the function
(1 - t-y-i e^
is regular at all points of the i-plane in the interior of tho contour OPQBCO ; and therefore the integral
/'
taken round this contour, is zero.
w. A. 19
290 TRANSCENDENTAL FUNCTIONS.
We can write this relation in the form
[chap. XII.
/ +f ./ W +/ =„.
J OP J J'Q J QJB J BC J CO
Now the part of the integral due to PQ tends to zero with p, and the part due to BG tends to zero as k becomes infinitely great, while the part due to CO is purely imaginary. Thus we have
Real part of I = — Real part of I ,
.' OP J QB
and so J„ (z) = Real part of In this integral write
2"-
(-»■
i^ QB
QB
(1 -0""*e"'ti<.
lU
z so that u varies between the limits 0 and co when t describes the line QB;
and
and therefore I = 2"~> le
' QB
Thus we have
t/„ (z) = Real part of
1
J QB Jo
-.(l.g)-'...
(-^)
' •' r
or J„ (z) =
("4)i}//-»-'i(>4r-(-£rH''
cos iZ
+ Sin -^2:— ?l +
1\ tt)
2; 2
( g-" It"-
-^I'-sr-o-sn--^-
This is the integral-expression required. It is easily seen to furnish an approximate value of J„ (z) for large positive values of z ; for as z becomes indefinitely large, the two integrals in the expression tend respectively to the
limits 2r ( n + - j and zero ; and therefore the function /„ (z) approximates
for large positive values of z to the value
2\J — cos iz
TTZj
-("+2)1
BESSEL FUNCTIONS.
291
jL 160]
The evaluation of Jn{z) when 2 is large will be considered in fuller detail in the next article.
I
Tlie result of this article civn also he obtained in tlio following quite different manner, which connects it more closely with the general theory. We have seen in § 148 that Bcssel'a differential equation is a limiting case of the general hypergeometric equation, represented by the function
/" 0 00 c '
e" Limit P
-« i 0 ;
71 i?-2jc 2ic- 1
I
-3-y,
or (putting t= — ivz)
Since the differential equation of the P-function
■ 0 00 c P- a li y
. "' H' i
is (§ 142) satisfied by the integral
2« (Z - C)1' [ «3+1'+»'-l it - cr+^+l-'-' {Z - t)- taken between suitable limits, we see that Bessel's equation is satisfied by the expression Limit e"i-" [ V-h\\-^-\-''-\''''-^' (z-ty-ldt,
or e^z-^\v'-\e-'^^^{z-tY'\dU
gii^-n / j,n-} jn-i (z + m)""* e-'^'zdv, or e"z" I (i)+«V)""Je-2tp»(;j,
The limits of the integral can be taken to be 0 and oo , since these satisfy the conditions for the limits found in § 142 ; and hence it follows that
e"i» I (w + iv^Y - i e-2« dv
is a solution of Bessel's equation.
Similarly the quantity
e-^'z" / (v-u'2)»
is a solution of Bessel's equation.
The solution J„ (z) must therefore be of the form
i e-'^'"dv
J„{z) = Ae^z'' f {v + iv-y-ie-^'^dv-^-Be-":" j {v-w^)''-ie-''"dv,
19—2
292
TRANSCENDENTAL FUNCTIONS.
[chap. XII.
where ^■l and B are constants independent of z. This is substantially the form given above, but the determination of the constants A and /J is a matter of some difficulty, for which the student is referred to a memoir by Schafheitlin, Crelle's Journal, cxiv. p. 39.
Example. Shew (by making the substitution u = 2z cot <j} in the integral found above, or otherwise) that
on + l^n /■ 2
J„(z)= 1 e-2"<''''''cos»-i0coseo2"*i(^cos{z-(?i-i) d)}cW).
r(n + i).'jT^J 0
161. The Asymptotic Expansion of the Bessel functions.
The Bessel functions can for large values of the argument be represented by asymptotic expansions. We shall here consider only the asymptotic expansion of J„ (z) for positive real values of z ; this was discovered by Poisson (for n = 0) and Jacobi (for general integer values of m). The theorem has been considered for complex values of z by Hankel * and several subse- quent writers.
We shall derive the asymptotic expansion from the integral-expression
1
/„(^) =
(27r^)4 T(n+l
cos iz — [11 + + sin iz — \n +
INtt
2/ 2
1\-J7
2 2
zj
-('-I)
U \"-i
n-i-
du
Idu,
found in the last article.
It is first necessary to find the asymptotic expansion of the integral
e-«„Mi+ -J du, {k>0),
which we shall denote by the symbol I. Now we have
, iuY I 7 *M Jc(k—l)/iu
+ ... +
'k(k- l)...(k-n + l) fiu
Tz)
k{k-l)...(k-n) p' /ill \",,
dt.
Math. Ann. I.
161] BESSEL FUNCTIONS. 293
Therefore
/=re-»»*d.+ i A^(^-i)-(^->-n)(iyr, -,.,...,,
Jo r=i rl \->z) Jo
or
/=r(i + i){i + iM:ii)ii^(^r!L!±)Q''(A. + ,-)(Z+r-i)...(/.-4- 1)| +ij,.,
where
7?,. = ^ L^ .- (2-) j ^ e- «^ rf« J^ (a - V)" (1 + 2-J cfe
Now as ^ becomes iutiiiitely large, n having any definite finite integer value, the remainder-term R„ tends to the limit
«! Vz^/ Jo .'0
„ k{k-\)...(k-n)( iy*' r „ ,. „^, , (?i + l)! ylz) .'„
_ A^(^■ - \)...{k-n) T{k-n + 2) ( i V+' (n + l)l \2z) •
It follows from this that
Limit 3"i?„ = 0,
2 = 00
and therefore the series
vik^ 1) ji + |^(^- + ^-)(^- + ^--i)-(^->-+i)0|
is the asymptotic expansion of the function
[ e-'M*('l + ^ydM (A,'>0).
Substituting this result in the expression already found for J„(^), we see that
1 / .n^lL. ^("-^^)("-^'•)•^•("+^-0^- + (-^)1" ..inL-f„ai^Uv("-^"'-)("-^^^)-("-^^^)^--(-r
2; 2 J I -, ;•: (2zX
294 TRANSCENDENTAL FUNCTIONS. [CHAP. XII.
cos iz
is the asymptotic expansion of the Bessel function Jn{z) for large positive values of z.
Even when z is not very large, the value of /„ (2) can be computed with gi'eat accuracy from this formula. Thus for all values of z greater than 8, the first three terms of this asymptotic expansion give the value of J^ (z) and t/j (z) correct to six places of decimals.
162. The second solution of Bessel' s equation when the order is an integer.
We have seen in § 149 that when the order n of Bessel's differential equation is not an integer, the general solution of the equation is
aJn (z) + 0J^n (^).
where a and /3 are arbitrary constants.
When however n is an integer, we have seen that
J„(^) = (-!)"/_„ (4
and consequently the two solutions J,i (z) and J_n (2) are not really distinct. We therefore require in this case to find another particular solution of the differential equation, distinct from Jn(z), in order to have the general solution.
To obtain this second solution, we write
y = uJn (z),
where m is a new dependent variable, in Bessel's equation
d^w 1 du /, w=N
Remembering that Jn(z) is a solution of Bessel's equation, the differ- ential equation for u becomes
162] BESSEL FUNCTIONS. 295
dh(, d Jn (2)
or -J- + 2 +- = 0.
du ,/„ (z) z
dz Integrating this equation, we have
log T- + 2 log Jn (z) + log z = constant,
or T- = , r , s^ . where 6 is a constant,
dz z [Jn {z)}-
, P d
dt
where o and h are arbitrary constants.
The complete solution of Bessel's equation can therefore be written in the form
y = aJ„iz) + bJ.(z)j ^jj-^,.
To find the nature of the solution thus obtained, we observe that in the vicinity of the point t = 0 the integrand
IS of the form
<■"-""' (constant + powers of t')~-,
■which when n is a positive integer can be expanded as a Laurent series in the form
The function
j't-'{Jnit)}-'dt has therefore the form
-^j«. +>.-= +...+— + d\ogz + d,z-+...,
where the quantities rf_m. (^-m+a. ■•• are definite constants.
It thus appears that the complete solution of Bessel's equation can be written in the form
y = AJ„ (z) + B {J„ (z) log z + v],
where v is the result obtained by multiplying together J^ (z) and a Laurent series of the form
296 TRANSCENDENTAL FUNCTIONS. [CHAP. XII.
whore the quantities rf_..,,i, d-..n+-,, ... are definite constants, and A and B are arbitrary constants. The expansion oi Jn{z) being known, we see that the product V has the form
z~" X a power-series in z^ ;
and thus a second solution of Bessel's differential equation, in the case in which n is an integer, can be taken of the form
Jn (z) log z + z~"' (flo + a^z- + a.,z* + a^z" + ...),
where the quantities «„, Oi, a^,... are definite constants. These quantities a are not however all of them strictly speaking definite, since by adding a multiple of J„ {z) (which will leave the expression still a solution of Bessel's equation), it is possible to change all the quantities a after a„_,.
This solution will be denoted by Kn{z)*.
The coefficients ao, a,, a.^, ... may theoretically be determined by substi- tuting this expansion in the differential equation, and equating to zero the coefficients of successive powers of z. A better method is however the foUowing-f-.
We have seen that when n is a positive integer, J^^{z) reduces to (— Vf'Jniz); in fact, if in the equation
/_,„_, {z) = ( 2 j ^S^ Y{-n + e+p + \)T{p+l) Uj
we suppose the quantity e to tend to zero, all the terms of the series vanish as far as p = n, since T{—7i+p + l) is for these terms infinite. Changing the meaning of the index of summation p in the other terms, we have
'"-"^^''"W ;:,r{-n + e+p + i)r{p + i)[2)
p=aT(e+p + l)T{n+p + l)\2j
and when 6 = 0, the first of these partial series is zero and the second is
(-!)»/„ (4 Since the quantity
(- 1)" /_,„_,, (^) -/,„_., (^)
vanishes with e, we can take as a second solution of Bessel's equation the limiting value of the quotient
(-1)"J_,^„(^)-J,^.,(^)
* In referring to memoirs it must be borne in mind that different writers have taken different definitions of. the Bessel functions of the second kind. t Due to Hankel, Math. Ann. i. p. 470 (1869).
I
162] DESSEL FUNCTIONS. 297
Substituting the above values for the Bcssel functions, this becomes
(z N-'"-i "^> (- 1)"+" 1 (iY , pr ? (_ iNp/ii) (i]''
where /(e) represents the expression
^^^^" r(H+p + i)r(p + €+i)V2J ~ r{n+p-e+i)r(p+i)\2) •
The limiting value o(f(e)le, as e tends to zero, is
1 .g, z 1 r'ip + i)
r (n +p + i)r(p + i) ^^ 2 r{n + p + i){rip + i)p
1 r'(«+;j+l)
r(p+i){r(n+^+i))^
Also, since
1 1
= — T {n — e — J)) sin {— n + e + p+l)7r,
r{-n+€+p+l) TT
we have Limit -^, ■, x = (- 1)"+"+' T{n- p).
.=0 eT(-n + e+p+l)
Consequently we obtain, as a second particular solution of Bessel's equation, the expression *
_ /^ ,-" "^1 r{n-p) /z_Y /£y ? (-1)^ M^
\2) p:o'r(p + l)\2) '^\2j ,Zor(n+p + l)r{p + l)[2j
9 in„ ^ _ r'jn+p + i) _ r(p + i)
" ° 2 Tin + p + l) r(p + l)
The coefficient of log^ in this expression is 2J„(z). So, dividing the expression by 2, we have the second solution in the form
^„(^)log.-2.:;^^i p^^y -/„U)log2
/A" I (-ly p\^ 1 ( _ r(n+p+i) _ r(p + iy
"^V2/ pror(n + p + l)r(;) + l)V2/' 2| r(n+;j + l) r(;) + l) It is convenient to add to this expression a term
J„(.)|log2+I^^^^^^^},
* This is Hankel's second solution Y„ (z). It is really
dn ** ' dn •
298 TRANSCENDENTAL FUNCTIONS. [CHAP. XII.
which is itself a solution of Bessel's equation ; so the second solution now takes the form
J-„(.)Iog.-2j^— 1^ y
2\2j ptor{n+p + i)r{p + i)\2j \r{n+p + i)^rip + i) r(i)|'
This is the solution K,^ {z) which we take as our standard. Since, when r is a positive integer, we have
r'(r + l) r'(l) ,11 1
— !^ — I — L 1-2 = 1-1 1 1- +-
r(r+i) r(i) ^2^s^ ^r'
we can write K^iz) in the form
Z„(.) = J„(.)log.-2y X^' ^, (^)
1 /zY 2 (- ly (, 1 1 1 -, 1 1 ] fzyp
-2[2),li^hw^.Y^2 + S-'---'-p+''--2^---'-^p\k) ■
When 71 is an integer, the two independent solutions of Bessel's differ- ential equation are J„ (z) and if,, (z).
Example 1. Shew that the function A',, (i) satisfies the recurrence-formulae 7iK„{z) = hz{K\^,{z) + A'„ . , (z)},
These are the same as the recurrence-formulae satisfied by J„ (z). Example 2. When the real part of z is positive, shew that the expression I sin(2sin^-?M^)c;(^- I e-«8inhe|e«e + (_i)ng-nej ^^^ is a second solution of Bessel's differential equation of integer order n.
(Schlafli.)
Example 3. Shew that the expression is a second solution of the Bessel equation of order zero.
163] BESSEL FUNCTION'S. 299
163. Neumann's expansion; determincution of the coefficients.
We shall now consider* the expansion of an arbitrary function f{z), regular at the origin, in a series of Bessel functions, in the form
fiz) = aJo (-') + a, J, (z) + a,J., {z) + ...,
where the coefficients a„, a,, a.,, ... are independent of ^.
Suppose first that such an expansion is possible, and let us try to determine the coefficients, by expanding both sides of the equation as power-series in z and equating coefficients of the several powers of z. Since
an
fiz)=A0)+2 (^|j/'(0)+ ^, (|) f"(0)+~ (Ij /"'(0)+ ...
^ ^■'^'^ = i(^f{'-lW^){i)'-'-2Hn + l)in + 2)& -■■■}' we have on comparing coefficients the equalities
/(0) = «„, 2/'(0) = a„
27'" (0) = - 2^0 + «., etc., from which without difficulty we find «o = /(O),
a,. = 2 |/(0) + |V" (0) + !^K^^)_^-iv (0) + . . . + 2-V I"' (0)| (n even),
+ >M»'-P)(n'-30^-,., (0) + _^ 2"-/ <") (0)} (« odd).
These coefficients take a simpler form, if we introduce functions Oi(z), 0«{z), Oiiz), .... defined by the formulae
^ , , 1 n= nUn^-2-) 2"-' re! , . ,
On(z) = -,+ ' ^ '+^ ^ ,^+. ..+__- (nodd);
for then it is easily seen that a„ is twice the residue of the function On(t)/(t)
' C. Neumann, Thcorie der Bessel'schen Functionen. The exposition here given follows Kapteyn, AnnaUi del'EcoU Normale (3) i. p. 106 (1893).
300 TRANSCENDENTAL FUNCTIONS. [CHAP. XII.
at the point t = 0. The two formulae for 0„ {z) can be united by reversing the order of the terms ; thus
+ S— TT^T- STT^; 7T+---
2(2?i-2) 2 . 4 (2?i - 2) (2?t - 4) the series terminating with the term in 2" or 2"~\ We thus have Neumann's expansion
f{z) = Oo Ju (z) + a, Ji lyz) + 02 Jj («) + ..., where a„ =/(0),
and (Xn (« > 0) is twice the residue of 0„ {t)f{t) at the point i = 0, so that
an = -.\ 0,dt) fit) dt,
TTl J y
where 7 is any simple contour surrounding the origin.
164. Proof of Neuviann's expansion.
The method by which this result has been found cannot be regarded as a proof, since the possibility of the expansion was assumed. We can, however, now furnish a proof by determining directly the sum of the series obtained.
From the definition of On (z), we can at once obtain the identities 0„+, (z) + 2 ^^^ - On-, (z) = 0, (n > 0),
Oo{z) = l.
Writing the first of these equations in the symbolic form
0„+, - 2D0n - 0„_, = 0, where -D = ^ ,
and solving the series of recurrence-equations obtained by giving n integer values, in the same way as if D were an algebraic quantity, we obtain for 0„ the symbolic expression
On(z) = ^[[-D + (D'+\)i}'' + {-D-{D'-+ l)i)"] ^^'
This symbolic expression can be transformed into a definite integral in the following way.
104]
We have
BESSEI. FUNCTR)NS.
t Jo
301
where the ui)I)<t limit must be understood to mean that direction at infinity which makes the real part of tu positive and infinite ; and therefore
.'o or, writing tu = x,
On (.<)=[ h <"""' e" [{^ + (*•' + <-')*i" + {x-ij^ + «»)»)"] dx, Jo
where the upper limit now means the real positive infinity, so that the integration may be regarded as taken along the real axis of a;.
Writing this in the form A-=» -21 Jo
0„(n= Limit -
' +(— l)"-^ -— — -r-if e^ da;,
we have
0,{t)J,{z) + 2:iOn{t)Jn{z) n = l
Jn{z)
■ e~* dx,
»
(by § 146) ■ =^ Limit 2e^' *• *+(x«+i=)M . e-*rfir
= - Limit I e ' dx.
t Ji= oo Jo
z — t In order that this integral may have a meaning, the real part of — —
must be negative, a condition which is fulfilled when
If this inequality is satisfied, we have therefore
0„ (0 J, (^) + 2 i 0„ {t) J„ {z) = — .
M = l t—Z
From this result Neumann's expansion can at once be derived ; for let f{z) be any function which is regular in the interior of a circle C whose centre is at the origin, and let i be a point on the circumference of the circle. Then if z be any point in the interior of the circle, the condition | ^ ] < | i | is satisfied, and therefore we have
-^ = Oo (0 J, (z) + 2 i 0„(0 J„ {z). t—z „=i
302 TRANSCENDENTAL FUNCTIONS. [CHAP. XII.
Thus /(.)=iJ/W^'
= a„ /„ (z) + a, Ji (2) + a^J. {z) + ..., where Oo =/(0)
and an = —.{ 0„ {t)f{t) dt (n > 0).
TTl .,' c
This establishes the validity of Neumann's expansion for points z within the circle C.
Example. Shew that
cos z=Jo (2) -2.72(2) + 2-^4 (2)- ••■) aim = 2Ji{z)-2J3{z) + 2J^{z)- ....
165. Schlomilch's expansion of an arbitrary function in terms of Bessel functions of order zero.
Schlbmilch * has given an expansion of a quite different character to that of Neumann. His result may be stated thus :
Any function f{z) which is finite and continuous for real values of z between the limits z = 0 and z = ir, both inclusive, may be expressed in the form
f{z) = Wo + a^Jt, {z) + «2-^o (2^) + ttsJo i^z) + ..., where a„ = / (0) + - [ m f (1 - <")-*/' {ut) dt du,
TTJo JO
«n = - M COS nu\ (1 - t-)-if' (ut) dt du (n > 0).
TTJo Jo
Schlomilch's proof is substantially as follows.
Suppose that F and / are two functions connected by the relation
f(z) = - [\l-s'^-iF(zs)ds.
Then we have
/' (z) = - [\l - sn-i sF' (zs) ds.
TTJo
Zeitschrift fiir Math. u. Physik, u. (1857).
I
165] 15ESSKL FUNCTIONS. 303
111 this equation, write zt for z, multiply both sides hy z {\ — t-)~^ dt, and integrate with respect to t between the limits < = 0 and t = \. Thus
- f \ 1 _ t-y^f {zt)dt^^ ! (I - t')-i dt f (l- s-)"* sF' (zst) ds
Jo TT J 0 Jo
= - (z'-x'-f)-iF'{a:)dxdy,
TT.'o ^0
where x = zst, y = zs{l — t-)K
Performing the integrations, we have
zf {l-t')if'{zt)dt = F(z)-F(0). Jo
Now by the definition of the function/, we have
/(0) = F{i)).
Thus F(z) =/(0) + z (\l - f-)-if' (zt) dt.
Jo
This equation expresses the function F explicitly in terms of the function /, whereas in the original definition f was expressed explicitly in terms of i^.
In order to obtain Schlomilch's expansion, it is merely necessary to apply Fourier's theorem to the function F(zs). We thus have
2 |"i {If" 2 " f" )
f(z) = - j (I - s"-)~^ ds \- I F{u)du + - S cos mt cos ms F(ii) dti[
1 /'" 2 "^ C"
= - F{u)dn. + - 2 I cos nu F {u) J„ (nz) du.
T" Jo TT n = l Jo
Id this equation, replace F{u) by its value in terms of /(«). Thus we have
f{z) = ^ JJ |/ (0) + u j\l - «')-*/' (ut) dt^ du
+ - i Jo (nz) r cos nu 1/(0) + u ( {I - t')-i/' (ut) dt] du,
'"■ n=l Jo [ J 0 )
which is Schlomilch's expansion.
Example. Shew that if 0 $ z $ tt, the expression
304 TRANSCENDENTAL FUNCTIONS. [CHAP. XII.
is equal to z ; but that, if n- ^ ^ < 2n-, its value is
z+27rcos-i--2(.-2-jr2),
z
where cos~* — ia taken between 0 and - . z o
Find the value of the expression when z lies between in and Stt.
(Cambridge Mathematical Tripos.)
166. Tabulation of the Bessel functions.
Many numerical tables of the Bessel functions have been published. Meissel's tables (Berlin, 1889) give the functions Joiz) and Ji{2) to 12 decimal places for real values of z from z = 0 to z= 1.5^, at intervals of 001.
Tables of the second solution F„ {2), defined by the equation
F„ (z) = /„(2) log^ + J, ~ (^ + 2) 2^= + •••'
from z = 0 to ^ = 10'2, are given by B. A. Smith, Messenger of Math. xxvi. (1897).
The British Association Reports for 1889, 1893, 1896, contain tables of the functions Ini^), which are solutions of the differential equation
(Pu 1 rfw / n-\ _ dz- z dz \ z-
so that /„ {z) = t~" /„ {iz).
A table of the first 40 roots of J^ {z) is given by Wilson and Peirce, Bull. Amer. Math. Soc. iii. (1897).
Miscellaneous Examples.
1. Shew (e.g. by multiplying the expansions for e''^ i' and e ^^ t' ^ and equating the terms independent oit) that
Ko {")} ' + 2 {^1 {z)] H 2 { J, (J)}2 + 2 {^3 {z)} 2 + . . . = 1,
and hence that, for real values of z, J^ (z) can never exceed unity, and the other Bessel coefficients of higher order can never exceed 2~i.
2. Shew that, for all values of ^ and v,
» (-1). (''+';+ 2»)(^,).+.+.„
J^{z)Jy{2)= 2
=0 V(fi. + 7i + \)V{v + n-k-l) '
MISC. EXS.] 15ESSEL FUNCTIONS. 305
3. Show that
4. Shew that
5. Shew that
/M ^cU,{z) 3<r-J„{z) d^Joiz) J„(z) 7i + \- n+2- 71 + 3- ..."
/-MW^M-.(--) + ^-..,(--)-^.(-') = 't'/"-
6. If "^"; '/f Ix; denoted bv <?„ (2), shew that
'%P = l-'J^^Q^izHz{QA^)y
7. Shew that
(Lommel.)
8. If the function
rj,r (22 cos 8) de = n{Jr (z)V.
1 f
— I 2' cos* ?« cos {mu — z sin !«) du T j 0
"(which when /t is zero reduces to a Bessel function) be denoted by J^'' (2), shew that
J„t(2)= 2 ^,(i2)"i\^_,„,l,p,
where iV.^, i, p is the " Cauchy's number " defined by the equation
1 f-" y-m,k,,. = ^ / e-"""(e"' + <i-'»)*(t>'"-e-'")PrfM. 2jr _/ 0
Shew further that this function satisfies the equations
and zJ*""' (2) = 27ny*"" (2) - 2 (/• + 1 ) {^i_i (2) - ^^+, (2)}.
9. If quantities v and jVare connected by the equations
M=E-es\n E, cosi)=, ,-. . where |e|<l,
1 - e cos A '
shew that v = i/'+2(l-e2)i 2 2 (Ae)' J„,*(me) - sin mJ/,
m=l *=0 '"
1 /■"■
where <^m*W = ~ I (2 cosM)*cos(7n«-zsin«)c?M.
"■Jo
(Bourlet.)
10. Prove that
^."•(003 5) = ^^ J-„{(j.'2+y2)J II ^n
where 2 = r cos tf, x^+y- = f'^sm^ 6, and Cn" is a numerical quantity.
(Cambridge Mathematical Tripos, Part II, 1893.) w. A. 20
306 TRANSCENDENTAL FUNCTIONS. [CHAP. XII.
11. Shew that, if n is a positive integer and («i. + 2m + l) is positive,
Jo Jo
(Cambridge Mathematical Tripos, Part I, 1899.)
12. Prove that
2 f
Ja{^) = ~ I sin (z cosh m) 6?t*. "• y 0
(Cambridge Mathematical Tripos, Part II, 1893.)
13. Prove that
and if Y„ (z) is Hankel's second solution of Bessel's equation, defined by the equation
1 r„ (.) = Limit ■^-^(--)-'^n(^)cos>»^ ^ "■ »i=integer Sm "■"■
Shew that ^rn«=2„..„/^(n + i)0 + £)" f?')"
14. Shew how to express «^"^2n (') in the form
J/j z) + BJ^{z), where A, B are polynomials in z ; and prove that
^,(6^) + 3J„(6*) = 0,
3/6(30i) + 5J^,(30^) = 0.
(Cambridge Mathematical Tripos, Part II, 1896.)'
1 5. Prove that, if J„ (n^) = 0 and J„ (01) = 0,
P scJ^ {ax) J„ {M d.v=^0, and [{-v { J„ (ax)} 2 rf.r = i|2 ..^„ ^ ^ („|)}2.
Hence prove that the roots of J„{x) = 0, other than zero, are all real and unequal.
(Cambridge Mathematical Tripos, Part I, 1893.)
16. Shew that
x-'' + -'"J„{ax)dx = 2'''*"'a"-"'-^
2n + l >?/i ^ -1.
(Cambridge Mathematical Tripos, Part I, 1898.)
MISC. EXS.] BESSEL FUNCTIONS. .SO?
IT. Shew that
(Lomruel.)
18. Shew that the solution of the differential equation
where ^ and i/^ are ai-bitrary function.s of z, in
19. Shew that
(^y j'J^{zsm6)sm^*i0de=z-iJ,„^i{z). 20. In the equation
the quantitj' n is real ; shew that a solution is given by
( _ l)m j2m cog (u^_ jj log z)
(Hobson.)
COS (n log z)- 2
- 22"'ni!(l+«2)*(4 + H2)i (ra2 + jt2)i '
where !<„, denotes
tan-i v + tan-i'^ + ... + tan-i - . 1 2 /;(
(Cambridge Mathematical Tripos, Part II, 1894.)
21. Prove that the complete primitive of the differential equation d^u 1 du /, 7n-\
where m is a positive integer, is
u = AI^{z) + BIi„{z), where, for real values of z,
^•" (^^=1.3. 5. ..(2m - 1) . J „ ""'^ (-' '"' "^^ «'""" "^ '''^'
^•»(^)=lT^H2;n-l) jo <^-"«"'*-"''^'"*'''^- Prove also that
K„{z) = l.3.')...{2m-l){-z)'" j {ii:'+z-)-"'-hcosudu.
20—2
308 TRANSCENDENTAL FUNCTIONS. [CHAP. XII.
Shew that fur very .small values of z,
A'o (.-)=- log I --577..., and that for very large values of z,
(Cambridge Mathematical Tripos, Part II, 1898.)
22. If C be any curve in the complex domain, and m and n are integers, shew that
l^^^m ('■)•/„ (2)^2 = 0,
J^O„.(^)0„(2)C/Z = 0,
where ^- = 0 if the curve does not include the origin ; and, if the curve does include the origin,
^• = 0 if m + n,
t^iwi if m — n.
CHAPTER XIII. Applications to the Equations of Mathematical Physics.
167. Introduction: illustration of the general metliud.
Tlie functions which have been introduced in the three preceding chapters are of very great importance in the applications of mathematics to physical investigations. Such applications are outside the province of this book : but most of them depend essentially on one underlying circumstance, namely that by means of these functions it is possible to construct series which satisfy certain partial differential equations, known as the partial differential equations of mathematical plii/sics ; and in this chapter it is proposed to explain and illustrate this fundamental projaerty.
The general method may be explained by considei-ing first the solution of the partial differential equation
V^+V^, = 0 (1);
ox- oy-
a solution which, while resting on the same principles as those to be developed later, does not require the use of any but the elementary functions of analysis.
Consider any solution V{x, i/) of this equation (1). Near any point at which a branch of the function V(a:, y) is a regular function of x and y, and which we may without loss of generality take as origin of coordinates, this branch of the function V {x, y) can by Taylor's Theorem be expanded as a power-series of the form
V{x, y) = «(, -I- aiX + b^y + a.,T- + h.jcy + c.y- + a^x' + ;
on substituting this value of V in equation (1), and equating to zero the coefficients of the various powers of x and y, we obtain the relations
a., + Co = 0,
3«3 + Cj = 0,
3d, + 63 = 0,
310 TRANSCENDENTAL FUNCTIONS. [CHAP. XIII.
Fixing our attention on thosH terms in V which are honiogeneoiis of the wth degree in a; and y combined, it is clear that the equalities just written will furnish (/( — 1) relations between the (m + 1) coefficients of these terms of degree n. When these etjuations are satisfied, there will therefore remain only ((n + 1) — (n — 1)} or 2 coefficients really arbitrary in the terms of the «th degree in V.
Now the expressions
V=(x + iy)"'
and V={x- iijY
satisfy equation (1), and therefore if An and B,i are any arbitrary constants, the expression
An{x + iyY + B„{x-iyY
satisfies equation (1), and is homogeneous of the «th degree in x and y, and contains two arbitrary constants. It therefore represents the most general form of the terms of the nth degree in V ; and so the general solution of equation (1), regular at the origin, can be expressed in the form
V{x,y) = A„ + A, {x + iy) + B,{x- iy) + A^ {x + iy)- + B,(x-iyf+ (2),
where the quantities Ao-, A^, B^, A.., ... are arbitrary constants.
This expansion furnishes the general solution of equation (1) ; what is however in general needed is the particular solution of equation (1) which satisfies some further conditions. As an example of the conditions most frequently occurring, we shall suppose that the value of the required solution V{x, y) is known at every point of the circumference of a circle, whose centre is at the origin and whose radius is any quantity a ; it being supposed that this circle lies wholly within the region for which V is regular. This being given, we shall shew that the constants A^, A^, B^, ... can be found, and the solution can be completely determined.
For writing
x = r cos Q, y= r sin 6,
the value of V is known when r = a, as a function of 6, say f{6). Let the function /(0) be expanded as a Fourier series in the form
f{d) = af, + tti cos 0 + 6i sin ^ + 0.2 cos 20 + 6, sin 2^ + (3),
where the coefficients fto, a^, b^, a.,, ... are given by the formulae
\
1 f'^" .
dt
1 /■-" a,i = - f{t)cosntdt y (4).
1 f^' h„ = - fit)
TT Jo
sin iitdt
11)8] APl'LR'ATIONS Tn THE EQUATIONS OF MATHEMATICAL PHYSICS. 311
Consider now the expression
Oo + - (a, cos 0 + b, sin 0) + [-] (a-, cos 29 + b., sin 20)+ (5).
This expression (">) reduces to (3), i.e. to f(0), when r = a; and since we have
r» cos nd = g {(a; + iyY + {x - vy)"},
>•" sin nd=^. {{x + i;/)" - (*• - iy)"] ,
it is clear that the expression (5) is of the form (2), i.e. that it is a .solution of the equation (1).
It follows that the solution V of equation (1), which is characterised by the condition that it has the value V=f(d) when r = a, is given by the expansion
F = a„ + - (((i cos 6 + bi sin $) + [-] (a., cos 20 + b. sin 26) + ...,
where
I 1 f-"
1 f-"
0,1 = — I f(t) COS ntdt,
1 r-"
bn= f(t)smntdt.
The principal object of this chapter will be to obtain theorems analogous to this for the other jjartial differential equations of mathematical physics; the method followed will be in most respects similar to that by which this result has been obtained.
168. Laplace's equation; the general solution ; certain particular solutions. The partial d'flerential equation
da^'^ dy^'^ dz-~
is known as Laplace's equation, or tiie potential-equation, and is of importance in tlie investigations of inatheuuitical physics.
The general solution of this equation was given by the author in 1902. ^•^ mav be written
V = /(a- cos t + ys'\nt + iz, t) dt.
312 TRANSCENDENTAL FUNCTIONS. [CHAP. XIII.
where / is any arbitrary function of the two arguments :i:cost + y sin t + iz and t. The sohition is effected in Monthly Notices of the Royal Astron. Soc, Vol. LXii. In this chapter however we are concerned not so much with the (jenerul solution as with the particular solutions which satisfy certain further conditions. To the consideration of these we shall now proceed.
Let the equation be transformed by taking instead of the independent variables x, y, z, a new set of independent variables r, 6, (f>, connected with them by the relations
' X — r sin 6 cos (p,
y = r sin 6 sin ^,
yz = r cos 6.
It is found without difficulty* that Laplace's equation becomes
Let us seek for particular solutions of this equation, of the form
where R, ©, <I>, are functions respectively of r alone, 6 alone, and </> alone.
Substituting, we obtain
1 d I „dR\ I d f . .d@\ 1 d"-^ ^
R dr V dr)^ (fi sm 6 ddK dO ) ^ siu= 6 d(j>' '
Now the quantity
Id/., dR\
Rdr\ dr) does not involve 6 or <p\ and since by this equation it is equal to
0 sin e dO r'" do) * sin^ 6 d<j>^ '
it clearly cannot vary with r: it is therefore independent of r, 6, and <^, and so must be a constant ; this constant we shall write in the form n (?i + 1).
We thus have
^^[r^f)-nin+l)R = 0.
Write r = e", so dr = e"du. Then this equation becomes
e-^~{e^~\-n{n+\)R = 0 du \ diij
d^R dR , ^. ^ ^
or -j--+-r--n{n+l)R = 0.
du^ du
* The work is given in full in Edwards' Differential Calculus.
1G8] .Vri'LICATIONS TO TlIK KQUATIOXS OK MATllEMATICAI, I'UYSICS. SV^
This is a linear diftVrential C(]nati<)n of the second order with constant coerticients; its sohition, found in the usual way, is
where .-1 and B are arbitrary constants.
The most general turni uf the funrtiun A' is therefore
R = Ar" + B>-"-K
Considering next the function <t>, it can in the same way be shewn that the quantity
is independent of r, 0, and c^, and so must be a constant. Writing this constant in the form —//(-', we have for the determination of <J> the equation
of which the general solution is
(S> = a cos iu(f> + 6 sin in(j>, where a and b are arbitrary constants. It thus appears that the expressions
?•" cos mcf) @ and ?•" sin yiKp ©
are particular solutions of Laplace's equation, if n and m are any constants and 0 is a function (of 0 only) which satisfies the equation
1 d f . ^ d&\ m- ^ & sm 6 d6\ dd J s\n-d
Writing cos ^= 2, this becomes
But when m is a positive integer, this is (§ 12,')) the equation which is satisfied by the associated Legendre functions of order n and degree m ; so a particular solution is the function
Pn"'{z), or P,.'«(cos^).
Hence generally we see that the {in-'r 1) expressions
r"P„(cose), r"cos<^P„'(cos^), 7-"eos2<^P„=(cos ^), ..., ?'"cos«(/)P„" (cos ^),
r" sin <f> P„> (cos 0), ?•" sin 20 P„= (cos 0), .... r" sin 7i(j> P„" (cos 0),
where n is a positive integer, are particular solutions of Laplace's equation.
314 TRANSCENDENTAL FUNCTIONS. [CHAP. XIII.
Moreover, since P„"'(cos^) is of the form sin™ 6 x a polynomial of degree (n — m) in cos 6, it is easily seen that each of these quantities, if expressed in terms of .r, y, z, becomes a polynomial, homogeneous of degree n, in x, y, z. It can in fact be easily shewn, by using the result of § 132, that
?•» cos m4> P„'» (cos 6)
is a constant multiple of
/:
{x COS t + y sin t + izY cos rnt dt,
' 0
and that
?•" sin mcf) P„'" (cos 0)
is a constant multiple of
(x cos t + y sin t + iz)" sin int dt,
I
from which their polj'iiomial character is evident ; these forms have the further advantage of exhibiting these particular solutions as cases of the general solution given at the beginning of this article.
Example. If coordinate.s r, 6, (f> are defined b}' the equations
ix=rcos 8, y = (r^ - 1 )- sin 5 cos (^, 2= (,-2-1)* sin ^sin <^, shew that the function
T'=P„"' (r) P,,'" (cos 6) cos wk^
is a solution of Laplace's equation
dx^ 3y2 332
169. The series-solution of Laplace's equation.
The particular solutions of Laplace's equation, which have been found in the preceding article, enable us to express the general solution, in the form of an infinite series involving Legendre functions. This series-solution will of course be really equivalent to an expansion of the general solution
J (
f(x COS t + y sin t + iz, t) dt
already mentioned; but the series-form is (as will appear from § 170) more convenient in determining solutions which satisfy given boundary-conditions.
For let V(x, y, z) be any solution of Laplace's equation
'bx'- dy- dz-
170] APPLICATIONS TO THE EQUATIONS OK MATHEMATICAL IMIYSICS. 31o
Then in the neighbourliood of any ordinary point, which wc may take as the origin of coordinates, V can be cx])anded in the form
V = (t^ + ((,,1- -t- t,y + i\z + (/o.c- + l).,f + c.,z^ + d-^yz + e.z.v + f...rij + a.,x^ + ....
Substituting this expansion in Laplace's equation, and e<juating to zero the coefficients of the various powers of x, y, z, we obtain an infinite number of linear relations between the coefficients a^, a,, 6,, c,, lu, —
There are -^n{n — \) relations of this kind between the .^(w + !)(« + 2)
coefficients of terms of degree n in the expansion of V : and so only
-j'.-5 ()( + !)(/( + 2) — .^ H (/i — D- or (2(( + 1 ) of the coefficients of terms of
degree n in the expansion of V are really independent. But in the last article we have found {in + 1) independent polynomials of degree n in x, y,z, which satisfy Laplace's equation, namely the quantities
?-"P„ (cos e\ r" cos (/)P„' (cos 61), , r" cos «<^P„" (cos 6),
r" sin (f>P„' (cos ff) ?•" sin «^P„" (cos ^).
It follows that the terms which are of degree n in x, y, z in the expansion of T"" must be a linear combination of these (2« + 1) q\iantities ; that is, V must be expansible in the form
F= ^0 + r ;^iP, (cos 0) + 4,' cos <^Pi' (cos (9) + P/ sin ^P,' (cos 6)]
+ ?•= {ii.,P, (cos 0) + A.^ cos (/)P,' (cos 6) 4- .•!,- cos 20 P.,- (cos Q)
+ B.} sin <^P„' (cos e) + P.,- sin 2* P.? (cos 0)]+ ...,
where the quantities A^, A-^, A^, P,\ ... are arbitrary constants.
170. Determination of a solution of Laplace's equation which satisfies given boundary conditions.
In order to determine the unknown constants A„, j4,, .4,\ P,', ..., which appear in the exjiansion just found, it is necessary to know the remaining Conditions which the function V is required to satisfy. A condition of frequent occurrence is that V is to have certain assigned values at the points of the surface of a sphere, which we may take as being of radius « and having its Centre at the origin. This sphere will be supposed to lie entirely within the region for which V is a regular function of its arguments x, y, z. When r = a, V is therefore to be equal to a given function f{d, cj)) of 6 and 0.
The constants A,,, Ai, .4,', P,' are therefore to be determined from the
equation
/(0, (f)) = A,, + a {AiP, (cos 0) + Ai' cos <f> P,' (cos 0) + U,' sin ^P,' (cos 0)]
-t- a» [A„P. (cos 0) + 4,> cos ^P,' (cos 0) +...)+ ...
316 TRANSCENDENTAL FUNCTIONS. [CHAP. XIII.
In order to obtain the value of one of these constants, say .4,t'", from this equation, we multiply both sides of the equation by P,,'" (cos ^) cos //k^, and integrate over the surface of the sphere. On the left-hand side we thus have
I " ( "/(^, <p) P,,'" (cos 6) cos iu(f> sin 6 cW dcj).
As to the right-hand side, we know that
I COS. m<j} COS 7'(j) d<f) Jo
is zero except when r = in, and that
cos in(f) sin r<p dcji is always zero ; and also (by § 130) that
P,'" (cos 6) P,r (cos d) sin 6 dO
is zero except when r = n. It follows that on the right-hand side, every term vanishes except the term
a" A,,'"- if jP,™(cos e)}-' cos'- m<j) sin 0 dd d(f).
rzir Since I cos'' 'm(j)d4) = Tr,
Jo
and (by § 1 SO) j'^ [P,r (cos ^)p sm 6 dO = ^^^ ^^^^^ ,
this term has the value
2n + 1 (n — m)l We have therefore the formula
An"" = -4^, ■ y : f(d. <i>) Pn'" (cos B) cos m.cb sin 0 dO dd>,
Stto" {n + m)Jo.'o ■' ^ ^' ^ ^ ^
which determines the coefficients 4„"' in the expansion of V.
The coefficients B,/'^ can be similarly determined : and so finally the solution V of Laplace's equation, which has the value f(d, (^) at the surface of the sphere, is given for points in the interior of the sphere by the expansion
F= S '^1- /(^', </,') P„(cos0')P„(cos^)
n = 0 ^TT \Uj .1,1 Jo (
-f 2 S y^'HL' ■ pm (cog 0'^ pm (cQs 0^ cog ,„ (^^ _ ^'^ gi^ 6' d0' d(t>'.
,„=i (n + m): J
171] APl'LICATIOXS TO lllE EQUATIONS OK MATUK.MATKAL I'llVSICS. 317
This result may bt- roQ^anlcd as a three-dinionsidnal analogue of the two- (liiuoiisional result of § KiT.
E.mmj>le 1. Shew, by applying the expansion-theorem just given, that P„ {cos 6 cos ^ -I- sin 5 sin 5' cos (0 - 0')| = J'„ (cos 0) P„ (cos 5')
+ 2 2 feS; ^n*" (COS 6) P„'" (cos ^') cos m (cp - 4>').
Example 2. Prove that if the product of a homogeneous poljTiomial of degree n in X, II, z and the function /'„ {cos^Jcos^' + sin^sin ^cos(<^-(^')j be integrated over the surface of the sphere, the result is 47r/(2n+l) multiplied by the value of the polynomial at tho jioint {ff, (f)').
(This can be proved by taking 0' to lie zero, which involves no real loss of generality, and expanding the polynomial by the theorem of this article.)
171. Particular solutions of Laplace's equation ivhich depend on Bessel functions.
It is possible to construct solutions of Laplace's equations in series in several ways, of which that which has been given, and which depends on Legendre functions, may be taken as representative. A full discussiort of the other methods would be beyond the scope of this book, but a general idea of them may be inferred from the result which will next be established, namely that the Bessel functions furnish a group of particular solutions of Laplace's equation, just as the Legendre functions do.
When Laplace's equation
daF dy' dz"
is expressed in terms of the "cylindrical coordinates" s, p. <j), where p and ^ are defined by the equations
{^ = p cos (j>,
\y = p sin 4>, it takes the form
Let us seek for particular solutions of this equation, of the form
V = ZPO, where Z, P, <I>, are functions of ^ alone, p alone, and (f) alone, respectively.
On substituting this value of V, Laplace's equation becomes 1 d'Z 1 /f;=P I rfP\ 1 d"-^
Z dz' ^ VKdp''^ p dp ) "*■ p=* dji" ^'
318
TUANSCENDENTAL FUNCTIONS.
[chap. XIII.
This equation shews that the quantity
Id'Z Z dz'
must be a constant independent of z, p, and (f>; let this constant be denoted by ^■^. Then on solving the equation
dz"- - ^ ^' we have the particular solutions
Z = e*^ and Z = e-*^ Similarly the quantity
(t> d(j>- is a constant, which may be denoted by — ni^ ; on solving the equation
d'<i>
d^-
+ «i-<J> = 0,
we obtain the particular solutions
<t> = cos VK^ and <t> = sin m(^. The equation to determine P is now
dp- pdp V p-J On putting kp = y, this becomes Bessel's equation of order m,
df y dy \ y-l ~ ' a particular .solution of which is
P = '/m(i/).
It follows that the expressions
e'^''^ cos 7n(f)J,„(kp) and e'^''^ sin m(f>Jm(kp),
tuhere k and m are arbitrary constants, are particular solutions of Laplace's equation.
172. Solution of the equation
..—„ + ^ + v=o.
ox- oy- We now proceed to consider another partial differential equation.
lis] Al'l'LlCATlOXS TU THE KgUATIONS Of .MATUEMATICAL I'UY.SICS. 31()
Wo have seen in thr last article that Laplace's e(iiiation
Par" 9//' 3z- is satisfied by the particular sohitinns
e* J„ (r) cos nO and e-' J„ (?•) sin nO, where x = rcos0, // = ?'siii^.
But if \vi' write Tr=e^F,
where V is a function of .c and // t^mly, the Laplace's equation for W becomes
It follows that, for all \alues of », the quantities
J„ (r) cos )i6 and J„ (r) sin /i^
(ire particular solutions of this latter equation.
From these particular solutions, as in the case of the solution of the equation
dx- dy already described, we can build up the general solution of the equation
ox- dy in the form V = - J,, (/•) (a,, cos n6 -\- 6„ sin nO),
11 = 0
where «o, «i, u.,, ..., hi, L, ..., are arbitrary constants.
173. Solution of the equation
cfV PF d'V ^^^
dx^ dy- dz-
In order to solve the equation
c-^v ?=r ?r-V ,, ^
car dy- cz-
which is likewise of great importance in the investigations of mathematical physics, wo first express the equation in terms of new independent variables- r, d, ^, defined by the equations
Ix= r sin d cos <^, y = r sin 6 sin <^, z = r cos d,
320 TRANSCENDENTAL FUNCTIONS. [cilAP. XIII.
and then endeavour to find particular solutions of the form
where R, f), <I>, are functions resj)eclively of r alone, 6 alone, and <p alone. Proceeding as in § 1(58, the differential equation becomes
"^Rd^A'' di-J^Ssinede V'"" ^ do) + 0 sin= C '■' ~ "•
d-<l> e d^'
This equation can be solved by the process used in § 1G8 for finding particular solutions of Laplace's equation ; the quantity
- 1 A ( -^
'^"^ R dr \" d>
")
must be a constant, which we shall denote by n {n + 1). If in the resulting equation
i{r"-f)^{r'-n(n.l)}R = 0, we write y — Ri-^, it becomes
which is Bessel's equation of order ( « + ;5 1 .
The quantity R can therefore be taken to be
R = r-i J„+i (r).
The equations for 0 and <P are now found to be the same as those which occur (§ 168) in the solution of Laplace's equation; and proceeding as in ^ 169, we find that the general solidion of the partial differential equation
da;- dy' dz- '
regular near the origin, can be expressed in the form
iAnPn (cos 6) + 4„' cos c^P„' (cos 9) + ... + ^„" COS ?i^P„" (cos 61)] ( + 5,.' sin 0P„i (cos 6)+ ...+ Bn" sin ?i(/)P„" (cos 0)\
Avhere the quantities A and B are arbitrary constants.
17.S] Al'l'LICATIONS TO TllK EQUATIONS (iK MATHEMATICAL PHYSICS. 321
WliL'u !i particular scilutinn l' ot' tlie e(|uatiiiii is ti) lie (leteriniiiL'tl b}' the condition that it is to take prescribed values at all jjoints on the surface of a sphere, the constants A and B are determined exactly as in § 170.
Exantple. Shew, as a case of the general expansion of this article, that gir C0B«_ i in (2,r)i (2« + 1 ) ;•" 4 /'„ (cos $) ./„ + » (r).
^ote. The ^lartial difl'erential equations of §§ 172, l";i, possess general solutions analogous to that of Laplace's equation. The solution of tlio equation of § 172 is
where /is an arbitrary function ; and the solution of the equation of § 173 is
where / is an arbitrary fvuiction. For the proof of the.se results, reference may be made to papers by the author.
Miscellaneous Examples.
1. If a solution 1' of Laplace's equation be .symmetrical with respect to the axis of z, and have the value V=f{z) at points on that axis, shew that its value at any other point of space is
r=i ['f{z + i(x'-+f)i cos 0} dcp. "■ y 0
2. Deduce from the result of Example 1 that the potential of a circular ring of mass M, whose equation is
3. Lot P (.V, !/, .:) be a pfiint in space, and let the plane through P and the axis of z make an angle <f) with the plane z.>:. Let this plane cut the circle whose equations are
I z = 0, x'-+f~=F;
in the points a and y, and let the angle aPy be denoted by 6 and log (Pa/Py) by o-.
If <r, S, 0 be regarded as coordinates defining the position of the pt>int /', shew that Laplace's equation
32 r 32 r 32 F_ 3a;2 dj/- 3^2 ~ takes the form
3_ f sinh o- an ^ f sinh o- dV\ 1 ^!Z-0
3<r lc08h(r-cos^ So-/ dd (cosh a- - cos d d6 j 8inh2 o- (cosh o- - cos 6) 80'- '
and that the quantities
V= (cosh <r - cos 6)^ cos n6 cos m0 P'^_ , (cosh <r)
are solutions of it.
W. A. 21
CHAPTER XIV. The Elliptic Function' p (z).
174. Introduction.
If f(z) denote any one of the circular functions sin z, cos z, tau ^ ... , it is well known that
/(Z + -llT) =f(z),
and hence that
f{z + 2«7r) =f{z),
where n is any positive or negative integer.
This fact is generally expressed by the statement that the circular functions admit the period 2ir. They are on this account said to be periodic functions ; and in contradistinction to other classes of periodic functions, which will be introduced subsequently, they are called singly -periodic functions.
It will in fact be established in this chapter that a class of functions exists possessing the following properties : if f{z) be any function of the class, then f(z) is a one-valued function of z, with no singularities other than poles in the finite part of the ^-plane ; moreover, _/' (2) satisfies, for all values of z, the equations
/(^ + 2a,,)=/(^),
f{z + 2cc,)=f{z),
where &)i and oja are two quantities independent of z. Functions f{z) of this class are said to adviit the quantities 2<»i and 2(i>2 as periods, and are called doubly -periodic functions or dliptic functions. The two periods 2£Ui and 2co„ play the same part in the theory of elliptic functions as is played by the single period 27r in the theory of circular functions.
By repeated application of the formulae written above, we obtain as the characteristic equation of all elliptic functions the equation
/(« + 2wia>, + 2n(o„) = f{z),
where m and n are any integers.
174. 17.t] the KLUll'TlC FUNCTION ^ (z). 323
175. Definition of p (z).
The elliptic functions may, as we have just seen, be regarded as a generalisation of the circular functions. It is natural therefore to introduce them into analysis by some definition analogous to one of the definitions used in the theory of circular functions.
One mode of developing the theory of the circular functions is to start from the infinite series
1 *J? 1
-.+ s
m =
r±i {z — jmry '
It can be shewn that this series converges absolutely and uniformly for all values of ^ except the values
z = 0, ± TT, ±2ir, ±3-77...;
and that it admits the period i-n: If now its sum be denoted by (sin«)~^ and this be regarded as the definition of the function sin z, then from this definition we can derive all the properties of the function sin ^, and thus a complete theory of the circular functions can be developed.
Similarly, as the basis of the theory of elliptic functions, we form the
infinite series
Cb / , r 2-2 -f S {{z - 2»ia), - 2H(ao)-= - (2mQ)i + 2?ia)o)-=j,
f
where a-, and o>„ are any two quantities, independent of z, whose ratio is not purely real, and where the summation extends over all integer and zero (except simultaneous zero) values of m and of n.
It has been shewn in | 11 that this series is absolutely corivergent for all values of z, except the values z = 0, + (Ui, + tD„, + ta, + o)„, ± 2ci, + ooo, ....
By comparing the series with the convergent series S (m- + n-)~i as in § 11, it is seen that this convergence is also uniform (§ 52). The series therefore represents a one-valued function of z, regular for all values of the variable z except the values z = 'Imwi + 2na).,; and at these points, which are the singularities of the function, it clearly has poles of the second order.
We shall denote this function by the symbol ^{z). Its introduction is due to Weierstrass.
There are other ways of introducing both the circular and elliptic functions into Analysis ; for the circular functions, the following may bo mentioned :
(1) The geometrical definition, according to which sin z is the ratio of one side to the hypotenuse, in a right-angled triangle of which one angle is z. This is the definition usually given in the introductory chapter of treatises on Trigonometi-)' : but from oui' point of view it is defective, as it applies only to real values of z.
(2) The definition by means of the infinite product
21—2
f
324 TKANSCENDKNTAL FUNCTIONS. [CHAP. XIV.
(.3) The definition liy the inversion of a definite integral,
/"sink's Mt\
.' 0 We shall see subsequently that alternative definitions of the elliptic functions exist, analogous to each of these definitions (1), (2), (3), and that tliey may if desired be taken as fundamental in the theory.
Example. Prove that
/ TT \2 fl " , 2?JB., 1
where C'= - „ - {^+ 2 cosec -^ -n-y .
\2li>J (.i „=_„ Ml J
176. Periodicity, and other properties, of ^ {z).
The function iffiz) is an even function of ^, i.e. it satisfies the equation
J{z)=j{-z).
For if —z be substituted for z in the series which defines g)(2), the
resulting series is the same as the original series, except that the order of
the terms is changed. But since the series is absolutely convergent, this
change in order does not affect the value of the sum of the series ; and
therefore we have
^{z)=^){-z).
Further, the function g) {z) admits the quantity 2&), as a period.
For
1^ {z + 2a),) - ^ (z)
= {z + 2o)i)-'--z-^ +'l{(z + 2a)i - 2m&)i - 2)i(o„)-- -{z- -Imu), - 2nw.,)--\
= 'S. \{z — 2 (m — 1) lUi — 2iia>^)~'- — (z — 2?«(Di — 2?!.&)2)^-),
where the last summation is extended over all integer and zero values of m and 71 without exception. But this last sura is zero, since its terms destroy each other in pairs. Thus we have
p(z + 2co,) = ^{z). Similarly ^ (^ + 2&).,) = ^ (z),
and generally P(^ + 2?h&), + 2n(02) = ^J (z),
where m and n are any integers.
Therefore the functioii g? (z) admits the two periods 2a)i and 26>g.
Differentiating the above results, we see that ^' (z) is an odd function of z, and admits the same periods as g) (z).
177. The period-parallelograms.
The study of elliptic . functions is much facilitated by a method of geometrical representation which will now be explained.
176 — 178] THE ELLIPTIC FUNCTION ^ (s). 325
Suppose that in the plane of the variable z we mark the points z = 0, z = 2a),, z = 2a).;, : = 2&), + 'Icd.^, ... and generally all the points cuniprised in the formula z ='2iiia), + '2n<o.,, where hi and n are any positive or negative integers or zero.
By joining the point :r = 0 by a straight lino to the ])oint 0 = 2&), , then joining the jjoiat 2(Ui to the point 2&), + 2a);., then joining the point 2&), + 2&j„ to the point 2u).,, and lastly joining the point 2a)2 to the point z = 0, we obtain a parallelogram in the 2-plane, which we shall call the fundamental period-jHtralleloffraiii.
It is clear that the whole .:-plano may be covered with a network of parallelograms, which are each similar and equal to this parallelogram, and which can be obtained by joining the other marked points by straight lines. These parallelograms will be called perivd-parallelorframs.
Then if t be any quantity, the points
z = t, z = t+ 2(Bi, z= t+ 2q)„, ..., z= t+ 2ma)i + 2no}o,
manifestly occupy corresponding positions in these parallelograms ; these points are said to be congruent to each other.
It follows from the fundamental property of ^{z) that the functicm ^{z) has the same value at all points which are conr/i-uent with each other ; and hence that the values which the Junction ^{z) has in any period-parallelogram are a mere repetition of the values which the function has in any other period- parallelogram.
178. Expression of the function f {z) hy means of an integral.
We shall now obtain. a form for ^{z) in terms of an integral, which will be found to be of great importance in the theory of the function.
The quantity p {z) - z"^,
or S {{z — 2>Hcui — 2)ia)o)~- — (2;kwi + 2n(i)^~'^\,
is a regular function of z in the neighbourhood of the point ^ = 0, and is an even function of z. It can therefore by Taylor's theorem be expanded, for points z near the origin, in the form
i,(.)-z- = g.= + |.'+...
1
where clearly we shall have
^ = 3S(2ma), + 2Ha),)-^ g = 5S(2w«, +2hw,)-«.
Thus ^(2) = ^-»+^^. + |^^ + .
-7
20 28
326 TKANSCEN DENTAL FUNCTIONS. [CHAP. XIV.
Forming the square anil the derivates of" this e.vpansion, we have
Therefore g)- {z) — g (j)"(^) = jo^'z + terms involving z* at least. It follows that the function
is regular in the neighbourhood of the point 2 = 0; and as it is doubly-periodic (for clearly any power or derivate of an elliptic function is likewise an elliptic function) it must be regular in the neighbourhood of each of the points
z = 2m&)i + 2)1(02.
But the only singularities of ^(z) are at these points : and therefore the only possible singularities of the function
f-{z)-lp"{z)
are at these points. The latter function is consequently regular for all values of z; and so by Liouville's theorem (§ 47) is independent of z, and therefore
is equal to the value which it lias at the point z= 0, which is j^S^- We have therefore the relation
Multiplying by 3^' (z) and integrating, we have
where c is a constant ; on substituting the expansions in this equality, we find that c = ^ffs-
Thus, finally, the function p (z) satisfies the differential equation
where ^2 and gs (called the invariants) are given in terms of the periods of p(z) by the equations
jj. = 60S {2mco, + 2no},)-\
g,= 1402 (2nia), + 2n(o.)-'.
178] THE EI.LII'TIC FUNCTION ^ (z). 327
This differential equation can be written in the form
where t= ^ (z),
and therefore (since ^(z) is infinite when z is zero) we have
z=( (it-'-g.,t-g,)-idt,
which is the required expression of <^{z) in terms of an integral.
The preceding theorems luay be illustrated by the results which correspond to them in the theory of the circular functions. Thus we may in the following way discuss the properties of a function f{z) (really cose<? z), which we shall take to be defined by the series
/(2) = J-2 + (.--n-)--' + (r + ,r)-2 + (---27r)-2 + (i + 2n-)--' + (.--.37r)-2 + ....
This series is clejirly infinite at the jioints : = 0, n, —n, 2n, ... ; for other values of ; it is absolutely and uniformly convergent, as is seen bj' comparing it with the series
l + l--+l---' + 2-2 + 2-2 + 3---' + 3--+....
The effect of adding any multiple of tt to .: is to produce a new series whose terms are the terms of the original series, arranged in a different order ; this does not affect the sum of the series, since the convergence is absolute ; and therefore / (s) is a periodic function of r, with the period n.
By drawing parallel lines in the i-plane at distances n- from each other, we therefore divide the plane into strips, such that at points occupying corresponding positions in the different strips, /(z) has the same value. In each strip, f{z) has only one singularity, namely at that one of the points 0, tt, — tt, Stt, —2n, ... which lies within the strip. The function is not infinite at the infinite ends of the strip, because the several terms of the series for f{z) are then small compared with the corresponding terms of the comparison- series
l+l-2+l---f2--' + 2-- + 3-2 + 3-2 + ....
Now near the point z = 0, the function /(z) can be written in the form
/(.) = .-2 + .-^(l-i)"+.-2(l+i)-V(2.)-^(l_A)-%...
= ^-2 + »r-2 (1 + 1 +2-2 + 2-2 + . ..) + 7r-'.-2 (.3 + 3^.3. 2-4 + 3. 2-<+. ..) + ... = .-. + 0.-3. ^ + .-,,..3.2.^ + ...
=.-+! + 1.2+...
Differentiating and squaring this equation, we have /"(r) = 6^-' + l + ...,
It follows that
f"{z)-6fHz) + 4f{z)
328
TRANSCENDENTAL FUNCTIONS.
[chap. XIV.
is a series containing no negative jiowors of z; it has therefore no singularity at tiie point J=0, and therefore (since that is the only possiljle singuhxrity) no singularity in the strip which contains ^ = 0, and therefore (on account of the periodic property) no singularity in any strip. It is therefore, by Liouville's theorem (§ 47), a constant : this constant must be equal to the value of the function at the point z = 0, which (on substituting the expan- sions) is found to be zero. We have therefore
/"(.-)-6/2(.-) + 4/(2) = 0. Multiplying by 2/' (z) and integrating, we have
fHz) = 4f^z)-4fHz) + c, where c is a constant. On substituting the expansions, c is found to be zero, and therefore
/'=(z) = 4/2(^){/(.)-l}
or ^^ y = 4«2 (< _ 1 ), where < =/ (z),
which gives 2z= j t-^{i-l)-hdt
as the expression of/(z) by means of an integral. Example. If y = ^J{z), shew that
1 d^ ■? \ dz^ 1 "i *?
\dz) \d~z)
where e^, e„, e^ are the roots of the equation
For we have P(2) = 4P(0)-5'2&J(z)-5'3,
/rfu\2 and so V&j " "* ^ ~ ^^^ ^ ~ "'^ ^ ~ ^'^'
Differentiating logarithmically, we have
I
\dz)
= (2/-ei)"' + (y-«2)-'+(2/-e3)"'-
Differentiating again, we have
/dyy fdyy \dz) \dz)
^-(^-ei)"^-(y-«2)"^-(y-«3)'
Adding the last equation, multiplied by |-, to the square of the preceding equation, multiplied hj ^, we have the required result.
It may be noted that the left-hand side of the equation is half the Schwartzian derivative of z with respect to y ; and hence the result shews that z is the quotient of two solutions of the equation
|J + [il J_ iy - er)-' -ly{y- e,)-^ {y - e,)-i (^ - e,)-^^ v = 0.
179,180] THE EI.LU'TIC FUNCTION ^0 (z). 329
179. Tlie homogeneity of the function jp (z).
When the Weierstnissian elliptic function is considered as depending on its arguments and periods, it has a certain property of homogeneity, which wiU now be investigated.
Let ^ (z, ' j denote the function formed with the argument z and periods 2ct)i and 2it).,. Then we have
u (\z, ^'"') = X--Z-- + S {(Xz - 2»iX&)i - 2)i\&),)-= - (2»(\w, + 2h\(o.^-"-}
It follows that the effect j^jrmdtijily in; i the (iiyiunent and the periods hy the same quantity X is equivalent to midtijilyuif/ the function by X~-.
This relation can also be expressed in terms of the quantities g., gs.
For let ^{z: g., g-^ denote the function formed with the invariants g., and g.^. Then we have
g,= 605:(2mcoi+2;i(u,)-',
(73= 1402 {Imw, + ■2nai.,)-\
The effect of replacing q), and a^ by X&>i and \w., respectively is therefore to replace g„ and g^ by \~*gi and X~*'(7. respectively; and thus we have
= \'^{\z; X-»5f,,, X-'g^),
which expresses the homogeneity-property in terms of the invariants.
Example. Deduce the last result directly from the equation
z=l {4fi-g^t-g^)-idt. J i>M
180. The addition-tlieorem for the function ^{z).
The function ^{z) possesses an addition-theorem, i.e. a formula which gives the value of ^{z + y) in terms of the values of ^{z) and ^{y), where 2 and y are any quantities.
330
TRANSCENDENTAL FUNCTIONS.
[chap. XIV.
To obtain this formulca, consider the expression
fiy)
f' (^ + y)
t»'(2/)
as a function of z.
Since it is compounded of doubly-periodic functions, it is itself a doubly- periodic function ; and the only points at which it can have singularities are the points at which the functions ^{z + y) and g> {z) have singularities, i.e. the points 2 = 0, z = —y, and points congruent (§ 177) with these.
Now for points z near the point 2 = 0, we can write the determinant in the form
1 ^{y) + z^'{y) + \z''if{y) ■{-... - ^' (ij)- z^i" {rj)- ..
^~"- + W.9^-'"- + ■
- -19.
+ u,g^.^ + -
1 ^iy) \i>'{y)
Expanding this determinant, we find that the terms involving negative powers of z destroy each other; the determinant can therefore, in the neighbourhood of the point 2 = 0, be expanded as a series of positive powers of z ; that is, the function represented by the determinant has no singularity at the point 2=0; and therefore (by the periodic property) it has no singularity at any of the points congruent with 2 = 0.
Considering next the neighbourhood of the point z = — y, write z = — y + x. The determinant can be written in the form
I
1
1 ^{-y) + x^'{-y) + ... <^' {-y)+xi^"{-y)+ ../['
1 p(2/) ^'(y) I
and on expansion this is found to contain no negative powers of x. The function represented by the determinant has therefore no singularity at the point 2 = — t/ or any of the congruent points.
The function has therefore no singularities, and so by Liouville's theorem (§ 47) is independent of z. But it vanishes when z has the value y, since two rows of the determinant are then identical, always zero.
We thus have the formula
The determinant is therefore
<pi^ + y) f{y)
ip'iy)
= 0,
180] THK EM.II'TIC FUNCTION ^ (z). 331
true tor all values of ^ and (/. Since, by § 178, p' (z + y), p' (2), p'(y) are at once expressible in terms ot' ^(2 + y), ^(2), ^(l/), respectively, this result really expresses ^{z+i/) in terms of ^(z) and p(^). It is therefore an addition-theorem.
The addition-theorem may also be obtained in the following way.
Take rectangular axes 0.v, Ou, in a plane ; and consider the intersections of the cubic curve
«2 = 4.r-'>-jr,,v-<73 with a straight line
u = )n.x + n.
The abscissae j,, .v.^, x^ of the points of intersection are the roots of the equation
(^ (.1-) = 0, where
(^ {x) = {mx + iif - 4.r^ -I- (/jT +^3.
The variation hxr in one of those abscissae, consequent on small changes bm and S« in HI and n, is therefore given by the equation
whence
<t>' {Xr) Sxr + 2 (mxr + ») {x^diH + Sk) = 0,
I tor ^ „ I x^m + S7>. r=imxr + n ,.=1 ^'(^r)
= 0, by a well-known theorem in partial fi-actions.
3 Therefore 2 {■lx/—g^Vr-gi)~iSxr = 0.
r=l
Now when n is infinite, the abscissae .c,, .?•,, x^ are all infinite : we may therefore integrate the last equation over the series of positions of the straight line ^ = ni.i'-|-«, and obtain the result
3 /»
2 I (,4x^^-g^-r-ff3)-idXr=0.
r=l J x^
If we write
.ri = iJ(j), x^=^{2/), X3=p{w),
we have therefore z+y + io=0.
But the ordinates of the three points of intersection are
«1 = F(2)> «2=&*'(i'). %=^'(«')'
Since the three points are coUinear, we have
= 0,
= 0,
1 |
*3 |
«3 |
1 |
a;, |
ttj |
1 |
^i |
»2 |
and therefore 1 ^ (.:+;/) —^' {!+>/)
I 1 ^(') S>'W
! 1 ^(j/) ^'iy)
which is the addition-theorem.
332
TllANSCENDEMTAL FUNCTIONS.
[chap. XIV.
181. Another form of the addition-theorem.
The detenninantal form of the addition-theorem given in the last article may be replaced in the following way by a simpler, though less symmetrical, formula.
Consider the equation
1 ^(x) f'(x) =0.
1 p(y) f'iy)
If in this we replace f' {x) by its value in terms of f) («), and expand, we have
= W (^) [9 (*) - 9 i'j)] + 9' iy) \<iP (^) - ^ (^))P-
This may be regarded as a cubic equation in the quantity ^ {x). One of its roots is g) («) = ^{z + y), by the addition-theorem ; and the other two roots are ^{x) = ^{z) and g)(a')= ^{y), since the determinant vanishes when z or y is substituted for x. We have therefore
^ {z) + ^{y) + ^{z-Vy)= Sum of roots of cubic
= - (Coefficient of §)"(«)) -f- (Coefficient of ^ (a?))
= i {&>' (^) - f' iy)]' \v{^)-<& (.y)]~^
and thus we have ^____
^<-^>=yfi^T-''«-^«
^
which is a new form of the addition-theorem.
Example 1. Prove that the expre.ssion
i {<P' (^) - r Wr W (--) - (,•' (y)} "'-f W - <P (-- +y),
considered as a function of z, has no singularities : and deduce the addition-theorem for f{z).
For the given expression, from the mode of its formation, can clearly have no singu- larities except at the points s = 0, z=y, z= — y, and points congruent with these.
Consider then first the neighbourhood of the point 2 = 0. The expression can be expanded in the form
i{-2.-3-J^(y)+T^^^ + ..
-■-i^(:y)+^g-^"+-
and this on reduction is found to coutain no negative powers of z, the first non-zero term being jJ(y). The expression has therefore no singularity at the point z = 0.
181, 182] rilK. KLI.llTlC KfNCTION ^(2). 333
Considering next the noighboiuluioil of the point :=y, we tjiko : = )/ + x ; the expression becomes
-p(2^)-^g)'(2</)-...,
and this on reihiction is found to coiiUiin no negative powers of .c; there is therefore no singularity at the point z=y.
The case of the point := —y c^vn be similarly treated.
The given expression has therefore no singularities, and so by Liouville's theorem is independent of .-. But its value at the point i = 0 has been shewn to be p(y). Wo have therefore, for all values of r,
HF'(')-^'(y)}Mi!'W-P(y)}-^-^W-&'(^+^)-P(2')=o,
which is the addition-theorem. Example 2. Shew that
^(-^+y)+P(^-y)={PW-fr'(y)}--[{2j>W^(2/)-i^2}{P(2)+&'(j/)}-sr3].
For by the addition-theorem we have
(>;)
Replacing p'2{z) by if^z) - g.jp {z) - g^, and replacing ^'^(y) by il^^l3)-g.fp{y)-gi, and reducing, we obtain the required result.
182. The roots e,, e., e^.
Let n denote any one of the periods of p {z), namely the quantities 2&),, -lui^, 2a)i-l- 2&).,, i(Oi-iw.„ -2w, -2a), Then
^' L5 n j = p' (2 n - 11 j , since ^\z) has the period O,
= — ^y f ., n j , since jp' is an odd function of z.
It follows from this that unless Al is itself a period (in which case ^'f.^nj i.'i infinite), jj'Uflj is zero.
We have therefore
^'(a),) = 0, jf)'(o.,)=0, ^J'(co3) = 0, where w, stands for — (o), -I- co.j).
334 TRANSCENDENTAL FUNCTIONS. [CHAP. XIV.
Now denote the (|nantities ^(wi), p(o).), &>(<«3) by gj, e., e-j, respectively. Then the equation
p - (&),) = 4.^' (wi) - fj,^ (to,) - r/3,
or 0 = 4ei'-goei- ff.,,
shews that Cj is a root of the cubic equation
Similarly e, and e^ are roots of this equation.
Moreover, the quantities e^, e„, e, are distinct roots of the equation; for if for example we had 61 = 63, we should have j,)(a)3) = g)(&)i), and therefore
0,3 = + (u, + a period,
which is not the case.
We see therefore that the three roots of the cubic
are e^, e^, e,, where
ei = io{(6i), e„ =§)(&).,), 63 = (,^(0)3), and . (Ui + o), + t»)3 = 0.
The quantities e,, e,, e^ therefore satisfy the relations
ei + e2 + e-j = 0,
e.e, + e.,ei + ete., = -^g2, e,e.e,= -^g,.
183. Addition of a half-period to the cwgument of ^(z). From the addition-theorem we have
^j (2 + to,) + iJ (z) + e, = j iJ'' (z) [p (z) - e,}-^-
= {& (z) - e,} [p (z) - e,} {^ (z) - e,} [jrf (z) - e,)-= = {i^(-) - e^l {i-> (2) - e,} {p (z) - e,}-\ or |g> (^ + &),) = e. + (e, - eo) (e. - e^) [ jp (2) - eipjj
This formula expresses the result of adding a half-period to the argument of the Weierstrassian elliptic function.
183, 184] THE ELLIPTIC FUNCTION ^ (z). 335
Example 1. Shew that
is a imiltiple of the discriminant of tlic equation For we have Differentiating, we liave
Therefore
^' (-■)&>' (' + <-■) ^J' (-' + "■.) F' ('' + "3)
= («, - e,j- {e, - e3)2 (e, - «.)2 ^'* (z) {^{z)-e,}-^{^ {z) - e,} " 2 {^ (j) - 63} " =
which is a multiple of the discriminant of the equation
4 (.« - 61) {x — e,) (^ - «3) = 0. Example 2. Shew that (P (2-') -e^Wm-e,} + {^ {2z) -e,}{p (2z} -e,}+{f(2i)- e,} {p {2z) - e,} = p (£) - p {2z).
184. Integration of {aod^ + ^ha? + 6cx- + ^dx + e)"*.
Wo shall now shew how certain problems in the Integi-al Calculus, whose solution cannot be found in terms of the elementary functions, can be solved by aid of the function p^z).
Let the general quartic polynomial be written
/(«) = aa--* + \h£ + 6cj;' + 4cLc + e. Let its invariants* be
</.> = ae — ibd + 3c-,
jr^= a b c =ace + 2bcd - c? - ad" — fcV. : b c d c d e 1 etitsHggsian-be-
h(a:) = {ac - b') a-* + 2 (ad -bc)a^ + (ae + 2bd - 3c=) a,-= + 2 (6e - cd) X + (ce - d-), and let its sextic covariant be
t(^) = l\-f{a-)h'{x)+h(x)f'(x)]
= (a=d - 3«ic + 2i') a-« + . . . .
* The student who is not already familiar with the elements of the theory of binary forms is referred to Burnside and Pautou's Theory 0/ Equations, where the invariants and covariants of the quartic are discussed.
i
336 TKANSCENDENTAL FUNCTIONS. [CHAP. XIV.
Then it is known that
t" (*•) = - 4/.» {w) + (j,f- {x) h (,r) - g.p {x).
If we write s = — li {x)jf(x), this relation becomes
r-{x)=f'(x){is'-ff,s-g,).
>T J h(x)f'(x)-h'(x)/{x).
Now ds = L, , -^ dx
= ^^hx
and so {W — g.ys — g^a)"* ds = 2 {f(x)}-i dx.
Let Xn be any root of the equation f(x) = 0 ; then to the value x = x^ corresponds s = x ; and hence, if we write
2=rif(a;)}-idx.
J rr„
we have 2z = I (W — gJ, — ^3)"* dt.
It follows that thssquatioti -.
is an integrate^form-Jifthe p.guntinn
z = I [ax* + 4^ha? + Qcx- + Mx + e}~* dx.
Example 1. Shew that (with the same notation)
r(2z;s'2,S'3)=+<W{/W}-3. Example 2. Shew also that, if
2/= f"{/W} -*<;<, then g) {s+y) and p{z — y) are the roots of the equation
where F {x, u) = axV + 2bxu (x + u) + e{x'' + 'ixu+u^) + 2d{x + u)+e,
and H {x, u) is derived from h (x) in the same way as F{x, u) iroxaf{x).
(Cambridge Mathematical Tripos, Part II, 1896.)
185. Another solution of the integration-problem.
The integration discussed in the last article may also be effected in the following way.
As before, let
z=r{f{x)]-idx.
J Xt,
185] THE ELLIPnC FUNCTION ^ (z). 337
whore f(x) = ax* + ibaf + Qcar' + idx + e,
;iiul let a-,, be a root of the equation /(w) = 0. Then, by Taylor's theorem, we have fix) = (x - .r„)/' (.r„) + l{x- x.yf" (a;,) + \ (.v - a-„)\f"' (*■„)
+ ~{x-.T,yf""{^„)
Writing (./• - .7\,)~' = ?". we have
/(^•)=?-^{/(^v)f'+ir(^-o)?=+^/"'(«-o)?+^4/""M.
and so --=1' {/'(.Or + |/"(^'.)r + o/"'(^'o)? + i/""(^"*^r-
Writing f = 4 [/' (.r„)l-' 6, we have
^ = [J {4^» + 1/" (x„) ^^ + i/' {x,)f"' (o-o) 5 + 24-16/'= {x,)f"" (.•„)}■ '*c?^. Now take a new variable of integration s, defined by the equation
e = s-y"{x,);
this substitution destroys the term involving the square of the variable of integration in the denominator, and we thus have
J n
■where
9-. = is/"' (*■«) - 24/' (^0)/'" («o).
^3 = w. {/'(-^o)/" (.^•o)/"' (^0) - I /"' M - 1 /'= (•^„)/"" O^o)! •
It cau easily be verified that these latter quantities are the same as the invariants g„ and ^3 of the last article.
We have therefore
s = &>(-'; g-2, 9-,),
and therefore 6 = ^{z) — 04/ ' (^0).
?=4i/'(^-„)j-'{s>(^)-.^/"W}. and finally x = x, + \ f (x,) |^ (z) - ^ f" (a-„)| "' .
This last equation is the integral-equivalent of the equation
=r{/wi-
\-^dx w. A. 22
338
TRANSCENDENTAL FUNCTIONS.
[CHAI>. XIV.
It may be observed that
(,y (z) = (4,5' - r/,,s - r/,)J = i/' (x„) {f(x)]i r=, and hence that
1/(^)1* = ■
f'{.r„)ip'(z)
Example. Shew that the integrated form of the equation
J X,
{/(..)} -idr,
where .r„ is any constant (not necessarily a root of f{.v)), and f(x) is any quartic function
where ^ is the Weierstrassian elliptic function formed with the invariants ^., and g^ affix).
Shew further that
and «7,w/_Z(:^_l />Ll/ia.^ f /(..p) 1 /' (.r„) I
and ^ (.)- |^^^_^^^3 ^ ^^__^^^,| /i (.t„) - |^__^3 - - ^--^^1 /J (.,
.V).
186. TJniformisation of curves of genus unity.
The theorem of the last article may be stated somewhat differently thus:
If two varigMss-Jl. and .c are connected hi/ on equation of the form ^ = aai' + 46ar' + 6cx' + idx + e,
then it is possible to express them in terms of a third variable 2 by means ofjhe equatio)is
|y = i. /"(.*..) &>'(^){F(^)-i/"0'4".
where /(*') = (i^ + 46*' + Gee- + Mx + e,
a^o w any root of the equation f{x) = 0, and the function ^{z) is formed with the invariants g.^ and g^ of the quartic f{x) ; moreover, tJie quantity z is defined- Ml the equation
z=r{f{x)]-idx.
. Lik . ■
Now y is a two- valued function of x, since the quantity + (aar* + ibx" + 6cx- + 'id.v + ef-
i;^^^)
n
18G] THK ELLIPTIC Fl-X(TIOX ^{z). . 339
may take either sign ; and .r is a four- valued function of y, since the equation in .r
(u-^ + ■ibx' + Qcaf' + 4da; + (e - y=) = 0
has four roots. But on referring to the equations which express x and y in
terms of s, we see that x and y are one-valued functions of z. It is this fact
which gives importance to the variable z\ ^ is called the uniformisiny variable
of the equation
y- = «,*■* -I- ^h.c' + Gcx- + ^dx + e.
The stucleiit who is acquainted with the theory of algebraic phmc curves will be aware that cur\-es are classitietl according to their geniis*, a number which may be geometrically inteipreted as the difference between the number of double points possessed by the curve and the niaxiniiim number of double points which cjin be possessed by a ctirve of the same degree as the given curve. Curves whose genus is zero are called iiniciirsal curves ; if /(.r, y) = 0 is the equation of a luiicursiil curve, it is known that .c and y can be expressed in the form
\y=^{z)
where <j> and ^ are rational fimctions of their argument ; since rational functions are always one-valued, it follows that the variable z thus introduced is the uni/onnmng variable for the equation /(.(•, y) = 0 ; i.e., although if is in general a many-valued function of x, and x is a many-valued function of y, yet .r and y are one-valued functions of z.
Considering now ciu'ves whose genus is not zero, let
W a curve of genus imity. Then it can be .shewn that .c and y can be expressed ia the form
ix = <l>(z)
where (f> and i/^ arc now elliptic functions of their argument z ; x and 1/ are thus expressed as one-valued functions of z, and z is the uniformising variable of the equation f(x, i/) = 0. This result is obtained by writing
V = G(|,,)'
where /"and G are rational functions of theii- arguments, and choosing F and G in such a ' ly that the equation /(.i-, y)=0 is transformed into an equation of the form
vo can then write
land .V and y will thus be expressed as one-valued functions of z.
When the genus of the algebraic curve
f{x,ff) = 0
is greater than unity, the iniiformisation can lie effected by means of avtomorphic functions. Two classes of automorphic functions are known by which this unifomiisiitiou
* In French genre, in German Geschlecht.
22—2
340 TRANSCENDENTAL FaNCTIONS. [CHAP. XIV.
may be eftocted : namely, one which was first given by Weber in Gottiiiger Nachrichten, 1886, and one wliich was first given by the author, Phil. Traiu., 1898. In the case of Weber's functions, the " fundamental polygon " (the analogue of the period-parallelogram) is "multiply-connected," i.e. consists of a region containing islands which are to be regarded as not belonging to it. In the case of the functions described in Phil. Trans., the fundaineiital-polygf)n is " simply-connected," i.e. is the area enclosed liy a polygon. This latter class of functions may be regarded as the immediate generalisation of elliptic functions.
Miscellaneous Examples.
1. Shew that
fp{z+y)~i^){z- y) = - ^J' (2) ^' {y) {^ (z) - §) (y)} - 2.
2. Prove that
where, on the right-hand side, the subject of differentiation is symmetrical in :, y, and w.
(Cambridge Mathematical Tripos, Part I, 1897.)
3. Shew that
r(^-y) r(y-"-') rc'"-^) I
f{^-y) mif-y) ^i^o-z)
f{z-y) <p{y-w) <^{w-z) ' 11 1 I
(Trinity College Scholarship Examination, 1898.)
4. If simplify the expression
y = ^i)(2)-ei, / = ;!'
■ i^' (^ " 4 ^2 ^''^' ^') "^ ''"'^ " '^^^ ^"^ ~ n ' where t'l, e.,, e.^ are the values of <p {z) for which ^' {z) = 0.
(Cambridge Mathematical Tripos, Part I, 1897.)
5. Prove that
S{^(z)-e}{^{y)-il){w)}^{p{y+w)-e}i{p{y-v')-e}i = 0,
where the sign of summation refers to any three arguments z, y, lo, and e is any one of the quantities e,, e.^, e^.
(Cambridge Mathematical Tripos, Part I, 1896.)
6. Shew that
(Cambridge Mathematical Tripos, Part I, 1894.)
7. Prove that
ff(2z)-^{o,,) = {^'{z)]-^p{z)-p{i<o,)Y{^(z)-^J{o,, + U,)y\
(Cambridge Mathematical Tripos, Part I, 1894.)
MISC. EXS.] THE ELLIPTIC KrNCTION ^{z}. 341
8. If lit he any constimt, prove that
-I'll'
J
wlici-c the .-.uiiiiiiutinii refoi-s to the values of tf> (c) for which ^' (c) is zero ; and the integrals
are indefinite.
(Cambridge iMatheniatical Tripos, Part I, 1897.)
9. Let
R (.r) = .-1 .1-* + B.t^ + C.i-2 + Dx + E,
and let | = <^(.r) be the function defined by the equation
/■f
.'■ = j {/? (1^1-4 </|, where the lower limit of the integral is arbitrary. Shew that
2<t>' (a) ^ (t,'{a+y) + 4>'(a) _^ <f>' {a->/) + <f>' (a) _ <j> {a + y) - <i>' (x)
0(«+y)-*(«) <^(a+y)-<^(a) "^("-y) -<;!>(«) <^(«+y)-</>W
(j>(a-i/)-<l>ix) '
(Hermite.) 10. Shew that when the change of variables
c =^) 1 - —
1 1
is applied to the equations
r+';(i +/'!) + 1^=0,
I 2, + l+/>^ '
they transform into the similar equations
rV^ + r,'(l+y;f) + P = 0,
shew that the result of performing this change of variables three times in succession is a return t<i the original variables f, r; ; and hence prove that if ^ and tj be denoted as functions of u by E{u) and F{u) respectively, then
where A is one-third of a period of the functions E{u) and F(u). Shew that £iu) = ^-fp(u ; g,_, <,,),
'■•bere ffi^^P + ^^P*' ^'= " ^ " ^^'~2T6^°-
(De Brun.)
CHAPTER XV.
The Elliptic Functions sns, cnz, dnz.
187. Construction of a doubly-periodic function with two simple poles in each period-parallelogram.
The fuuctirm [J{z), which ha^ been considered in the previous chapter, is a doubly-periodic function of z, with a single pole of the second order in each period-parallelogram, namely at the point congruent with the origin*. We shall next introduce a doubly-periodic function which differs from ^ (2) in having ttuo poles, each simple, in every period-pai'allelogram.
Consider the series
f(z) = t[{z + 2m<o, -1- (2n -f 1) (UoJ-i - {inm^ -\- (2n -f- 1) o),}-'
-{z + (2m -\r I) CO, + (2» + 1)0),!- -^ !(2m -)- 1) o), -I- (2/i -|- 1) o).)-],
in which the summation extends over all positive and negative integer and zero values of m and n.
When the modulus of (2;/ta), -I- 2»a)2) is large (and we may suppose the series arranged in order of ascending values of | 27«&)i -)- 2/i&>2 ] ), the terms of the series bear a i-atio of approximate equality to those of the series
l[-z {2mco, 4- (2» -H 1) &),}-^ + z [(2m -I- 1) «, + (2n -f- 1) a),)-^],
or -zl {2mw, -\- (2n -\- I) to '"= fl - { I + „ -■,'"' , ,
' L I. 2m(Bi-l-(2n+ 1)0),
and these terms bear a ratio of approximate equality to those of the series
- 2zcoit{2m(o, -I- (2« + 1) fo,!"',
which again bear a finite ratio to those of the series
S (2?ftQ)i -1- 2na>2)~^,
which was shewn in § 11 to be an absolutely convergent series.
* In the network of paralleloKrams described in § 177, the poles of (j) (z) are not within the parallelograms, but on their bounding lines. We may however suppose the whole network slightly translated so as to bring the poles within the parallelograms.
187,188] TiiK Ki.Lii'Tic FUNCTIONS am, en 2, i\uz. 343
It folldws that the .series wliich ropicseiits /'(;) is absDlutely convergent for all values of 2, except for the exceptional values included in the formula
z= incoi + {'2n + l)<u..,, (//(, 11, integers)
for which the several terms of the series are infinite, and which have been tacitly excluded from the foregoing discussion of convergence.
Moreover, since the terms of the comparison-series are independent of z, tlie convergence is (§ 52) not only absolute but unit'oini.
By a discussion similar to that in § 17G, we can shew that/(^) is a doubly- periodic function of z, whose periods are io), and iojo ; it is an odd function of z, so that
f(z) = -f(-z);
and its sinjularities are at the points
z = mwi + (2« + 1) (U.J,
where )/; and 7i may have any .integer or zero values ; these singularities are simple poles, with the residues + 1. There are two of these singularities in each period-parallelogram.
188. E.rpression of the function f {z) by means of an integral.
The singularities of f{z) in the fundamental period-parallelogram are, as we have seen, at the points 2 = Wo and 2 = to, -|- w.,.
Consider now the neighbourhood of the point z = (t3.,.
Writing z = &>.. -t- ;r, we have
f{ai.,-'rj) = —f(—(o.. — x), since /' is an odd function,
= — /'(2(».j — a).. — ic), since 2<u... is a period,
= -f(<c.,-u:),
from which it follows that /(wj + ,r) is an odd function of x; the expansion oi f(z) in ascending ])owers of ./■ will therefore cmitain only odd powers of «.
Now
f{z) = S [[.7; + 2»(c, -f- (-In + 2) ft).,}-' - \imm, + {In -f- 1 ) o).,}"'
- \x + (2w + 1) 0)1 4- (2« + 2) w..'-' + [(2m +\)o), + (In -1- 1) o.,j-'],
where the summation extends over all positive and negative integer and zero values of in and n.
In this expres.sion, replace all expressions of the form (^4 4- .<)""' by their expansions A~^ — A~-x + A~'x'— ..., x being supposed small. A term
344 TUANSCENDENTAL FUNCTIONS. [cHAP. XV.
in .«"' will arise from the pair of values (;« = 0, /< = — 1), anrl we thus have
where ii = S [- 12//(&), + 2«a)»|-=+ ((2»i + 1 ) m, 4- 2h&),,]--],
the summation being in this case extended over all positive and negative integer and zero values of m and n, excluding simultaneous zeros in the first term.
If now by means of this expansion we express the quantity
as a series of powers of x, it is found that the negative powers of x destroy each other; this quantity has therefore no singularity at the point z = o).2.
Consider next the neighbourhood of the point 2 = &>] + &)„.
Writing ^ = &>, + m.^ + y, we have
f{a)i + o}2 + y) = —/{—(Oi — (o., — i/), since /" is an odd function,
= — /'(wi + (Uo— y), since (2&), + 2g)„) is a period.
It follows that /'(&)] + (U2+7/) is an odd function of y; its expansion in powers of 2/ will therefore contain only odd powers of y.
Now expanding_/'(2) in powers of //, in the same way as/'(-2) was formerly expanded in powers of x, we find that
./» = -U5'y + Cy+...,
where B' = l[- {{2m - 1) a)i + 2nay„]-^ + {2w&), + inccJj-"-].,
the summation extending over all positive and negative integer and zero values of m and n, excluding simultaneous zeros in the second term.
Comparing this with the expansion of B, we have
B' = -B,
so f{z) = -^-By + Cy + ...,
and, as before, the quantity
f"^(z)-/^{z) + (5Bp{z)
has no singularity at the point ^ = coi + a).,.
Now the points z = 03„ and z = w^ + a., are the only possible singularities of this quantity in the period-parallelogram ; it has therefore no singularity in the parallelogram, and therefore (since it is doubly-periodic) no singularities
ISy] THE ELLIPTIC FUNCTIONS SU Z, CU 5, (111 2. 345
in the whole z-plane ; it is therefore by Linuville's theorem (§ 47) a constant independent of z, siiy ^l.
The function /'(«) therefore satisfies a differentia! equation
Rephicing B and ^-1 by new constants k and fi, we can write this in the form
/'H.) = {^-/M.)}{^-/M^)};
so that, as/(z) is zero when z is zero,
We see therefore that the odd doubli/-periodic function f(z), which has penods 20), and 2a).. and simple poles at all points congruent with s = &), and z = (1), + Wo, may be rer/arded as defined by the equation
where k and fi, are constants depending only on cO] and &).>.
189. The function sn z.
The function f{z) discussed in the last two articles can be expressed in terms of another function, which we shall denote by sn z, in the following way.
Replacing the variable t of integration by a new variable s, defined by the equation fcs = fit, we have
J 0
(1-sn-Hi -k"-s')-ids.
Now define the new function sn z by the relation
fifi/iz) = />• sn z ■ then we have
2= (I - s')-i (I - ts')-'' ds.
Jo
This last equation can be regarded as the definition of the function snz in ternts of its argument z and the constant-parameter k, which is called the modulus; it is analogous to the definition of the function sin 2 by the relation
y*sin ; J a
)-i ds.
346 TRANSCENDENTAL FUNCTIONS. [CHAP. XV.
From the equation
fJif(^z) = k Sll z,
it is clear that the function sn^ has the same general properties as f(z), namely, it is an odd one-valued doubly-periodic function of z, with two poles in each period-parallelogram, the distance between the poles being half of one of the periods. The two periods will be connected by a relation, as they depend only on the single constant /■.
190. Till', functions en z and dn z.
We now proceed to introduce two other functions, either of which may be regarded as bearing to the function sn^ a relation similar to that which the function cos z bears to sin z.
rsnz
Since z= ( 1 - s"-)-^ ( 1 - ^■^s•■')-i ds,
J 0
dz
we have -yj , = ( 1 — sn- z)~i (1 — k- sn- z)^i
d{snz)
or -j-{snz) = (l — sn- 2')^ 1 — /,■- sn- z)K
Now sn z is a one-valued function of z, so its derivate must be also a one- valued function. It follows that
(1 -?n-a)i(l -k-sn-z)i
can have no branch-points (§ 46), considered as a function of z; and therefore either
(a) Each of the quantities (1 — sn=2)* and (1 — k'-sn-z)^ is a function of z which has no branch-points, or
(/3) The functions (1— sn'-2)^ and (1— ^-sn-^)* have branch-points, but are such that their product has no branch-points.
Now the alternative {/3) could be true only if the functions (1 — sn= z)i and (1 — k- sn'^ z)^ had their branch-points at the same places ; but this is not the case, since (1— sn'-^)i has branch-points at the places when sn^^^=l, and (1 —k-sn-zy has not. The alternative (/9) being thus ruled out, we see that the alternative (a) must hold.
If now we write
en z = {1 — su=^)=^,
dn^ = (1 — k'-sn- z)i,
where it is supposed that each of these functions has the value imity when snz is zero, then since cn^: and dn^ have no branch-points, and have definite values at the point z = 0, it follows that the functions en z and dn z are one- valued functions of z.
100, 1!)1] TflK F.T.I.Il'TIC FIINTTKlXS 8112, cn 2, du Z. 347
Tliey obviously satisfy the relations
sii'^ + cii-2 = 1, k-si\- z + (Iti- J = 1.
The functions sn z. en z, dn z are often called the Jacobian elliptic functions.
The function cosz is in the same way a one-vahieJ function, although the occurrence of the radical in (1 -sin*;)' might lead us at first sight to suppose that it possessed hranch- points.
191. E.vpressio)i of cu z and dn z by means of integrals.
We shall next find, fur the functions ci\ z and dxiz, integral-expressions similar to that found in § IN!' for sn2.
Differentiating the eijuation
c\\- z = 1 — sn'^,
we have en ^ -r- en 2 = — sn z en ^ dn z, dz
d ,
so -7- en 2 = — sn z <inz
dz
= - {(1 - en- z) {k'- + k- cu- z)]^-, where ^■'- = 1 — k-.
Thus if cu 2 = t, we have
dz = -{l- f)-i (k'- + kH-)-i dt, and therefore (since en 2 = 1 when 2 = 0)
z=l' (1 -t')-H'^^'"■ + '^'n-idt.
■ cnz
In the same way we can shew that
-^ dn 2 = — k- sn z en z, dz
and 2=1(1- t-)-i (t- - k'-)-i dt.
■1 Example 1. If cs 2=cn ^/su 2, shew that
J cnz Example 2. If sd c = sn zjdn z, shew that
z=l'*'{\- fit^yi (1 +if:»<«)-i dt
we have
or
348 TRANSCENDKNTAL KUNCTIONS. [CHAP. XV.
192. The addition-theorem for the function dn«.
We shall ne.xt .shew how to find dn.«, where
X = y + z,
in terms of the sii, en, and dn, of y and z: the result will bo the addition- theorem for the function dn.
Suppose that y and z vary, x remaining constant, so that
dz _ dy"
Introducing new variables u and v, defined by the equations
. u = en 2 en y,
V = sn z sn y,
dv , , dz , ^j- sn 2 en y dn y + sn ;/ en z dn z t- dv _ dy J J ^ ^
dii du , , dz '
'-r- — cn 2 sn 11 dn 7 — en w sn 2 dn ^ -,- dy J J a ^y
dv sn z en y dn y — sn 3/ en 2 dn 2 du en y sn 2 dn 2 — en 2 sn y An. y'
From this we obtain the equations
T- J — \—k- (sn- y — sn- z)- (en y sn 2 dn 2 — en 2 sn y dn y)~°,
V — u-j- = (sn y en 7/ dn 2 — sn 2 en 2 dn ?/) (en _y sn ^ dn 2 — en 2 sn 7/ dn t/)""',
( -5- J ~ ( '^ ~ '* ;7^ ) = (su- y — sn- 2)- (en y sn 2 dn 2 — en 2 sn y dn ?/)"-, and consequently
This equation is the equivalent, in the new variables, of the equation
dy It is a differential equation of Clairaut's type, and its integral is therefore
k- (v — ucf = 1 - k'^c-, where c is an arbitrary constant.
102] THE ELLIPTIC FUNCTIONS sn Z, Cll 2, dn z. 349
Tliiis the equation
i' (sn z sn y — c en s en y)- = 1 — h'-c^ must be equivalent to the equation
y-¥ z = x, \vhere c is some function of x.
To determine c in terms of a;, put y = 0 ; then we have
i-^c" cn= a; = 1 — A'-c", which gives c" = dn~-.r = dn~- {z + y).
Now the integral c([uation can be written in the form
c- ( 1 — k- + k- en- y en- z) — ick~ sn y sn z en y en 2 + (i-- sn- y sn" ^ — 1 ) = 0. '
Solving this equation in c, we have
_ i'' sn y sn 0 en y en £: + (Ar* sn^y sn- ^ cn^ y cn^ 2— (1 — ^+^ cn= yen'' 2)(A;-sn' 2/ sn* ^— 1))*
1 — A-^ -f k- en- y en- ^ '
/-•^ sn ?/ sn 2 en w en ^ + dn V dn 2
■^i" c= ^^, t:^ — jT, — .. - „ .
I — k- + k- en- y en- z
Since
A-* sn= y sn- ^ cn= y en- 2 — dn-y dn- z = (1 —k'- + k- en- y en- 2) (A'- sn- y sn" 5 — 1),
this e(juation can be written
k" sii' y sn- z — 1 Jtr sny sn z cuy en z + dny dn z'
, , . -f dn (/ dn 2 -t- k- sn (/ sn z en y en z
or dn (^ -t- y) = ■:^ — ^— ~ v — v^^^; ^ ■
'^ 1 — A- sn- y sn- ^
The two ambiguities of sign in this equation remain to be decided. Taking z = 0, it is seen that the first ambiguous sign must be -f ; so
, , , dn ?/ dn 2 -I- k^ sn « sn ^ en w on «
dn (2 -f- y ) = ^ r^H-^ — r , •
1 — k' sn- y sn- z
Now suppose that y is a small ([uantity ; expanding both sides in ascend- ing powers of y, and retaining only the terms involving the first power of y, we have
dn 2 + y J- dn 2 = dn z ± k?y sn 2 en 2.
Since dn 2 = - ^•= sn 2 en 2,
dz
350 TRANSCENDENTAL FUNCTIONS. ' [CHAP. XV.
it is clear that the ambiguous sign must be — . We thus finally obtain the addition-theorem for the fiiiictiuii dn, namely
. , . dn 2 dn ?/ — k- sn z sn }i en s en (/ dn {s + }j) = i — rj — I ~ .
Example 1. Shew that
, , , , , , dn^ « - P en- (/ .sn^ z dn(.+y)dn(.-y)^ 1-^^Bn^ysn^z "
Example 2. Prove that
l+dn22 = .
193. The addition-theorems fur the functions sn z and en z. To obtain the addition-theorem for the function sn z, we have
sn(5 + y)= + ^[l-dn=(2+y)l*.
Substituting for dn {z-\-y) from the result of the last article, this equation after some algebraical reduction gives
sn z en !/ dn y + ?,u y en z d,n z
sn{z + y) = ±'
1 — h?SD?z sn- y
On putting y = 0 in this formula, it is seen that the ambiguous sign is -I- ; we thus obtain the addition-theorem for the function sn, namely
sn z en y dn w + sn ii en ^ dn ^
sn {z + y) = ^ f, — - — ^4 .
■^ \ — k-sn-z sn- y
Similarly for the function cn^ we obtain the addition-theorem
en z en )/ — sn « dn ^ sn y dn y
en (0-1-2/)= y ,., ., TT^ -.
\ — k- sn- z sn- y
These results may be regarded as analogous to the addition-theorems for the circular-functions, namely
sin {z -\- y) = sin z cos y ■{■ cos z sin y,
cos {z -\- y) = cos z cos y — sin z sin y,
to which, indeed, they reduce when k is put equal to zero.
Example 1. Prove that
, . , , , sn^z-sn^?/ sn(:4-y)sn (z- y) =- rs — r, t— ,
, , , , cn'^ V - dn- y sn^ 2 cn(. + y)cn(2-^) = ^-^^^-,--.
Example 2. Shew that
' „ 1 - en 20 sn^=-; — :; — ;;-• l-(-dn20
litS-l!)")] THE ET.T.IPTIC KUXCTIONS till 2, Cn Z, cltl ;. 351
194. The constant K.
W'r shall <lfniitt' tiu' intrtjral
' (1 -t-rH\ -k-t-)-''dt
l\v K \ it is cioai'ly a cdiistaiit (K^peniling only on the niodulus k. The ambiguity of sign in the radical will be removed by the supposition that at the lower limit of integration the integrand has the value 1.
From the equation
^=r""(i - v)-H\ - k'V)-^ dt,
J (I
we see that sii A' = 1 ,
and hence cn K ={\ — sn- K)^- = 0,
dnA'=(l-A-^sn^/f)4 = i-'.
Example. Prove that
sniA'= (! + /(•')-*,
195. The periodicity of the elliptic functions luith respect to K.
It will now appear tliat the constant K is intimately connected with the periodicity of tlie elliptic functions sn2, cn.j, dn^.
For by the addition-theorem, we have
sn z cn A' dn K + sn K cn ^ dn 2 cn 2
sn {z + A' ) =
\ — kr sn- z sn- K Am'
, sa.z
Similarly cn(2 -f- A')= — /■; ,
k' and dn {z -f /v ) = -j- .
dn z
rr / -,7>-, cn (Z + K)
Hence sn (2 + 2A ) = ,^ =.' = - sn z,
dn (2 + /t )
and similarly cn {z -f 2K) = — cn 2,
dn(2-f-270 = dn2; and finally sn {z + 4A') = — sn {z + 2K) = sn z,
cn (z + 4Ar) = cn z,
dn (2 + 4^") = dn 2.
352 TRANSCENDENTAL FUNCTIONS. [CHAP. XV.
This 4Ar is a period fur tlie fanctiuns sn z and cm, and 2K is a period for the function dii z.
Example. If cs i = en z/sn 2, shew tliat
ctizcs{K-z) = k'.
196. The constant K'.
We shall denote the integral
\\l-t')-^-{\-lcH')-idt
J (1
by K'.
The ambiguity of sign in the radical will be removed by supposing that at the lower limit of the integration the integrand has the value 1.
Write ■ s = (1 - //=<-)"*■
Then (i_,=)-i=l-(i_r^i-^)i, and (1 - A;V)-i = ^!-^^^*,
and ds=il- l-'H')-^k'-tdt.
I
Therefore K' =:-i(\l- s-)-i (1 - A;V)-i ds,
and so
+ iK' = f (1 - s')-i (1 - A-^s-^)-i ds,
I A
or • sn (K + iK') = j ,
ik' -whence dn (A' + iK') = 0 and en (K + iK') = ±-r ■
To determine the ambiguous sign in the last equation, we observe that the sign of i must be understood in the light of the relation
■which was used in the transformation ; putting
s = sn{K+iK') = ^, t=l,
we have . yy- = ( 1 - s'-)~* = ! ,' ,
en {K +iA ) k
ik' and so en (K + iK) — — ;- .
k
Example. Shew that cn^(A'+iA") = (l-)') ( -7 )' .
19t)-19>S] THE F.T.T.IPTin FUNCTIONS sn 2, Cm, 6m. n.-)3
197. The periodicity of the elliptic functions with respect to K + I'A".
The quantity A" introduced in tho last article is of importance in eonnexion with the second period of the functions sn z, cii z, dn z.
For by the addition-theorem, \vc have
, ,- , ,,,v silken (A''+ iA")(hi(yv'-|-iA") + sn(/v +t/r')cn«dna ^ ' l-k'sn'zsn'(K+iK )
dnz
kcnz'
Similarly
en (z + K + I'A") = —
ik' 1 k cm'
and
dn(2+A'+2-A") = *'^"-.
By repeated application of these formulae we have
sn {Z + 2K + 2iK') = -snz, cn{z+2K + 2iK' )= cm, (\n{z + 2K + 2iK') = -dm,
i"ifl [ sn {z + -iA' + UK') = sn z,
- cn{z + 4A + 4iA') = en z, dn(z + 4-K + UK') = dn z.
Hence it appears that the function en z admits the period 2K + 2iK', and the functions sn z and dn z admit tlie period 4 A + 4tA''.
198. The periodicity of the elliptic fvmctions with respect to iK' .
By the addition-theorem, we have
sn {z + iK) = sn{z+K+ iK' - K)
sn (^ + A" -I- iK') en A dn A - sn Kcn(z + K+ iK) dn (g -^ A + iK') 1 - ^•' sn' A" sn» (2 -f- A -I- tA') 1 k&m'
Similarly we find the equations
.,^,^ t dn^
en (5-l-iA ) = — r ,
^ K sn 2
dn {z -f tA") = - i
en z saz
w. A.
23
354 TRANSCENDENTAL FUNCTIONS. [CHAP. XV.
By repeated application of these formulae wo obtain
sn (z ■+ 2iK') = sn z, en (2 + 2iK') = — en z, dn (z + 2iK') = - dn z,
and [ sn (^^ + ^xK') = sn z,
- cn{z + 4iiK') = en z, dn {z + UK') = dn 2,
so that the function sn z admits 2iK' as a period, and the functions en z and dn z admit 4iK' as a period.
199. The behaviour of the functions sn z, en z, dn 2, at the point z = iK'.
J^or points in the neighbourhood of the point 5 = 0, the function sn z can be expanded by Taylor's theorem in the form
sn 2 = sn 0 + 5 sn' 0 + 2 ^" sn" 0 + . . . , where accents denote derivatives.
Since
the expansion becomes
Hence
and
and therefore
sn 0 = 0,
sn'0 = cnOdnO = l, sn"0 = 0,
sn"'0 = -(1 + k-), etc.
saz = z-\{l+k')z''+ .... en 5 = (1 — sn-^)^
= 1-^2=+...,
dn^ = (1 —k-sn-z)^ = \-l¥z"-+...-
sn (z + iK') = T
ksnz
\{\+k^)z"- +...]'
1 1 + ^•=
kz
Qk
19n--201] I'liK KLLHTic FUNCTIONS sn z, cn^, dn 2. 355
— i 2A^ — 1 and similarly en (z + iK') = tt ^ STT^ *^ + • • •
i 2 — lc' .
and dn (z + iK') = — H .- iz + ... .
^ z 6
It follows that at the point z = iK', the functions sn z, en z, dn z have simple
poles, with the residues
1 i
k' k' *•
respectively.
200. General description of the functions sn z, en «, dn z.
Siiinniarizing the foregoing investigations, we can describe the functions sn z, en z, and dn z, in the following terms.
(1) snz is a one-valued doubly-periodic function of z, its periods being 4A' and 2iK'. Its singularities are at all points congruent with z = iK' and z = ^A' -I- iK' ; they are simple poles, with the residues k"'^ and — A;"' respectively; and the function is zero at all points congruent with z = 0 and z = 2K.
It may be observed that no other fuuctioii than sn i exists which fulfils this description. For if <f) (?) be such a function, then
<f> (2) - sn z
has no singularities, and so by Liouville's theorem is a constant independent of z ; but it is zero when z = 0, and therefore the constant is zero ; that is,
<^ {z) = sn z. When k- is real and positive and less than unity, it is easily seen that K and K' are real, and sn z is real for real values of z and purely imaginary for purely imaginary values of z.
(2) en z is a one-valued donbl3'-periodic function of z, its periods being •itK and 2K + 2iK'. Its singularities are at all points congruent with z = iK' and z=2K + iK' ; they are simple poles, with the residues ik~' and —ik~^ respectively ; and the function is zero at all points congruent with z = K and z = 3K.
(3) dn z \^ a one-valued doubly-periodic function of z, its periods being 2A'' and -iiK'. Its singularities are at all points congruent with z = iK' and z = 3i7v' ; they are simple poles, with the residues — i and -|- i respectively ; and the function is zero at all points congruent with z = K + iK' and z = K + SiK:
201. A geometrical illustration of the functions snz, cnz, duz.
The Jacobian elliptic functions may be geometrically represented in the foJ lowing way.
Let the position of a point, on the surface of a sphere of radius unity, be defined by (1) its perpendicular distance p from a fixed diameter of the
23—2
356 > TRANSCENDENTAL FUNCTIONS. [CHAP. XV.
sphere, which we shall call the polar axis, and (2) the angle yfr which the plane through the point and the polar axis (the meridian plane) makes with a fixed plane through the polar axi.s.
Then if ds denotes the arc of any curve traced on the sphere, we clearly have the relation
(dsY- = p"- {dfr- + (1 - p"-)-^ (dpy.
Let a curve (Seiffert's spherical spiral) be drawn on the sijlicre, its defining-equation being
df = kds,
where k is a constant. We have therefore for this curve
{dsr{\-kY)={i-p')-^{dpr-,
and so if s be measured from the pole, or point where the polar axis meets the sphere, we have
s=\\i-p^)-'Hi-i^p"-)-idp,
Jo or p = sn s,
the function sn being formed with the modulus k.
The rectangular coordinates of the point s of the curve, referred to the polar axis and an axis perpendicular to it in the meridian-plane, are p and (1 — p')i, and can therefore be written sn .9 and en s ; while dn s is easily seen to be the cosine of the angle at which the curve cuts the meridian. Hence if K be the length of the curve from the pole to the equator, it is obvious that sn s and en s have the period 'iK, and dn s has the period 2K.
202. Connexion of the function sn z with the function ^(z).
We shall now shew how the functions considered in this chapter are related to the elliptic function of Weierstrass.
Let ei, Bj, ek denote the quantities gi, 62, 63, taken in any order.
In the integral
r 1
2=1 i, (x - e^)-^ (x - e..)~i (x - 63)-* dx, let the variable of integration be changed by the substitution
Thus
1*
or
(1 - <=)-* [(e;. - ej) + (cj - e,) t'}-i dt, Jo
(ei -ej)iz=\ (1 - tT^ (1 - kH^i dt,
Jo
202,203] THE ei-liitic functions sn^', en ^, dn 5. 357
where k- ■■
gfc- ej
€i 6j
This is clearly equivalent to the equation
-^J^_ = snme,-ej)iz}.
We thus obtain the result that the /unction ^(z), formed with any periods, can be expressed in terms of the function snz by the equation
f (z) = ej + „ ,^' ~ ^^ ■ . , , the function sn being funned with the modulus
\ei - ejj Example. Shew that this relation can be written in either of the forms
and e,^e,dnme,-e,)^z}
203. Eaypansion of sn z as a trigonometric series.
Since sns is an odd function of ir, admitting the period -iii' (which we shall for our present purpose suppose to be real), it can by Fourier's theorem be expanded in a series of the form
. . irz J . 2Trz , . Sttz sm = b,&m^j^ + b.sm ^ + b,smjj^^ + ...,
where (§ 82) b,. = ;> I snt sin ^yrrdt.
This expansion will (§ 78) be valid for all points in the ^-plane contained in a belt parallel to the real axis and bounded by the lines whose equation is
Imaginary part of 2 = + iK',
since within this belt the function sn z has no singularities.
We have now to evaluate the integrals 6,.. We shall follow a proof due substantially to Schlomilch.
Let OARSCBQPO be a figure in the plane of a variable t, consisting of the rectangle whose vertices are the points
0{t = 0), A(t = 2K), C(t = '2K+2;K'). B(t = 2iK'),
with a very small semi-circular indentation PQ around the point t = ilC, and another small semi-circular indentation RS round the point t = 2K + iK'.
358 TRANSCENDENTAL FUNCTIONS. [CHAP. XV.
Consider the integral
r iVTrt
I .sn t e^ dt, taken round this contour.
Since the integrand is regular everywhere in the interior of the contour, we have (§ 36)
/ +/ ■'I +/ +f +/ +/ 4 =»■
J OA J AR J RS J SC J CB J BQ J QP J I'O
Consider first the integral along the semi-circular indentation QP. Writing t — iK' + Re^^, we have
sn < e 2A' dt= sn {iK' + Re'') e '^*" e^^ Re'' idO
J QP J -n
2
= ye 2-^ -r^-^T—idd, Since sn (lif' + ^) = ^
k J^ sn^ite'") A;
2
= -j-e '^^1 (1 + positive powers of R) dd
2
= — ^ e ^-^ , when R tends to zero. Similarly we have
|. irirt • TirK'
snte'^dt = (-iy~e '^ .
J RS n-'
Since sn {z + 2i7i') = sn ^, we have
I =-^'"'' I '
and since sn {z + 2K) = — sn 0, we have
f =(-!)'•[ , and f =(-l)'f
J. -lis J PO J SC J J
AR J PO J SC J SQ
We thus have
Now equate to zero the imaginary parts of this equation. Since
irivi
snte '^^ dt
2(K>] THE KI.MPTIC FUNCTIONS suz, cn ^, dn ^. 359
is ie;il when t is purely imaginary, there is no imaginary part arising from
J BQ J J'O
(1 - e""^') " sn « sin ^ rf( = y e" '-a' (1 - (- l)'}
' BQ
Therefore
irA"
Writing 5 = e ^ ,
this eqnation gives
(l-,/)A7v = j9Ml-(-l)'l.
r 6. = ^,/j*^^^^ifrisodd,
and lir = 0 it' ?• is even.
Thus finally we have the expansion of sns as a trigonometric series,
27r / 7* . 7ra o' Stt^ o- y-=^ I — :* SI n — _ _ -I ■* sin — L^ -I i—
Example. Prove that
sn 2 = y-=^ —3 — sin — ,. + ^^ sin ^-.^ + —^ — - sin ,, „ + kK \\-q 2a 1 - (f •2K \—q^ 2K
ni .^7. n* a^, rS
1. Shew that
2. Shew that
3. Prove that
4. Prove that
5. Prove that
C. Prove that
2ir f o* TTS o* Stt? o' Sttz 1
2A- liT^ '^°^ 2A' + r+7^ '°' w + r+? '^^^ 2A- + -j ■
Miscellaneous Examples.
^ = - ["" \l - ^2) - i (1 - F^S) - i (/<.
1
/"CD C
{l±cn(.+y)} U±cn(.-y)}= j^-^^^.
l-Fsn-2sn-y
, . /{r'2 + dn22 + /t2cn22
an* 2= ; — -. — .
1 +dn 2z
sn {z -«)dn (.-+y) = ''"^^"^''"y-^"y'^"y'^»-'
360 TRANSCENDENTAL FUNCTIONS. [CHAP. XV.
7. Shew that
8. Shew that
jj, _ dnz-cn2 / , 1 E^ ,,,,,,, /-'snj+cnzdnz
9. Prove that
■ r ■ -1 I ,.■,,.■ II / ^,^ 2sn^cn^dIly sin[sm i{sn(^+^)}+sin-i{.sn(i-y)}] = ^— p^^jT^-
10. Shew that
r ._,,,, ,, . ,, , ,,, cu^ 1/ - sn^ ?/ dn^ z
cos sin 1 {sn(z + y)} -sin~'{sn (s-v)]\ = —; H^ — ^— .
"■ I \ ' ^/) t V ff/ij l_F.sn-zsn-^
11. Shew that the quarter-periods K and I'K' are sokitions of the equation where2 = i(;2.
12. Shew that the quarter-periods A" and iK' are Legendre functions of the argument (1-2F), of order -J.
13. Shew that
cn/3cnysn (j3 — y)dna-|-cny cnasn (y-u)dn/3-t-cn acn/3 sn (u-/3) dny
-l-sn (3 — y) su (y - a) sn (a — (3) dn a dn /3 dn y = 0. (Cambridge Mathematical Tripos, Part I, 1894.)
14. I{u + v + r + s = 0, shew that
^•2 an usnvcnrcns-k^cn u en w sn r sn s - dn a dn v -I- dn r dn s = 0,
^2 sn M sn «) - i'2 sn rsns-|-dni«dn ocnrcn^-cn Men vdnr dns = 0,
sn M sn y dn r dn i' — dn M dn V su r sn s -|- en /■ en s - on m on v = 0.
(H. J. S. Smith.) . 15. Shew that, if a>A'>/ii>y, the substitutions
.r-y = (a — y)dn2M and .i'-y = (/3-y)dn"- v, where X,-2=(a — /3){a-y)"i, reduce the integrals
jiia-a;) (x - ^) {x - y )} " ^ d.v and ['' {{a-x){x- /3) (.r - y)} " ^ dx
respectively to the forms 2ii (a - y) ~ * and 2v (a - y) " * ; and deduce that, if ii + v = K,
1 - sn- !4 - sn- V -f- k'^ sn^ u sn^ v = Q.
From the substitution y = (a - «) (.r - /3) (.r-y)"', applied to the integi-al above with the limits /3 and n, obtain the result
I (a, cos2 e + \ sin- 5) " i d6 = f ^(a^ cos^ 5 -t- 6^ sin^ 6) " i
rf^,
where a^, 6, are the arithmetic and geometric means between a and h.
(Cambridge Mathematical Tripos, Part I, 1895.)
MISC. EXS.] THE ELLIPTIC FUNCTIONS Sl\ Z, CU Z, dll 2. 361
16. Shew how to express
I {{cufl +bx+c) {a'.v' + b'x + c')} - i dx
as an elliptic integral of the first kind, in the case when both quadratic expressions have imaginary linear factoi-s.
If
2=l{{x+l){.v^ + .v+\)\-idx,
express .r in terms of z by means of Jacobi's elliptic functions.
(C'ambiidgo Mathematical Tripos, Part I, 1899.)
17. The difl'ei-ent values of j satisfying the equation cn3i = a are «,, Zj, ... Zf,. Shew that
0 0
3)!-* n cnzr + ^''* n cn2,. = 0.
r=l r=0
(Cambridge Mathematical Tripos, Part I, 1899.)
18. Shew that
on Z 2»r f O* nz o' 3772 o- bnz ]
that
'c'aiiz in ( q^ . iTZ o' . 37r2 o- . bnz 1
Sx^'-rK [l^q «'" 2Z - IT? "'" M- + 1 +? ^■" 2A- - -j •
that
J n i-rr i a nz q^ 2n2 )
^" '=^+K \rk^ "'"^ K + r+T^ ''°' -K + ■■■]■
21. Prove that
3\ dni TrZ
5 2A'
■ 1 2^3 \2K) -IB \2K) j 1-q^ •""" 2A
^ am ^77. - }* 2A
1 2/C-3 V2A7 21-3 V^AVJ 1-
+ ....
(Cambridge Mathematical Tripos, Part II, 1896.)
22. Shew that
k-sn-z=^{z- {!{') + Constant,
where the Weierstrassian elliptic function is formed with the i)eriods iK and 2!'A''.
23. Shew that the difterential equation
fPti
^^ = {fi2sn2 2-i(H-F)}«
admits the general integral
« = |sni(6'-2)cni(C-2)dni(C-2)}-*{.4 + 5sn4(C-2)},
where .1 and B ai-e arbitrary constants, and C=2K+iK'.
CHAPTER XVI.
Elliptic Functions ; General Theorems.
204. Relation between the residues of an elliptic function.
In this chapter we shall be chiefly concerned with properties of more general elliptic functions than the special functions J.) (2), snz, en ^, and dn z, which have been discussed in the two preceding chapters.
We shall first shew that the sum nf the residues 0/ any elliptic function, with respect to those of its poles which are situated in any period-parallelogram, is zero.
For let f{z) be an elliptic function, and let 2&)i and 2a)2 be its periods. The sum of the residues is, by § 56, equal to the integral
l,\mdz
taken round the perimeter of the parallelogram.
Now in this integral, any two elementsy'(2') dz corresponding to congruent line-elements dz on opposite sides of the parallelogram, are equal in magnitude but opposite in sign, and therefore destroy each other. Hence the integral is zero, which establishes the theorem.
The number of poles or zeros of an elliptic function contained within a single period-parallelogram is often referred to as the number of irreducible poles or zeros.
205. The order of «» elliptic function.
We shall next shew that if c is any constant and f{z) is an elliptic function, the number of roots of the equation
f(^) = c contained within a pei'iod-purallelof/rarn depends only on f{z), and is inde- pendent of c, and is therefore equal to the number of irreducible zeros, and also to the number of irreducible poles.
204.-206] ELLIPTIC FUNCTIONS, GENERAL THEOREMS. 363
For the difference between the luiinhi'i- of zeros of the function
and the munbor of its poles, contained within the parallelogram, is (§ 60) eqnal to the value of the integral
2vijf{z) — c
taken round the perimeter of the parallelogram. But if P and Q are two points eongruent with each other, situated on opposite sides of the parallelo- giaiii, then the elements f'(z) {/(z) — c}~' dz arising from P and Q are equal in magnitude but opposite in sign, and so destroy each other. The integral is theivfore zero ; that is, the number of zeros of the function f{z) — c contained within the parallelogram is equal to the number of its poles, i.e. to the number of the poles of /(^) ; but this latter number is independent of c, which establishes the theorem.
The number of irreducible poles or zeros of an elliptic function is called the order of the function. It must be noted that a zero or pole, which is multiple of order n in the sense of "order" defined in §§39, 44, must be counted as ii zeros or poles for the purposes of this definition of " order."
The order is never less than two; for if an elliptic function had only a single irreducible simple polo, the sum of its residues within any period- parallelogram would not be zero, contrary to the theorem of the last article. This explains why the functions discussed in the two preceding chapters, which are of order two, are the simplest elliptic functions.
206. Expression of any elliptic function in terms of p (z) and ^' {z).
We shall now shew how any elliptic function can be expressed in terms of the Weierstrassian elliptic function which has the same periods.
Let f(z) be any elliptic function, and let ^(z) be the Weierstrassian elliptic function with the same periods 26), and 2&),. ; and let ^' (z) be the derivate of ^(z).
First, we can write
/(-)=! (/(^) + /(- -)) + Y-^^Y^=^ p' (-)•
Now the functions /(^)4-/(-^) and { f (z) — f {— z)] ^'-^ (z) are even elliptic functions of z : let <p{2) denote either of them : we shall now express <}> (z) in terms of j) (z).
Since <f>(z) is an even function, it follows that if a be one of its zeros in the fundamental period-parallelogram, then another of its zeros in the parallelogi-am will be congruent to —a: its irreducible zeros may therefore be aiTanged in two sets, say <»,, «..,, ... «„, and zeros congruent to
— a,, —a.,, ... - a„.
Elliptic F(
204 Relation between '
In this ch.aptiT we shi general elliptic functions dn^, which have been di-.
We shall first shew tli.ii ivitk respect to those of its /< ■ is zero.
For let f{z) bo an elli| The sum of the residues is, 1
taken round the perimeter o;
Now in this integral, any line-elements dz on opposite : but opposite in sign, and the^ zero, which establishes the tl
The number of poles or a single period-parallelogram poles or zeros.
205. The order of an el
We shall next shew thai function, the number of root.s
contained within a period-pa i pendent of c, and is therefore r to the number of irreducible po
AL rUNCTIONa
[chap. XVI.
w of 2Tn', the last expression must be
1 - A multiple of 2w„ )eorem.
I
0, tar which *,, ^, ... «n the irreducible poles, y* •* « one-ralucd function without
MathcuiAtic.tl TriiH^s Pjirt l]^ 1899.)
I), defined b^ iln- f(|uuti..n
'«ro when i « 0.
' P(*) i» uniformly convergent, it
ve
I
Jw. + 2ii«h)-*)] d*
+ 2n«#,)-' -f- 1 (2i»«», -f. 2mw,)-'!,
* •t««0. ia 8*ti»ficd by this
nmation i«, u usual, eitended
) values of m and n, . \r,|,r
ni«i« flnr!»'>"'"' i'l
, \' i^ i^ ■:'; '-.y
•■!, ^. >q .-:;■ *^ .ff j>fj .
,>■«■ 'i't/^^.
SCnOM; aC5(BA> rUCUREMS.
UtA with lb' f :t»»i.ti iTj* *, wbo« ex|Miinon is
• 1.
367
Uotaintn*! I nU-Kmlion.
• -«, in t "oiw respectively, we have
?(-•,) + 2% -<"-.» + 2.;,.
antA ij, and ?>,.
C(jr)+CWi' + f * -rW-0. (schottky.)
J M Um> •ddiUon-th**'" f-r the function f (i).
/■.4nc/i<M "-hen the principal part of its is ffivei
^„ wit iHTi.xls 2a), and 2a>,. Let its point --=«,. ...... a„; and let the
poit (tk he
"-^^-*r!ct..r*-'(^
-1)!
-fl*)|:
364 TRANSCENDENTAL FUNCTIONS. [CHAP. XVI.
Similarly its poles can be arranged in two set.s, say h,, b.,, ... 6„, and poles congruent to —61, —b.,, ... — 6„.
Now form the quantity
1 }^j(g)-^(«iVt \p(z)-^(a„)} ... {p(z)-p(an)] cj> (z) (g) (z) - s^ (6,)1 iiJ (^) - gJ {b,)\ ...lf(z)-^ (bn)\ '
where if one of the quantities a^ or b^ is zero, the corresponding factor [p(z)-^iar)} or {p{z)- ^{b,)} is to be omitted.
This quantity is a doubly-periodic function of z ; it clearly has no zeros or singularities in the interior of the parallelogram, except possibly at z = 0, and therefore either it or its reciprocal has no singularities in the interior of the parallelogram, and so has no singularities in the entire plane. It must therefore by Liouville's theorem be a constant independent of ^.
Thus
0 (.) - Constant ^^ ^^y_ ^-^^ ,^(,-)ir^(64 ... (^ (.) _ ^. (ftj} "
The i|uantities {/(z) +/{— z)} and \ f (z) — f (— z)} {^' {z)}~^ cim thus he expressed as rational functions of ^J (z) ; and thus we obtain the theorem that any elliptic function can be e.rpre.ssed in terms of the Weierstrassian function formed with the same periods, the e.cpression being linear in ^' {z) and rational in ^{z).
Example. Shew tliat
s,nzc\\zAnz=\k~^ ^' (s - iK'),
where the function ^' (2 - iK') is formed with the period.s 25" and 2iA''. /
207. Relation between any two elliptic functions iuhich admit the same periods.
We shall now shew that an algebraic relation exists between any tivo elliptic functions whose periods are the same.
For let /(2) and <f>{z) be the functions. Then by the last article, /(«) and (f) (z) can be expressed rationally in terms of ^ (z) and ^' (z). Eliminating ^ (z) and ^y (z) fi-om the three equations constituted by
p''(z) = 4'f(z)-g,piz)-g,
and these two relations, we have an algebraic relation between y(2) and <ji(z); which establishes the theorem.
It is easy to find the degree of this equation in / and cj). For if / be an elliptic function of order m, and if c^ be of order n, then each value of / determines m irreducible values of z, and each of these determines one value
207, "208] Ki.i.MTic functions; gionkhai, tiikohems. 365
of <f): so to each value of /' correspond m values of (f), and similarly to each value of <f> correspond ii values of /. The equation is therefore of degree Dt, in (f> and n in /!
Tims ^ {:) is of order 2, nnd jp" (s) of order 3. The relation between them, namely
p(z) = 4p(2)-£r,S»(j)-</3. should therefore bo of degree 2 in ^'(z) and 3 in ^{:) — as in fact it is.
An obvious consequence of this proposition is that every elliptic function is connected leith its derivate bi/ an algebraic equation.
Example. If t, ii, r are three elliptic funetions of the second order, with the same ])eriod8 and argument, shew that there exist in general between them two distinct relations which are linear with respect to each of them, namely
Atuv + Buv + Crt + Dtu + £i + Fu + 0'v + M=0,
A'ttiv + B'uv + C'vt + iytu + E't + F'u + G'v + H' = Q,
where A, B, ... , //' are constants.
208. Relation between the zeros and poles of an elliptic function.
We shall now shew that the sum. of the affixes of the irreducible zeros vf an elliptic function is equal to the sum of the affi.res of its irreducible poles, or differs from this sum only by a period.
For if /(s) bo the function, and 2fu, and 2&).j its periods, the difference in question is (§ 59) equal to the integral
1 {zf'{z)dz
2-mJ
/(^)
taken round the perimeter of the fundamental period-parallelogram. This can be written
or
1 r r-"' \zf {z) (2a,, + z)f' (2a,, + z)\ ^_
Jo I }{z) f{2a>, + z) J
or — . {- 2a,, log 1 + 2a,, log 1 j ,
36() TRANSCENDENTAL FUNCTIONS. [CHAP. XVI.
and as log 1 is zero or .some multiple of 27ri, the la.st expression must be either zero or some quantity of the form
A multiple of 2wi + A multiple of 2(0o,
i.e. a period. This establishes the theorem.
Example. If F(z) is an elliptic funt-tion, for which 2,, z^, ... arc the irreducible poles, and .li, A.,, ... the corresponding rcsidue.s, and if /(s) in a one-valued function without singularitie.s in the parallelogram, shew that the integral
^.jmFiz),z,
taken round the period-parallelogram, is equal to 2J„/(z„).
(Cambridge Mathematical Tripos, Part II, 1899.)
209. The function ^ {z).
We shall next introduce a function ^{z), defined by the equation
with the condition that ^{z) — z~^ is to be zero when z = 0.
Since the infinite series which represents f {z) is uniformly convergent, it can be integrated term by term ; we thus have
^{z) = - {[z-- -\-^[{z- 2m(u, - 2iM„y - {2mo), + 2n(u.,)--j] dz
= 2-' + S {(z - 2mQ)i - 2nco„)-' + (2to&)i + 2no}.^-' + z {2mo)^ + 2ncor,)--},
since the condition, which ^{z) has to satisfy at 2 = 0, is satisfied by this choice of the constant of integration. The summation is, as usual, extended over all positive and negative integer and zero values of m and n, except simultaneous zero values.
When I 2m(0i + inw^ \ is large (and we can suppose the series arranged in ascending order of magnitude of | 2mo)i + 2wq)2 j), the quantity
{z - 2miOi - 2?!ft)j)"' + (2m&)i + 2n(o„)-' + z (2?n&)i + 2*10)2)-"
bears a ratio of approximate equality to the quantity
— z'' {2mo)i + 2n(02)~'*.
The series which represents ^(z) can therefore be compared with the series 2 {2m(Oi + 2nQ}2)~''\ and hence we see that it is absolutely convergent, except at the singularities z = 2ni<0i + 2n(02, and that the convergence is uniform.
It is evident from the series that at its singularities z= 2mw^ + 2no).,, the function ^{z) has simple poles with residues unity; and that }^{z) is an odd function of z.
20f) 211] KI.Lll'TIC Fl'NCTIONS; GENERAL THEOREMS. 367
'I'lic. t'unctiou f (j) may be compared witli the function cot :, whose expansion is
00
cotr=2-'+ S {(2-J"7r)-' + (m7r)-'}. ni=— »
The equation -j- cot z = — cosec^ z
eorresixjnds to the equation
£fW=-P(-').
210. The quasi-periodicity of the function ^(z). Since P (^ + So),) = jj (z),
we have J^^ f (z+ 2.,) = |^ r(4
or n^ + 2a,,) = f(z) + 27,,,
and similarly f (^ + 2(«.j) = ^(z)+ 2rj2,
where 77, and tj., are two constants introduced b}' integration.
Writing z = — Wi and z= — to., in these relations respectively, we have r(t«,) = r(- 0,,) + 2.,, = - ^(co,) + 27?,,
?(«.,) = r(- 0,,,) + 27?., = - r («,) + 27,.,,
whence •>?, = f(a),),
77.,= f(a).,), which determines the constants t;, and 772. l{ j: + i/+2 = 0, shew that
{fw+f(y)+fW}^+rw+ro/)+f'w=o.
(Schottky.) Thi.s result may bo regarded as the addition-theorem for the function f (z).
211. Expression of an elliptic function, when the principal part of its ej;pansion at each of its singularities is given.
Let f (z) be any elliptic function, with periods 2a), and 2a)o. Lot its irreducible singularities be at the points ^=0,, a,,, ... a„; and let the principal part of its expansion near the point n^ be
z-at (z-atY '" (■s -«*)■■» Then if we consider the function
E(z) = i L^{z - at) - c,X (z-a,}+...+^ ^^"J ] c,t"\-» (z - a A ,
d' where f" {z) denotes y^^(z), we see that
368 TRANSCENDENTAL FUNCTIONS. [ClIAP. XVI.
(1 ) When z is replaced by (2 + 2(u,), the function E{z) is increased by
n
*■=! n
But — Cjt, is zero, since the sum of the residues of /(2) within a period-
parallelograni is zero. Hence E {z) admits the period 2(i),. Similarly E{z) admits the period lw„. E {z) is therefore an elliptic function, with the same periods a,sf{z).
(2) Since the function 5'""' (^ — f't) has singularities only at a^ and congruent points, and its principal part at Uk is (— 1)™ m\ (z — a^.)"™"', we see that E{z) has the same singularities as /(2), and the same principal parts at them.
It follows from (1) and (2) that f{z) — E{z) is a function with no singularities in the whole plane ; and therefore, by Liouville's theorem, f{z)—E(z) is a constant. Thus the ftmction f(z) can be expanded in the form
f(z) = Constant + I S J-^^^' c*. ?"-" (^ - «*)•
Tlii.s theorem maj- bo regarded as analogous to the decomposition of a rational function into jjartial fractions, or the decomposition of a circular function into a scries of co- tangents (§ 76).
Example 1. Shew that
I: am = Ciz- iK') - f (2 — 2K—iK'}+ Constant, where the f-functions are formed with the periods iK and 2iE'.
Example 2. Shew that
1 p{x) p'{x)
1 p(z) r«
2 1 ^(x) ^Hx)
1 «>(y) r(y) =f(.^+y+^)-f(.'0-f(.y)-fW-
Extend this theorem to the case in which there are any number of variables.
(Cambridge Mathematical Tripos, Part II, 1894.)
212. The function a-(z).
We shall next introduce a function a (z), defined by the equation
^logo-(2)=f(2),
with the condition that a {z)jz is to be unity when 2 = 0.
212, il:?] Ki.i.ii'Tie Ki'NCTioNS ; okneuai, tiieokkms. 309
Since the convergence of" the infinite series which represents ^(z) is uniform, the series can be integrated term by term : \vp thus have
iog.(.) = log. + S{log(l-^2W.)
z I
2))iQ)i + 2n(o., 2 (2TOa»i + 2nci)^)-] '
(in cliDosing tlio constant of integration so as to satisfy tlio condition at ^ = 0 ; and therefore
I <r{z)-~n\^i 2mw,+ 2,1(0,)^
the product being, as usual, extended over all integer and zero values of m and n, except simultaneous zeros. The absolute convergence of this product follows from that of the series
vJl /. - ■g '\ f 1 z' )
~ I ''^ V iiiito, + 211(0., I "^ 2m<a, + 2nu>, 2 (2mw,+ 2n(o^ri '
which is established by comparison with the series
. _T ^
~ 3 (2mw, + 2«a)o)' '
since t iu' terms of tiio two series have ultimately a ratio of etiuality.
It is evident from the product-expression that o-(2) is an odd function of :, that its zeros are at the points z= 2m&), + 2/!a)o, and that 5^' (7(2) tends to the limit unity as z tends to zero.
The function <t{z) may be compared with tho function sin;, dchncd by the e.xpansiou
sm |
i=z n ^ 1 ,„ = _=c|.\ "iffy |
|
The relation |
J- log(siui) = coti dz |
|
corresponds to |
~ log <r{z) = C(z). |
213. The quasi-periodicity of the function a {z).
On integrating the equation
r(z + 2a,.) = ?(«) + 2i;,
we have log o- (2 + 2a),) = log o" {z) + 27?, 2 + Constant,
or (T(z+2ai,) = ce^''a{z),
where c is a constant. To determine c, write z = -o}i\ thus
o-(«,) = -ce-^'"'cr(a),),
or c = - e*"'-',
w. .K. 24
370 TRANSCENDENTAL FUNCTIONS. [CHAP. XVI.
and therefore (t(z + 2(u,) = - e-'''^-+'"''> a (z).
Similarly it (z + 2W2) = - e''i2(-+"')o- (z).
The behaviour of the function a (z) when its argument is increased by a period of ^ (z) is thus determined. By repeated application of these formulae wo can Hnd the value of cr {z + 2ma>i + 2»ia).j), where m and n are any integers.
A71 example sheiidng how the function a {z) may be expressed as a singly-infinite product. We have
/ \ Z ■ z'
tT(z) = zU I 1 — )e2m<u,+2ni.ij '(2mco|+2)ii«2)2
the summation being extended over all pcsitive and negative integer and zero values of m and 11, except simultaneous zeros. This can be written in the form
*» / z \ -'-- +i '^ ±» / z \ -5-4-1--^^
<t{z)=Z n (1-- )e2?mo/'(2mu,)2x n (1 i_ ) e2Mu,/' (2>«oj)!'
m=±i \ 2?na),/ „=±, V 'i.naij
*« » / 2 \ ' +1— -^?
X n n (1 1 e2m«)|+2nu, ^(2»iu,+2n<oj)2
m=±i 11=1 \ 2ma)j + 2«a)2/
±" » / z \ ~ I 1 ^'^
X n n (iH ] e2jiiu,+2ncoj '(2mu,+2ni»J".
m=±i 71=1 \ 2ni(»i + 2n(U2/
Now
0 IT fl_^^^-Ae2™^'^*('^^'».)'=— ■e*''''"=±>(2^^)'sin — ,
)a=±l \ 2TOa)]/ TT 2(Uj
and II n (1 ) e2«i«.,+2n«>j"'"H2m(»i+2n«.J
m=±l 71=1 \ 2»t(Oi + 2Mu2/
271U)3-Z
(2mio,+2n<ui)2
!+-„-" e 27M(..i -an-jZ . ,
^ jj ^ \ "''""i ^ g2mu., (2mo.,+2«<uj)"^*
,„ = ±i „=i / 2»(a).A -.f^" V 2»ia)iy
. (2?J(0,-z)7r ., , ,
_ 17 -^"i t «m=±i 127110), (2jn«,+27W)2) ''(2m»,+2)Moj)2J
^^"^^^^^^"^^^ ^^^^^^~ • «=l . a?i&).,7r , z
Sin — -^ 1 - „ — 22(0] zn&>2
Similarly
n n ( iH ) e2m(»,+2»uj '(2niu,+2Mioj)»
7tt=±l 71=1 \ 2»1<Ui + 2mm2/
. (2»»(a, + z) TT Sin^ ^-^ I " ±£ J 2BmaZ ■ z' ■<
^ ^^;M 1 m..±l I 2mi^i (2iiliui+2»<u,) (2llliu, + 2mu2)»J
.sin — -^- 1+5 —
21.'i] EI.I.IITIC l-MINCTIONS ; CENEHAI, TMKOUEMS.
'I'luMvfolv
. 2(., -
371
2(0,
X n
. X n
11=1
. (2na.,-z)Tr *, , ,
' +1 '■ am- *£ t -2>n.i>g ., z- ■>
2ii», '(8ii<u,>' -"i „. j;il2»i»i,(2m«i.+2>iu.) '(2»n«,+2ii<»,)»/
^ +j _'!_ .sin (?!!5^±i)- - r 2,..^ ^j ^_ ,
sin — i^ — '— sin ^ ' — —
Zb),
1 (2m^)2"*'„=±, (Snoi, +2m^
slll-
Now write y= e "i .
Then
(2Ha,,-.)jr (2ngj+£)^ / (2«„,--)':^l f ':^r2««.,+-')l 2cD, ° 2o,, (1-e "'/U-e"' /
sin- — "
{,
-e "' ]
l-2j2»cos — + 5^»
(1 -,/-■"
Now if the imaginary part of u/u, is positive, we have \q\ <\ ; and thus the infinite product
1 - 2(?-'" cos — + y^»
>,=i (1 -</";'
converges absolutely, since the series
2 o^"
converges absolutely ; and hence we can separate oft' the exponential factors, and can write
1 -2y2»cos-- + 5^'' c=' ^1 =;„ 'LL n "1
„ (z) = e"'' - ' sin i— n ^T .-^
where C is a constant.
The quantity (7 can be very simply determined I'rum tlie relation
<r(2 + 2»,)=-e-'''(-+"''<r(0);
for this gives
gCte + 2u,)>^gCi' + 2T,,U + co,)
(7= '■ 2a),'
372 TRANSCENDENTAL FUNCTIONS. [CHAP. XVI.
We have therefore finjilly an expression for o- (;) iis a singly-infinite product, namely ^,i» 1 - 27=" cos -' + '/"
aW=c-''"'^-"'sin^ n
whore q = e "i .
214. The integration of elliptic functions.
The integral of any elliptic function can he foiitid in terms of the functions ^(z) and a (z), by u.'fing the theorem given in §211, on the resolution of elliptic functions into a sum of f-functions.
In fact, in § 211 an expression
has been found for the elliptic function f{z) ; the indefinite integral of this expression is
cz+l c,.,log<7(2-a,)+ I S ^fnrN*'^"""^^-"*)' which is the required integral of f{z).
Example. The expression for p^ (z), found by the theorem of § 21 1, is
It follows that
^^ (2) dz=lf' (2) + ^JJ.^ + Constant.
/'
215. Expression of an elliptic function who.se zeros and poles are known.
We have already seen (§ 20.5) that the number of irreducible zeros of an elliptic function is equal to the number of its irreducible poles ; and that (§ 208) the sum of the affixes of the zeros differs from the sum of the affixes of the poles only by a quantity of the form (2/mq), + 2«ft).), where vi and /; are integers. By replacing the zeros and poles by others congruent to them, we can reduce this difference to zero. Suppose this done, so that for a given function f{z) the irreducible zeros are a,, a„, ... a„, and the irreducible poles are 6,, 60, ... b^, where
tt, + (to + . . . + Un = bi + k+ ... +b„.
If any of the zeros or poles is multiple, of order k say, it will of course be counted as if it were k distinct simple zeros or poles.
Consider now the quantity
a (z — Oj) a- (z — a^) (r(z — a„)
E{z)=^
a {z - b-,) a {z - b.^ a {z - b„) '
:JH. 21.5] Ki.i.iiTic functions; uenkkai. tiik(iukm.s. :iT.]
We have
E{z + 2a),) = e^''i"^"''i' + ('"''-''"*" ■■•■'"''"''"'"''"''''"•■■"<'"''»*! /i" (2)
= E{z).
Similarly E {z + 2a>,) = E (z).
Tims E{:) is an elliptic function, with the same periods ns f(z)\ and thiTetore t\z)iE{z) admits these periods.
But the function f{z)'E{z) clearly has no zeros or poles at the points
(/,, (».,, ... (/„, I),, ... b„,
and so has no zeros or poles at any point of the ^-plane. Therefore, by Liouville's theorem, t'(z);E{z) is a constant; and so finally
f( \- g (^ - ".) g (-g - "o) <r{z- a„)
■'^^' ''<r(z-b,)a{z-L) a-(2-6„)'
where c is some constant.
An elliptic function is therefore determinate, save for a multiplicative constant, when the places of its irreducible zeros and poles are known.
This i^^ analogous to the factorifwitioii of a ratioual function : if a rational function has zero.s at jH)ints «,, a,, ... «„, and poles at points 6,, b.^, ... b„, it can be expre.ssed in the form
(z-a^)(z-a,),..{z-a„)
U^-h,){z-b^)...(z-Ky
where c is a constant.
Example 1. Prove that
By diftei-entiating this formula, shew that and by further differentiation obtain the addition-theorem
Example 2. If
»-<«^)-^<.)-«'<'>^Uf5Fft)'
2 (ax-b^) = 0,
A = l
Shew that I .r(a.-6,)...^(a,-6.) ■a(a.-&.)^Q
\=l <r(aA-ff,)... 1 ...<r(aA-a„)
24—3
374
TKANSt:KNI)KNTAI, I'TNCTIONS.
[chap. XVI.
Miscellaneous Examples.
1. Shew tliat, if/i denote one of the functions sn 2, en z, dn z, and if y and •;• denote the other two, it in alway.s jjo.ssible to choose constants a, 6, c, such that
I pdz = a log (65' + cr
).
2. Shew that c\ei-y elhjitic fiuiotion of order n can Ije expressed as the quotient of two expressions of the form
«i&'('- + 6) + a2^^' (-^ + 6) + . ..+«„&*<"-'•{-' + *),
where h, a,, a.^, ... «„, are coastants. (Painleve.)
3. Prove tliat
^(.— a)JP(.-6) = i.>(a-6){^(3-a) + ^J(^-6)-j;J(a)-^(6)}
+ i->'(«-6){f(--«)-a-— i) + f(«)-fW! + ^->(")S-'(/')-
(Cambridge Mathematical Tripos, Part II, 189.5.)
4. Shew that
1 <^{x) fr)'(.r)
o- (x+y + 2) 0- {x-y) a{y-z)a (z -.r) ^ 1 <fi {x) a^ ii,) &i iz) 2
1 F(y) r(i') 1 f{i) r(')
Obtain the addition-theorem for the function ^<> {z) from this result. 5. Establish the identity
1 f (^o) f ('0) ...&«"- '1 (2o) 1 _ / _ 1 xi» (» - » 1 t 9 ' » • o-(2o+^i + - + 2„)na-(zA-2M) 1 &'(^l) &>'(--,)•■. P<"-"fe)
1 ^(O &>'(0-S-""-''W
where the product is taken for all integer values of X and /j from 0 to n, with the restriction
6. Prove that
n \ n t\ n i\syia o;\ <r(2-2(l + 6)w(2-26-(-o)
f(2-a)-f(2-6)-f(a-6)-|-f(2«-26) = -;,;'
a-(26 - 2a) o- (2 - a) o- (2 - 6) ' (Cambridge Mathematical Triixjs, Part II, 1895.)
7. Shew that, if 25+21 + 2.^-1-23 = 0, then
{2f (2^)F = 3 {2f (2a)} {2P (2;^)} + 2^' (2a), the summations being taken for X = 0, 1, 2, 3.
(Cambridge Mathematical Tripos, Part II, 1897.)
MISC. EXS.] El.I.iniC FrXCTIONS ; GENERAL THEOREMS. 375
M. I'rovo tliiit
^^' <r {2i + J (j, +20 + 23 + 2,)} .
is a doubly-iieruxlic function of :, .such that
^W+J^(i + 0>,)+Sr(j + (B..)+J^(.' + <U, + W2)
= - 2<r {i (.'o + .-3 - *-, - 24)} <r {i (23 + 2, - 22 - 2,)} <7 {J (2, + 22 - ■23 - 24)1-
(Cambridge Mathematical Trii">s, Part IT, 1893.)
9. If f(:) be a iloiibly-poriirtlio fum-tion of the thiril order, with poles at 2 = c,, z — c.,, 2 = (,-.i, and if <f> (j) be a doubly-j)eriodic function of the second order with the same iierioda and poles at 2 = a, 2 = 3, its value in the neighbourhood of 2 = 0 being
<^(2) = . -+\{z-a) + \.,[z-af + ..., * — a
prove that
A X- !/" («) -/" {»)} - X {/' (n) +/' (3)} 2 * (c,) + !/(«) -/(/3)} {3XX, + 20 (c^) 4> (C3)} =0.
1 1
(Cambridge Mathematical Tripos, Part II, 1894.)
10. If X(2) be an elliptic function witli two poles «;, ii„, and if 2,, z^, ... z^„, be 2n arbitrary arguments such that
2, + -\, + . . . + .'2„ = n (ai + a,), shew that the detenninant whose ;(th row is
1, X(2.), X2(2(), ...X"(2.), X,(2.), X(2i)X,(2,), X-^(2i)X,(2,-), ... X"--'(-)Xi(2i),
where ^i (2;)=^ X(2i),
vanishes identically.
(Cambridge Mathematical Tripos, Part II, 1893.)
11. Shew that, provided certain conditions of inequality are satisfied,
,\ ,\ e "• =5— cot -^ + cot -^+—2</^""'sm— m2 + ?«/),
where the summation applies to all jiositive integer values of m and n.
(Cambridge Mathematical Tripos, Part II, 1895.)
12. Assuming the formula
i.j* „ „ l-2<j^cos — +o<» <r(2) = e2"' . - ' sm — n -1 ,
prove that
o>, V2a),/ 2w, Vw,/ , l-^-* u '
on condition that.
■2«('^?')<ye(^W2/j(-?'A.
Vo>,/ \K0,/ Wl/
(Cambridge Mathematical TriiH)s, Part II, 1896.)
f
376
TRANSCENDENTAL FUNCTIONS.
[chap. XVI.
13. Shew that
C 1 , O- (2 - 2„) I , O- (z - ?'2o)
where
a;^ = a + K t,
6 |)2(2)-jf)='(2o)'
26
^3 = 0.
^■>^'o) =
6 (a - 6) ■
(Dolbnia.)
INDEX OF TERMS EMPLOYED.
{The numbers refer to the pities, where the term occurs for the first time in the
book or is defined.)
Absolute convergence, 12
,, value (modulus), 5 Affix, (i
Aniil.vtic fuiR'tioii, 4o Ar^niul ilia^'riiin, (i Associated Letiendie functions, 231 AsyiiiiUotic expimsion, KiS Autouiorphic functions, SH'.I
13einoulliuu numbers and polyncniials, 97 Bessel coefficients, 266
functions, 274, 294 Branch, branch-point, 66
Circle of convergence, 29 Coefficients, Bessel, 26G Complex numbers, 4 Conditions. Dirichlet's, 146 Congruent, 325 Contitj'uous, 260 Continuation, 57 . Continuitj-, 41 Contour, 47 Convergence, 10
„ absolute, 12
,, circle of, 29
,, radius of, 29
,, semi-, 12
uniform, 73 Cosine series, 1.H8
Definite integral, 42 Dependence, 40 Derivate, 51, .53 Determinants, infinite, 35 Diagram, Ari;and, 6 Dirichlet's conditions, 146
„ integrals, 191
Doable-circuit integrals, 258 Doubly-periodic, 322
Elliptic function, 322 Equation, associated Legendre, 231 Bessel, 269
,, hypcrgeometric, 242
,, Laplace's, 311
,, Legendre, 206
Essential singularity, 63 Eulerian integrals, 181, 189 Expansion, asymptotic, 163 Exponents of a singularity, 245
Fourier series, 127
Function, analytic, 45
,, associated Legendre, 231
„ automorphic, 339
Bessel, 274, 294 „ Gamma-, 174
elliptic, 322 „ hypergeometric, 242
„ identity of, ,59 „ Legendre, 209, 221
„ many-valued, 66
Gamma-function, 174 Genus, 339 Geometric series, 13
Hypergeometric series, 20, 240 „ function, 242
Identity of a function, 59 Infinite determinants, 35
,, products, 31
,, series, 10 Infinity, point at, 64 Integrals, definite, 42
„ Dirichlet's, 191
,, (li)Ml)lecircuit, 258
Eulerian, 184, 189 Invariants, 326
378
Irreducible, 302
Kiiiil of Legendre functions, 20'J, 221 Bessel „ 274, 296
Laplace's equation, 3U
Legendre associated functions, 231
I, equation, 206
„ functions, 209, 221
polynomials, 204 Limit, 8
Many-valued function, 66 Modulus, of complex quantity, 5
,, Jucobian elliptic functions,
Non-uniform convergence, 73 Numbers, Bernoullian, 97 II complex, 4
Order of Bessel t-oefBcients, 267 ,1 functions, 274 elliptic functions, 363 Legendre functions, 209
I, polynomials, 204
pole, 63, 65 zero, 55, 64
Parallelogram, period-, 325 Part, principal, 63 Period, 322
Period-parallelogram, 325 Point, regular, 45
II representative, 6
„ singular, 45
345
INDEX.
L Pole, 63, 65
Polynomials, Bernoullian, 97
,1 Legendre, 204
Power. series, 28 Principal part, 63 Process of continuation, 57 Products, infinite, 31
Quantity, complex, 4
Radius of convergence, 29 Regular, 45, 46 " Residue, 83 Representative point, 6
Semi-convergence, 12 Series, Fourier, 127
geometric, 13
hypergeometric, 20, 240
infinite, 10
power-, 28
sine and cosine, 13 s Simple pole, 63 Sine series, 138 Singly.periodic, 322 Singularity, 45
II essential, 63
of hypergeometric equation, 245
Uniform convergence, 73 Uniformisatiou, 338
Value, absolute (modulus)i 5 Zero, 55, 64
CAMBRIDGE : PRINTED BY J. & c. F. CLAY, AT THE nNIVEBSITY
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