THEORY OF FUNCTIONS
OF A
COMPLEX VARIABLE
CAMBRIDGE UNIVERSITY PRESS
C. F. CLAY, MANAGER
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THEORY OF FUNCTIONS
OF A
COMPLEX VARIABLE
BY
A. R. FORSYTH,
Sc.D., LL.D., MATH.D., F.R.S.,
CHIEF PROFESSOR OF MATHEMATICS IN THE IMPERIAL COLLEGE OF SCIENCE
AND TECHNOLOGY, LONDON : AND SOMETIME SADLERIAN PROFESSOR OF PURE
MATHEMATICS IN THE UNIVERSITY OF CAMBRIDGE
THIRD EDITION
CAMBRIDGE
AT THE UNIVERSITY PRESS
1918
331
I
PKEFACE.
AMONG the many advances in the progress of mathematical.
-j^- science during the last forty years, not the least remarkable
are those in the theory of functions. The contributions that are
still being made to it testify to its vitality : all the evidence points
to the continuance of its growth: And, indeed, this need cause no
surprise. Few subjects can boast such varied processes, based
upon methods so distinct from one another as are those originated
by Cauchy, by Weierstrass, and by Biemann. Each of these
methods is sufficient in itself to provide a complete development ;
combined, they exhibit an unusual wealth of ideas and furnish
unsurpassed resources in attacking new problems.
It is difficult to keep pace with the rapid growth of the
literature which is due to the activity of mathematicians,
especially of continental mathematicians : and there is, in con-
sequence, sufficient reason for considering that some marshalling
of the main results is at least desirable and is, perhaps, necessary.
Not that there is any dearth of treatises in French and in
German : but, for the most part, they either expound the pro-
cesses based upon some single method or they deal with the
discussion of some particular branch of the theory.
The present treatise is an attempt to give a consecutive
account of what may fairly be deemed the principal branches of
the whole subject. It may be that the next few years will see
additions as important as those of the last few years : this account
would then be insufficient for its purpose, notwithstanding the
breadth of range over which it may seem at present to extend.
My hope is that the book, so far as it goes, may assist mathe-
maticians, by lessening the labour of acquiring a proper knowledge
of the subject, and by indicating the main lines on which recent
progress has been achieved.
No apology is offered for the size of the book. Indeed, if
there were to be an apology, it would rather be on the ground
of the too brief treatment of some portions and the omissions
of others. The detail in the exposition of the elements of several
vi PREFACE
important branches has prevented a completeness of treatment
of those branches : but this fulness of initial explanations is
deliberate, my opinion being that students will thereby become
better qualified to read the great classical memoirs, by the study
of which effective progress can best be made. And limitations of
space have compelled me to exclude some branches which other-
wise would have found a place. Thus the theory of functions of
a real variable is left undiscussed : happily, the treatises of Dini,
Stolz, Tannery, and Chrystal are sufficient to supply the omission.
Again, the theory of functions of more than one complex variable
receives only a passing mention ; but in this case, as in most
cases,' where the consideration is brief, references are given
which will enable the student to follow the development to
such extent as he may desire. Limitation in one other direction
has been imposed : the treatise aims at dealing with the general
theory of functions and it does not profess to deal with special
classes of functions. I have not hesitated to use examples of
special classes : but they are used merely as illustrations of the
general theory, and references are given to other treatises for
the detailed exposition of their properties.
The general method which is adopted is not limited so that
it may conform to any single one of the three principal inde-
pendent methods, due to Cauchy, to Weierstrass and to Riemann
respectively : where it has been convenient to do so, I have
combined ideas and processes derived from different methods.
The book may be considered as composed of five parts.
The first part, consisting of Chapters I VII, contains the
theory of uniform functions : the discussion is based upon power-
series, initially connected with Cauchy's theorems in integration,
and the properties established are chiefly those which are con-
tained in the memoirs of Weierstrass and Mittag-Leffler.
The second part, consisting of Chapters VIII XIII, contains
the theory of multiform functions, and of uniform periodic
functions which are derived through the inversion of integrals
of algebraic functions. The method adopted in this part is
Cauchy's, as used by Briot and Bouquet in their three memoirs
and in their treatise on elliptic functions ; it is the method that
PREFACE Vll
has been followed by Hermite and others to obtain the properties
of various kinds of periodic functions. A chapter has been
devoted to the proof of Weierstrass's results relating to functions
that possess an addition-theorem.
The third part, consisting of Chapters XIV XVIII, contains
the development of the theory of functions according to the
method initiated by Riemann in his memoirs. The proof which
is given of the existence- theorem is substantially due to Schwarz ;
in the rest of this part of the book, I have derived great assist-
ance from Neumann's treatise on Abelian functio/is, from Fricke's
treatise on Klein's theory of modular functions, and from many
memoirs by Klein.
The, fourth part, consisting of Chapters XIX and XX, treats
of conformal representation. The fundamental theorem, as to the
possibility of the conformal representation of surfaces upon one
another, is derived from the existence-theorem : it is a curious fact
that the actual solution, which has been proved to exist in general,
has been obtained only for cases in which there is distinct
limitation.
The fifth part, consisting of Chapters XXI and XXII, contains
an introduction to the theory of Fuchsian or automorphic functions,
based upon the researches of Poincare and Klein : the discussion is
restricted to the elements of this newly-developed theory.
The arrangement of the subject-matter, as indicated in this
abstract of the contents, has been adopted as being the most
convenient for the continuous exposition of the theory. But the
arrangement does not provide an order best adapted to one who is
reading the subject for the first time. I have therefore ventured
to prefix to the Table of Contents a selection of Chapters that
will probably form a more suitable introduction to the subject for
such a reader ; the remaining Chapters can then be taken in an
order determined by the branch of the subject which he wishes
to follow out.
In the course of the preparation of this book, I have consulted
many treatises and memoirs. References to them, both general
and particular, are freely made : without making precise reserva-
tions as to independent contributions of my own, I wish in this
Vlll PREFACE
place to make a comprehensive acknowledgement of my obligations
to such works. A number of examples occur in the book : most of
them are extracted from memoirs, which do not lie close to the
direct line of development of the general theory but contain
results that provide interesting special illustrations. My inten-
tion has been to give the author's name in every case where a
result has been extracted from a memoir : any omission to do so
is due to inadvertence.
Substantial as has been the aid provided by the treatises and
memoirs to which reference has just been made, the completion of
the book in the correction of the proof-sheets has been rendered
easier to me by the unstinted and untiring help rendered by
two friends. To Mr William Burnside, M.A., formerly Fellow of
Pembroke College, Cambridge, and now Professor of Mathematics
at the Royal Naval College, Greenwich, I am under a deep debt
of gratitude : he has used his great knowledge of the subject in
the most generous manner, making suggestions and criticisms that
have enabled me to correct errors and to improve the book in
many respects. Mr H. M. Taylor, M. A., Fellow of Trinity College,
Cambridge, has read the proofs with great care : the kind assist-
ance that he has given me in this way has proved of substantial
service and usefulness in correcting the sheets. I desire to
recognise most gratefully my sense of the value of the work which
these gentlemen have done.
It is but just on my part to state that the willing and active
co-operation of the Staff of the University Press during the
progress of printing has done much to lighten my labour.
It is, perhaps, too ambitious to hope that, on ground which
is relatively new to English mathematics, there will be freedom
from error or obscurity and that the mode of presentation in this
treatise will command general approbation. In any case, my aim
has been to produce a book that will assist mathematicians in
acquiring a knowledge of the theory of functions : in proportion
as it may prove of real service to them, will be my reward.
A. R. "FORSYTE.
TRINITY COLLEGE, CAMBRIDGE,
25 February, 1893.
PREFACE TO THE SECOND EDITION.
IN issuing the second edition of this treatise, I desire to express
my grateful sense of the reception which has already been
accorded to the book. When it was first published, I could not
but fear that, if from no other reason than the breadth of range
which it covers, it would contain blemishes in the way of inaccuracy
and obscurity. During the preparation of the second edition, I
have had the advantage of suggestions and criticisms sent to me
by friends and correspondents, to whom my thanks are willingly
returned for the help they thus have afforded me ; my hope is that
improvement has been secured in several respects. The principal
changes may be indicated briefly.
Some modifications have been made in the portion that is
devoted to the theory of uniform functions : no substantial
additions have been made to this part of the book, but new
references are given for the sake of readers who may wish to
acquaint themselves with the most recent developments.
The exposition of Schwarz's proof of the existence of various
classes of functions upon a Riemann's surface has been considerably
changed. The new form seems to me to be free from some of the
difficulties to which exception has been taken from time to time :
the general features of the proof have been retained.
Several sections have been inserted in Chapter XVIII, which
are intended to serve as a simple introduction to the theory of
birational transformation of algebraic equations and curves and of
Riemann's surfaces. Moreover, as that part of the book is occupied
with integrals of algebraic functions and with Abelian functions, it
seems not unnatural that a proof of Abel's Theorem should be
given, as well as some illustrations : this has been effected in some
supplementary notes appended to Chapter XVIII. With minor
exceptions, these additions constitute the whole of the new matter
relating to algebraic functions and their integrals.
p. F. 6
X PREFACE TO THE SECOND EDITION
The chief omission from the contents of the former edition is
caused by the transference, to the second volume of my Theory of
Differential Equations, of the sections that discussed the properties
of certain binomial differential equations of the first order. The
space thus placed at my disposal has been assigned to the theory
of birational transformation ; and I have been enabled to keep the
numbering of the paragraphs the same as in the former edition
with only very few exceptions.
The increased size of the book has prevented me, even more
definitely than before, from attempting to discuss some of the
subjects left undiscussed in the first edition. The volume will
probably be regarded as sufficiently large in its present form :
I hope that it may continue to be found a useful introduction to
one of the most important subjects in modern pure mathematics.
A. R. F.
TRINITY COLLEGE, CAMBRIDGE,
31 October, 1900.
PKEFACE TO THE THIED EDITION.
THE differences between the present edition and the second
edition are not substantial.
The general plan of the book is unaltered ; and no change has
been made in the numbering of the paragraphs. Not a few
detailed changes have been made in places as, for instance, in the
establishment of the fundamental functions in the Weierstrass
theory of elliptic functions ; but some chapters remain entirely
unaltered.
The theory of conformal representation is important in particular
ranges of subjects such as hydrodynamics and electrostatics ; so
I have included a note giving some applications of that theory to
some branches of mathematical physics. It is intended only as
an introduction ; but it may suffice to shew that many analytical
results are common to these selected ranges, though they are ex-
pressed in the various vocabularies appropriate to the respective
subjects.
In passing from the first edition to the second, I omitted certain
sections which discussed the properties of certain differential equa-
tions of the first order. These sections are now contained in the
second volume of my Theory of Differential Equations, Owing
to their importance as illustrations of the theory of functions, I
have included a note stating the results.
Here and there, throughout the book, some further examples
have been added. At the end of the book, I have given a set of
some two hundred miscellaneous examples, which have been
collected from Cambridge examination papers. For making the
collection, I am indebted to Mr C. H. Kebby, B.Sc., A.E.C.S., a
demonstrator in the department of mathematics and mechanics in
the Imperial College of Science and Technology, London.
The Staff of the University Press have shewn to me the same
courteous consideration that I have experienced for many years ;
62
Xll
and they have achieved the task of printing the volume within a
brief period in spite of their grave depletion by the demands of
this world-wide war. To all of them, who have been concerned
with the book, I tender my most cordial and appreciative thanks.
A. R. F.
IMPERIAL COLLEGE OF SCIENCE AND TECHNOLOGY,
LONDON, S.W.
11 October, 1917.
CONTENTS.
The following course is recommended, in the order specified, to those who are
reading the subject for the first time : The theory of uniform functions, Chapters
I V; Conformed representation, Chapter XIX; Multiform functions and uniform
periodic functions, Chapters VIII XI ; Riemann's surfaces, and Riemann's theory
of algebraic functions and their integral*, Chapters XIV XVI, XVIII.
CHAPTER I.
GENERAL INTRODUCTION.
PAGE
1 3. The complex variable and the representation of its variation by points
in a plane 1
4. Neumann's representation by points on a sphere . . . . . 4
5. Properties of functions assumed known ....... 6
6, 7. The idea of complex functionality adopted, with the conditions necessary
and sufficient to ensure functional dependence ..... 6
8. Riemann's definition of functionality ....... 8
9. A functional relation between two complex variables establishes the
geometrical property of conformal representation of their planes . 10
10, 11. Relations between the real and the imaginary parts of a function of 2;
with examples 12
12, 13. Definitions and illustrations of the terms monogenic, uniform, multiform,
branch, branch-point, holomorphic, zero, pole, meromorphic . . . 15
CHAPTER II.
INTEGRATION OF UNIFORM FUNCTIONS.
14, 15. Definition of an integral with complex variables; inferences. Definitions
as to convergence of series 20
16. Proof of the lemma I \(^- - ~-\ dxdy = \(pdx + qdy), under assigned
conditions 24
xiv CONTENTS
PAGE
17, 18. The integral \f(z)dz round any simple curve is zero, when f(z) is
holomorpbic within the curve ; and I f(z) dz is a holomorphic
function when the path of integration lies within the curve . . 27
19. The path of integration of a holomorphic function can be deformed
without changing the value of the integral 30
20 22. The integral .r^-^-dz. round a curve enclosing a, is f(a) when
ziri J z a
f(z) is a holomorphic function within the curve; and the integral
| __ZW_ dz is t a ' . Superior limit for the modulus of
27rty (z a)" + 1 ! aa n
the nth derivative of /(a) in terms of the modulus of /(a) . . 31
23. The path of integration of a meromorphic function cannot be deformed
across a pole without changing the value of the integral . . 39
24. The integral of any function (i) round a very small circle, (ii) round a
very large circle, (iii) round a circle which encloses all its infinities
and all its branch-points . . . 40
25. Examples 43
CHAPTER III.
EXPANSION OF FUNCTIONS IN SERIES OF POWERS.
26, 27. Cauchy's expansion of a function in positive powers of z- a; with re-
marks and inferences 50
28 30. Laurent's expansion of a function in positive and negative powers of
z-a; with corollary 54
31. Application of Cauchy's expansion to the derivatives of a function . 59
32. Definition of an ordinary point of a function, of the domain of an
ordinary point, of (a pole) or an accidental singularity, and of an
essential singularity. Behaviour of a uniform function at and near
an essential singularity . . . . . . . 60
33. Weierstrass's theorem on the values of a uniform function in the imme-
diate vicinity of an essential singularity 64
34, 35. Continuation of a function by means of elements over its region of
continuity 66
36. Schwarz's theorem on symmetric continuation across the axis of real
quantities 70
CHAPTER IV.
UNIFORM FUNCTIONS, PARTICULARLY THOSE WITHOUT ESSENTIAL
SINGULARITIES.
37. A function, constant over a continuous series of points, is constant
everywhere in its region of continuity ...... 72
38, 39. The multiplicity of a zero, which is an ordinary point, is finite ; and
a multiple zero of a function is a zero of its first derivative . . 75
CONTENTS XV
PAGE
40. A function, that is not a constant, must have infinite values . . 77
41, 42. Form of a function near an accidental singularity 78
43, 44. Poles of a function are poles of its derivatives 80
45. A funct'ion, which has infinity for its only pole and has no essential
singularity, is a polynomial 83
46. Polynomial and transcendental functions 84
47. A function, all the singularities of which are accidental, is a rational
meromorphic function 85
48. Some properties of polynomials and rational functions .... 87
CHAPTER V.
TRANSCENDENTAL INTEGRAL FUNCTIONS.
49, 50. .Construction of a transcendental integral function with assigned zeros
a \, 2> a s> wnen an integer s can be found such that 2 | a n ~*
is a converging series. Definitions as to convergence of products . 90
51. Weierstrass's construction of a function with any assigned zeros . . 95
52, 53. The most general form of function with assigned zeros and having its
single essential singularity at z = o> 99
54. Functions with the singly-infinite system of zeros given by z=mm, for
integral values of m . ; 101
55 57. Weierstrass's o-function with the doubly-infinite system of zeros given
by z=m> + m'(0', for integral values of m and of m' . . . . 104
58. A uniform function cannot exist with a triply-infinite arithmetical pro-
gression of zeros 108
59, 60. Class (genre) of a function 109
61. Laguerre's criterion of the class of a function; with examples . . Ill
CHAPTER VI.
FUNCTIONS WITH A LIMITED NUMBER OF ESSENTIAL SINGULARITIES.
62. Indefiniteness of value of a function at and near an essential singularity 115
63. A function is of the form G ( 7- ) +P(z b) in the vicinity of an essential
\z b I
singularity at b, a point in the finite part of the plane . . . 117.
64, 65. Expression of a function with n essential singularities as a sum of n
functions, each with only one essential singularity .... 120
66, 67. Product-expression of a function with n essential singularities and no
zeros or accidental singularities . . . . . . . .122
68 71. Product-expression of a function with n essential singularities and with
assigned zeros and assigned accidental singularities ; with a note on
the region of continuity of such a function 126
Xvi CONTENTS
CHAPTER VII.
FUNCTIONS WITH UNLIMITED ESSENTIAL SINGULARITIES,
AND EXPANSION IN SERIES OF FUNCTIONS.
72. Mittag- baffler's theorem on functions with unlimited essential singu-
larities, distributed over the whole plane 134
73. Construction of subsidiary functions, to be terms of an infinite sum . 135
74 76. Weierstrass's proof of Mittag-Leffler's theorem, with the generalisation
of the form of the theorem 136
77, 78. Mittag-Leffler's theorem on functions with unlimited essential singu-
larities, distributed over a finite circle 140
79. Expression of a given function in Mittag-Leffler's form .... 146
80. ' General remarks on infinite series, whether of powers or of functions . 150
81. A series of powers, in a region of continuity, represents one and only
one function; it cannot be continued beyond a natural limit . . 152
82. Also a series of functions : but its region of continuity may consist of
distinct parts '153
83. A series of functions does not necessarily possess a derivative at points
on the boundary of any one of the distinct portions of its region
of continuity ........... 155
84. A series of functions may represent different functions in distinct parts
of its region of continuity ; Tannery's series . . . . .161
85. Construction of a function which represents different assigned functions
in distinct assigned parts of the plane 163
86. Functions with a line of essential singularity . . . . . .164
87. Functions with an area of essential singularity or lacunary spaces ; with
examples 166
88. Arrangement of singularities of functions into classes and species . . 175
CHAPTER VIII.
MULTIFORM FUNCTIONS.
89. Branch-points and branches of functions 178
90. Branches obtained by continuation : path of variation of independent
variable between two points can be deformed without affecting a
branch of a function if it be not made to cross a branch-point . 179
91. If the path be deformed across a branch-point which affects the branch,
then the branch is changed 184
92. The interchange of branches for circuits round a branch-point is cyclical 185
93. Analytical form of a function near a branch-point . ... . .186
94. Branch-points of a function defined by an algebraic equation in their
relation to the branches : definition of algebraic function . . 190
95. Infinities of an algebraic function . . 192
96. Determination of the branch-points of an algebraic function, and of the
cyclical systems of the branches of the function . . .197
CONTENTS XV11
PAGE
97. The analytic character of a function defined by an algebraic equation 203
98. Special case, when the branch-points are simple : their number . . 208
99. A function, with n branches and a limited number of branch-points and
singularities, is a root of an alge"braic equation of degree n . . 210
CHAPTER IX.
PERIODS OF DEFINITE INTEGRALS, AND PERIODIC FUNCTIONS
IN GENERAL.
100. Conditions under which the path of variation of the integral of a
multiform function can be deformed without changing the value
of the integral . . . . . . . . . . .214
101. Integral of a multiform function round a small curve enclosing a
branch-point 217
102. Indefinite integrals of uniform functions with accidental singularities ;
dz
f- i-
I z ' /]
218
1+2 2
103. Hermite's method of obtaining the multiplicity in value of an integral ;
sections in the plane, made to avoid the multiplicity . . . 219
104. Example^ of indefinite integrals of multiform functions ; \wdz round
any loop, the general value of J(l - z*)~^dz, of J{(1 - z 2 )(l - Fz 2 )}~^rfz,
and of I((z-e l }(z-e z )(z-e 3 )}~^dz 224
105. ' Graphical representation of simply-periodic and of doubly-periodic
functions ............ 235
106. The ratio of the periods of a uniform doubly-periodic function is not
real . . . . . . . . . . . .
107, 108. Triply-periodic uniform functions of a single variable do not exist .
109. Construction of a fundamental parallelogram for a uniform doubly-
periodic function .......... 243
110. An integral, with more periods than two, can be made to assume any
value by a modification of the path of integration between the
limits . 246
CHAPTER X.
UNIFORM SIMPLY-PERIODIC AND DOUBLY-PERIODIC FUNCTIONS.
2nei
111. Simply-periodic functions, and the transformation Z=e> . . . 250
112. Fourier's series and simply -periodic functions 252
113, 114. Properties of simply -periodic functions without essential singularities
in the finite part of the plane 253
115. Uniform doubly-periodic functions, without essential singularities in
the finite part of the plane 257
116. Properties of uniform doubly-periodic functions . . . . 258
XV111
117.
CONTKNTS
PAGE
The '/eros and the singularities of the derivative of a doubly-periodic
function of the second order 271
118, 119. Relations betwi-cn bomoperiodio functions 273
Note on differential equations of the first order having uniform integrals 283
CHAPTER XL
DOUBLY-PERIODIC FUNCTIONS OF THE SECOND ORDER.
120, 121. Formation of an uneven function with two distinct irreducible in-
finities ; its addition-theorem ........ 286
122, 123. Properties of Weierstrass's <r-function 291
124. Introduction of f (z) and of #> (z) 295
125, 126. Periodicity of the function %>(z], with a single irreducible infinity of
degree two ; the differential equation satisfied by the function ^ (z) 296
127. Pseudo-periodicity of f (z) 300
128. Construction of a doubly-periodic function in terms of (z) and its
derivatives 301
129. On the relation rj<a' /a>= $iri ' . . . 302
130. Pseudo-periodicity of <r(z} 304
131. Construction of a doubly-periodic function as a product of cr-functions ;
with examples 305
132. On derivatives of periodic functions with regard to the invariants
ff 2 and </ 3 309
133 135. Formation of an even function of either class 312
CHAPTER XII.
PSEUDO-PERIODIC FUNCTIONS.
136. Three kinds of pseudo-periodic functions, with the characteristic equa-
tions 320
137, 138. Hermite's and Mittag- Lender's expression for doubly-periodic functions
of the second kind 322
139. The zeros and the infinities of a secondary function . . . . 327
140, 141. Solution of Lamp's differential equation 328
142. The zeros and the infinities of a tertiary function .... 333
143. Product-expression for a tertiary function 334
144 146. Two classes of tertiary functions ; Appell's expressions for a function
of each class as a sum of elements 335
147. Expansion in trigonometrical series ....... 340
148. Examples of other classes of pseudo-periodic functions . . . 342
CONTENTS XIX
CHAPTER XIII.
FUNCTIONS POSSESSING AN ALGEBRAICAL ADDITION-THEOREM.
PAGE
149. Definition of an algebraical addition -theorem 344
150. A function denned by an algebraical equation, the coefficients of
which are algebraical functions, or simply-periodic functions, or
doubly-periodic functions, has an algebraical addition-theorem . 344
151 154. A function possessing . an algebraical addition-theorem is either
algebraic, simply-periodic or doubly-periodic, having in each in-
stance only a finite number of values for an argument . . . 347
155, 156. A function with an algebraical addition-theorem can be defined by a
differential equation of the first order, into which the independent
variable does not explicitly enter . 356
CHAPTER XIV.
CONNECTIVITY OF SURFACES.
157 159. Definitions of connection, simple connection, multiple connection, cross-
cut, loop-cut ........... 359
160. Relations between cross-cuts and connectivity 362
161. Relations between loop-cuts and connectivity 367
162. Effect of a slit 368
163, 164. Relations between boundaries and connectivity 369
165. Lhuilier's theorem on the division of a connected surface into
curvilinear polygons .......... 372
166. Definitions of circuit, reducible, irreducible, simple, multiple, compound,
reconcileable 374
167, 168. Properties of a complete ystem of irreducible simple circuits on a
surface, in its relation to the connectivity ..... 375
169. Deformation of surfaces 379
170. Conditions of equivalence for representation of the variable . . 380
CHAPTER XV.
RIEMANN'S SURFACES.
171. Character and general description of a Riemann's surface . . . 382
172. Riemann's surface associated with an algebraic equation . . . 384
173. Sheets of the surface are connected along lines, called branch-lines . 384
174. Properties of branch-lines . . . 386
175, 176. Formation of system of branch-lines for a surface ; with examples . 387
177. Spherical form of Riemann's surface ....... 393
XX CONTENTS
PAGE
178. The connectivity of a Riemann's surface 393
179. Irreducible circuits : examples of resolution of Riemann's surfaces
into surfaces that are simply connected 397
180, 181. General resolution of a Riemann's surface 400
182. A Riemann's ?z-sheeted surface when all the branch-points are simple 403
183, 184. On loops, and f their deformation 404
185. Simple cycles of Clebsch and Gordan 407
186 189. Canonical form of Riemann's surface when all the branch-points are
simple, deduced from theorems of Liiroth and Clebsch . . 408
190. Deformation of the .surface 412
191. Remark on rational transformations 415
CHAPTER XVI.
ALGEBRAIC FUNCTIONS AND THEIR INTEGRALS.
192. Two subjects of investigation 416
193, 194. Determination of the most general uniform function of position on a
Riemaun's surface . . . . . . . . . .417
195. Preliminary lemmas in integration on a Riemann's surface . . 422
196, 197. Moduli of periodicity for cross-cuts in the resolved surface . . 423
198. The number of linearly independent moduli of periodicity is equal to
the number of cross-cuts, which are necessary for the resolution
of the surface into one that is simply connected .... 427
199. Periodic functions on a Riemann's surface ; with examples . . . 428
200. Integral of the most general uniform function of position on a
Riemann's surface . 436
201. Integrals, everywhere finite on the surface, connected with the equa-
tion w 2 -S(z) = Q .'.... 438
202 204. Infinities of integrals on the surface connected with the algebraic
equation f(w, 2) = 0, when the equation is geometrically interpret-
able as the equation of a (generalised) curve of the nth order . 438
205, 206. Integrals of the first kind connected with f(w,z')=Q, being functions
that are everywhere finite : the number of such integrals, linearly
independent of one another : they are multiform functions . . 444
207, 208. Integrals of the second kind connected with f(w, 2) = 0, being func-
tions that have only algebraic infinities ; elementary integral of
the second kind 446
209. Integrals of the third kind connected with /(w, 2)=0, being functions
that have logarithmic infinities . . . . . . . 450
210, 211. An integral of the third kind cannot have less than two logarithmic
infinities ; elementary integral of the third kind .... 451
CONTENTS XXI
CHAPTER XVII.
SCHWARZ'S PROOF OF THE EXISTENCE-THEOREM.
PAGE
212, 213. Existence of functions on a Riemann's surface ; initial limitation of
the problem to the real parts u of the functions .... 455
214. Conditions to which u, the potential function, is subject . . . 457
^15. Methods of proof: summary of Schwarz's investigation . . . 458
216 220. The potential-function u is uniquely determined for a circle by the
general conditions and by the assignment of finite boundary values . 460
221. Also for any plane'area, on which the area of a circle can be con-
formally represented 477'
222. Also for any plane area which can be obtained by a topological com-
bination of areas, having a common part and each conformally
representable on the area of a circle ...... 480
223. Also for any area on a Riemann's surface in which a branch-point
occurs ; and for any simply connected surface .... 485
224 227. Real functions exist on a Riemann's surface, everywhere finite, and
having arbitrarily assigned real moduli of periodicity, whether the
surface .has a boundary or not . 487
228. And the number of the linearly independent real functions thus ob-
tained is 2/> 495
229. Real functions exist with assigned infinities on the surface and
assigned real moduli of periodicity. Classes of functions of the
complex variable proved to exist on the Riemann's surface . . 495
CHAPTER XVIII.
APPLICATIONS OF THE EXISTENCE-THEOREM.
230. Three special kinds of functions on a Riemann's surface . . . 498
231 233. Relations between moduli of functions of the first kind and those .
functions of the second kind . ' 500
234. The number of linearly independent functions of the first kind on a
Riemann's surface of connectivity 2jo + l is p . . . . 504
235. Normal functions of the first kind ; properties of their moduli . . 506
236. Normal elementary functions of the second kind : their moduli . . 509
237, 238. Normal elementary functions of the third kind : their moduli : inter-
change of arguments and parametric points . . . . .511
239. The inversion-problem for functions of the first kind . . . . 515
240. Algebraic functions on a Riemann's surface without infinities at the
branch-points but only at isolated ordinary points on the surface :
Riemann-Roch's theorem : the smallest number of singularities
that such functions may possess .519
241. A class of algebraic functions infinite only at branch-points . . 524
242. The Brill-Nother law of reciprocity . . . . . . .526
243. Fundamental equation associated with an assigned Riemann's surface 528
Xxii CONTENTS
PAGE
244. Appell's factorial functions on a Riemann's surface: their multipliers
t the cross-cuts ; expression for a factorial function with assigned
zeros and assigned infinities ; relations between zeros and infinities
of a factorial function ......... 531
245. Birational transformation of equations and Riemann's surfaces . . 537
246. Conservation of genus under birational transformation : moduli . . 542
247. Equations of genus 548
248. Equations of genus 1 554
249. Equations of genus 2 562
250. Equations of genus p (^ 3) . . ^66
251. Normal equivalents of equations for birational transformation . . 567
252. Birational transformation of any algebraic plaoe curve into an algebraic
plane curve having no singularities except simple nodes . . 569
I
SUPPLEMENTARY NOTES: ABEI/S THEOREM.
I. Proof of Abel's Theorem in general; with examples . . . 579
II. Application of Abel's Theorem to the normal elementary integrals
of three kinds on a Riemann's surface ..... 590
III. Proof that the sum of any number of integrals is expressible as
a sum of p integrals together with an additive function . 598
CHAPTER XIX.
CONFORMAL REPRESENTATION: INTRODUCTORY.
253. A relation between complex variables is the most general relation that
secures conformal similarity between two surfaces . . . . 602
254. One of the surfaces for couformal representation may, without loss of
generality, be taken to be a plane 606
255, 256. Application to surfaces of revolution ; in particular, to a sphere, so
I as to obtain maps 607
257. /iorne examples of conformal representation of plane areas, in par-
represented on the area
. 614
25^. Linear fiomographic transformations (or substitutions} of the form
= . : their fundamental properties. . 625
cz + a
259. Parabolic, elliptic, hyperbolic, and loxodromic substitutions . . . 631
260. An elliptic substitution is either periodic or infinitesimal : substitutions
of the other classes are neither periodic nor infinitesimal . . 635
261. A linear substitution can be regarded geometrically as the result of
an even number of successive inversions of a point with regard
to circles 637
I as to ouiaiii maps .....
/orne examples of conformal representation (
ticular, of areas that can be coriformally i
of a circle .......
NOTE: SOME APPLICATIONS OF CONFORMAL REPRESENTATION
TO MATHEMATICAL PHYSICS.
I. Applications to hydrodynamics 639
II. Applications to electrostatics 646
III. Applications to conduction of heat 649
CONTENTS XX111
CHAPTER XX.
CONFORMAL REPRESENTATION : GENERAL THEORY.
PAGE
262. Riemann's theorem on the conformal representation of a given area
upon the area of a circle with unique correspondence . . . 653
263, 264. Proof of Riemann's theorem : how far the functional equation is
algebraically determinate 654
265, 266. The method of Beltrami and Cayley for the construction of the
functional equation for an analytical curve ..... 658
267, 268. Couformal representation of a convex rectilinear polygon upon the
half-plane of the variable . . . . . . . 665
269. The triangle, and the quadrilateral, conformally represented . . 671
270. A convex curve, as a limiting case of a polygon 678
271, 272. Conformal representation of a convex figure, bounded by circular arcs:
the functional relation is connected with a linear differential
equation of the second order ........ 679
273. Conformal representation of a crescent ....... 684
274 276. Conformal representation of a triangle, bounded by circular arcs . 685
277 279. Relation between the triangle, bounded by circles, and the stereographic
projection of regular solids inscribed in a sphere .... 694
280. On families of plane algebraic curves, determined as potential-curves
by a single potential-parameter u : the forms of functional relation
z = <f)(u + iv) J which give rise to such curves 706
Supplementary note ; surfaces of constant negative curvature, and their
representation on a plane, in connection with 275 . . . 712
CHAPTER XXI.
GROUPS OF LINEAR SUBSTITUTIONS.
281. The algebra of group-symbols . . 715
282. Groups, which are considered, are discontinuous and have a finite
number of fundamental substitutions . . . % . . .717
283, 284. Anharmonic group : group for the modular-functions, and division of
the plane of the variable to represent the group . . . .719
285, 286. Fuehsian groups : division of plane into convex curvilinear polygons :
polygon of reference . . . . . . . . . .724
287. Cycles of angular points in a curvilinear polygon .... 729
288, 289. Character of the division of the plane: example 732
290. Fuehsian groups which conserve a fundamental circle .... 736
291. Essential singularities of a group, and of the automorphic functions
determined by the group 739
292, 293. Families of groups: and their genus 740
294. Kleinian groups : the generalised equations connecting two points in
space . . . ' . . . 743
295. Division of plane and division of space, in connection with Kleinian
groups 747
296. Example of improperly discontinuous group ...... 749
XXIV CONTENTS
CHAPTER XXII.
AUTOMORPHIC FUNCTIONS.
PAGE
297. Definition of automorpkic functions 753
298. Examples of functions, automorphic for finite discrete groups of sub-
stitutions ............. 754
299. Cayley's analytical relation between stereographic projections of posi-
tions of a point on a rotated sphere 754
300. Polyhedral groups; in particular, the dihedral group, and the tetra-
hedral group 757
301, 302. The tetrahedral functions, and the dihedral functions .... 762
303. Special illustrations of infinite discrete groups, from the elliptic
modular-functions ........... 767
304. Division of the plane, and properties of the fundamental polygon of
reference, for any infinite discrete' group that conserves a funda-
mental circle ........... 771
305, 306. Construction of Thetafuchsian functions, pseudo-automorphic for an
infinite group of substitutions ....... 775
307. Relations between the number of irreducible zeros and the number
of irreducible poles of a pseudo-automorphic function, constructed
with a rational meromorphic function as element .... 779
308. Construction of automorphic functions 784
309. The number of irreducible points, for which an automorphic function
acquires an assigned value, is independent of the value . . 786
310. Algebraic relations between functions, automorphic for a group :
application of Riemann's theory of functions .... 788
311. Connection between automorphic functions and linear differential
equations; with illustrations from elliptic modular-functions . 789
MISCELLANEOUS EXAMPLES
GLOSSARY OF TECHNICAL TERMS
INDEX
794
829
833
CHAPTER I.
GENERAL INTRODUCTION.
1. ALGEBRAICAL operations are either direct or inverse. Without
entering into a general discussion of the nature of rational, irrational, and
imaginary quantities, it will be sufficient to point out that direct algebraical
operations on numbers that are positive and integral lead to numbers of the
same character ; and that inverse algebraical operations on numbers that are
positive and integral lead to numbers, which may be negative or fractional
or irrational, or to numbers which may not even fall within the class of real
quantities. The simplest case of occurrence of a quantity, which is not
real, is that which arises when the square root of a negative quantity is
required.
Combinations of the various kinds of quantities that may occur are of
the form x + iy, where x and y are real, and i, the non-real element of the
quantity, denotes the square root of 1. It is found that, when quantities
of this character are subjected to algebraical operations, they always lead
to quantities of the same formal character ; and it is therefore inferred that
the most general form of algebraical quantity is x + iy.
Such a quantity x + iy, for brevity denoted by z, is usually called a
complex variable* ; it therefore appears that the complex variable is the
most general form of algebraical quantity which obeys the fundamental laws
of ordinary algebra.
2. The most general complex variable is that, in which the constituents
x and y are independent of one another and (being real quantities) are
separately capable of assuming all values from oo to + oo ; thus a doubly-
infinite variation is possible for the variable. In the case of a real variable,
it is convenient to use the customary geometrical representation by measure-
ment of distance along a straight line; so also in the case of a complex
variable, it is convenient to associate a geometrical representation with
the algebraical expression; and this is the well-known representation of
* The conjugate complex, viz. x - iy, is frequently denoted by z .
F. F. 1
2 GEOMETRICAL REPRESENTATION OF [2.
the variable x + iy by means of a point with coordinates x and y referred
to rectangular axes*. The complete variation of the complex variable z
is represented by the aggregate of all possible positions of the associated
point, which is often called the point z; the special case of real variables
being evidently included in it because, when y = 0, the aggregate of
possible points is the line which is the range of geometrical variation of the
i-eal variable.
The variation of z is said to be continuous when the variations of x and //
are continuous. Continuous variation of z between two given values will
thus be represented by continuous variation in the position of the point z,
that is, by a continuous curve (not necessarily of continuous curvature)
between the points corresponding to the two values. But since an infinite
number of curves can be drawn between two points in a plane, continuity of
line is not sufficient to specify the variation of the complex variable; and,
in order to indicate any special mode of variation, it is necessary to assign,
either explicitly or implicitly, some determinate law connecting the variations
of x and y or, what is the same thing, some determinate law connecting
x and y. The analytical expression of this law is the equation of the curve
which represents the aggregate of values assumed by the variable between
the two given values.
In such a case the variable is often said to describe the part of the curve
between the two points. In particular, if the variable resume its initial
value, the representative point must return to its initial position ; and then
the variable is said to describe the whole curve f.
When a given closed curve is continuously described by the variable,
there are two directions in which the description can take place. From
the analogy of the description of a straight line by a point representing a
real variable, one of these directions is considered as positive and the other
as negative. The usual convention under which one of the directions is
selected as the positive direction depends upon the conception that the curve
* This method of geometrical representation of imaginary quantities, ordinarily assigned to
Gauss, was originally developed by Argand who, in 1806, published his Essai sur une maniere
de representer les quantites imaginaires dan* leu constructions geometriques. This tract was
republished in 1874 as a second edition (Gautiiier- Villars) ; an interesting preface is added
to it by Hoiiel, who gives an account of the earlier history of the publications associated with
the theory.
Other references to the historical development are given in Ghrystal's Text-book of Algebra,
vol. i, pp. 248, 249; in Holzmuller's Eitifuhrung in die Theorie d'r isogonalen Verwandschaftm
und der conformen Abbildungen, verbundrn mit Anwendun/ien auf mathematische Physik, pp. 1 10,
31 23; iu Schlomilch's Compendium der hoheren Analysis, vol. ii, p 38 (note) ; and in Casorati,
Teorica delle funzioni di variabili complete?, only one volume of which was published. Iu this
connection, an article by Cavley (Quart. Journ. of Math., vol. xxii, pp. 270 308; Coll. Math.
Papers, t. xii, pp. 459 489) may be consulted with advantage.
f In these elementary explanations, it is unnecessary to enter into any discussion of
the effects caused by the occurrence of singularities in the curve.
THE COMPLEX VARIABLE
8
is the boundary, partial or complete, of some area ; under it, that direction is
taken to be positive which is such that the bounded area lies to the left of
the direction of description. It is easy to see that the same direction is taken
to be positive under an equivalent convention
which makes it related to the normal drawn
outwards from the bounded area in the same
way as the positive direction of the axis of y
is usually related to the positive direction of
the axis of x in plane coordinate geometry.
Thus in the figure (fig. 1), the positive
direction of description of the outer curve
for the area included by it is DEF.\ the
positive direction of description of the inner
curve for the area without it (say, the area
excluded by it) is ACB; and for the area
between the curves the positive direction of description of the boundary,
which consists of two parts, is DEF, ACB,
Fig. l.
3. Since the position of a point in a plane can be determined by means
of polar coordinates, it is convenient in the discussion of complex variables
to introduce two quantities corresponding to polar coordinates.
In the case of the variable z, one of these quantities is (x- + y^f, the
positive sign being always associated with it; it is called the modulus*
(sometimes the absolute value) of the variable and it is denoted, sometimes
by mod. z, sometimes by \z\. The modulus of a complex variable is quite
definite, and it has only one value.
The other is 0, the angular coordinate of the point z; it is called the
argument (and, less frequently, the amplitude) of the variable. It is
measured in the trigonometrically positive sense, and is determined by
the equations
x\z\ cos 6, y\z\ sin 0,
so that z\z\ e 6i . The actual value depends upon the way in which the
variable has acquired its value ; when variation
of the argument is considered, its initial value
is usually taken to lie between and 2-Tr or, less
frequently, between TT and + TT. The argu-
ment of a variable is not definite ; it has an
unlimited number of values differing from one
another by integer multiples of 2?r. This
characteristic property will be found to be of
essential importance.
Fig. 2.
* Der absolute Betrag is often used by German writers.
12
GREAT VALUES OF
[3.
As z varies in position, the values of \z\ and vary. When z has com-
pleted a positive description of a closed curve, the modulus of z returns to
the initial value whether the origin be without, within, or on, the curve.
The argument of z resumes its initial value, if the origin 0' (fig. 2) be with-
out the curve ; but, if the origin be within the curve, the value of the
argument is increased by 2?r when z returns to its initial position.
If the origin be on the curve, the argument of z undergoes an abrupt
change by TT as z passes through the origin ; and the change is an increase
or a decrease according as the variable approaches its limiting position on the
curve from without or from within. No choice need be made between these
alternatives ; for care is always exercised to choose curves which do not
introduce this element of doubt.
Later on, it will appear that, for the discussion of particular types of
functions of z, a knowledge of the actual value of z or the actual position
of z is not sufficient ; account has to be taken of the fact that the argument
of z is not uniquely determinate.
4. Representation on a plane is obviously more effective for points at a
finite distance from the origin than for points at a very great distance.
One method of meeting the difficulty of representing great values is to
introduce a new variable z' given by z'z=\: the part of the new plane for
z which lies quite near the origin corresponds to the part of the old plane
for z which is very distant. The two planes combined give a complete
representation of variation of the complex variable.
Another method, in many ways more advantageous, is as follows. Draw
a sphere of unit diameter, touching the 2-plane at the origin (fig. 3) on
the under side : join a point z in the plane to 0', the other extremity of
Fig- 3.
the diameter through 0, by a straight line cutting the sphere in Z.
Then Z is a unique representative of z, that is, a single point on the
sphere corresponds to a single point on the plane : and therefore the variable
4.] THE COMPLEX VARIABLE 5
can be represented on the surface of the sphere. With this mode of
representation, 0' evidently corresponds to an infinite value of z ; and points
at a very great distance in the z- plane are represented by points in the
immediate vicinity of 0' on the sphere. The sphere thus has the advantage
of putting in evidence a part of the surface on which the variations of
givat values of z can be traced*, and of exhibiting the uniqueness of
z = x) as a value of the variable, a fact that is obscured in the represent-
ation on a plane.
The former method of representation can be deduced by means of the
sphere. At 0' draw a plane touching the sphere : and let the straight line
OZ cut this plane in /. Then z is a point uniquely determined by Z
and therefore uniquely determined by z. In this new /-plane take axes
parallel to the axes in the z-plane.
The points z and / move in the same direction in space round 00'
as an axis. If we make the upper side of the z-plane correspond to the
lower side of the /-plane, and take the usual positive directions in the
planes, being the positive trigonometrical directions for a spectator looking
at the surface of the plane in which the description takes place, we have
these directions indicated by the arrows at and at 0' respectively, so
that the senses of positive rotations in the two planes are opposite in
space. Now it is evident from the geometry that Oz and O'z' are
parallel : hence, if 6 be the argument of the point z and 0' that of the
point /, so that 6 is the angle from Ox to Oz and & the angle from O'ac
to O'z , we have
<9 + 0'=2-7r.
Oz 00'
1 urther, by similar triangles, -^-, ^y-, ,
that is, 0*.OY=00' 2 =>1.
Now, if z and z be the variables, we have
z = Oz . e* z = O'z' . e ffi ,
so that zz'=0z.0'z' .eW+'K
= 1,
which is the former relation.
The /-plane can therefore be taken as the lower side of a plane touching
the sphere at 0' when the ^-plane is the upper side of a plane touching
it at 0. The part of the ^-plane at a very great distance is represented on
the sphere by the part in the immediate vicinity of (7. Conversely, this
part of the sphere is represented on the very distant part of the ^-plane.
Consequently, the portion of the sphere in the immediate vicinity of 0' is a
space wherein the variations of infinitely great values of z can be traced.
* This sphere is sometimes called Neumann's sphere; it is used by him for the representation
of the complex variable throughout his treatise Vorlcsungen uber Riemanris Theorie der Abcl'schen
Integrate (Leipzig, Teubner, 2nd edition, 1884).
6 CONDITIONS OF [4.
But it need hardly be pointed out that any special method of represent-
ation of the variable is not essential to the development of the theory of
functions; and, in particular, the foregoing representation of the variable,
when it has very great values, merely provides a convenient method of
dealing with quantities that tend to become infinite in magnitude.
5. The simplest propositions relating to complex variables will be
assumed known. Among these are, the geometrical interpretation of opera-
tions such as addition, multiplication, root-extraction ; some of the relations
of complex variables occurring as roots of algebraical equations with real
coefficients ; the elementary properties of functions of complex variables
which are polynomial, or exponential, or circular, functions ; and simple
tests of convergence of infinite series and of infinite products*.
6. All ordinary operations effected on a complex variable lead, as
already remarked, to other complex variables; and any definite quantity,
thus obtained by operations on z, is necessarily a function of z.
But if a complex variable w be given as a complex function of ./
and y without any indication of its source, the question as to whether
w is or is not a function of z requires a consideration of the general idea
of functionality.
It is convenient to postulate u + iv as a form of the complex variable w,
where u and v are real. Since w is initially unrestricted in variation, we
may so far regard the quantities u and v as independent and therefore as
any functions of a; and ?/, the elements involved in z.. But more explicit
expressions for these functions are neither assigned nor supposed.
The earliest occurrence of the idea of functionality is in connection with
functions of real variables: and then it is coextensive with the idea of
dependence. Thus, if the value of X depends on that of x and on no other
variable magnitude, it is customary to regard X as a function of x ; and
there is usually an implication that X is derived from x by some series of
operations f.
A detailed knowledge of z determines x and y uniquely ; hence the values
of u and v may be considered as known and therefore also w. Thus the
value of w is dependent on that of z, and is independent of the values
* These and other introductory parts of the subject are discussed in Chrystal's Text-book of
Algebra, Hobson's Treatise on Plane Trigonometry, Bromwich's Theory of infinite series, and
Hardy's Course of pure mathematics.
They are also discussed at some length in the translation, by G. L. Cathcart, of Harnack's
Klements of the differential and integral calculus (Williams and Norgate, 1891), the second and
the fourth books of which contain developments that should be consulted in special relation
with the first few chapters of the present treatise.
These books, together with Neumann's treatise cited in the note on p. 5, will hereafter be cited
by the names of their respective authors.
t It is not important for the present purpose to keep in view such mathematical expressions
as have intelligible meanings only when the independent variable is confined within limits. .
6.] FUNCTIONAL DEPENDENCE 7
of variables unconnected with z ; therefore, with the foregoing view of
functionality, w is a function of z.
It is, however, equally consistent with that view to regard w as a complex
function of the two independent elements from which z is constituted ; and
we are then led merely to the consideration of functions of two real
independent variables with (possibly) imaginary coefficients.
Both of these aspects of the dependence of w on z require that z be
regarded as a composite quantity involving two independent elements which
can be considered separately. Our purpose, however, is to regard z as the
most general form of algebraical variable and therefore as an irresoluble
entity ; so that, as this preliminary requirement in regard to z is unsatisfied,
neither of the aspects can be adopted.
7. Suppose that w is regarded as a function of z in the sense that it
can be constructed by definite operations on z regarded as an irresoluble
magnitude, the quantities u and v arising subsequently to these operations
by the separation of the real and the imaginary parts when z is replaced by
x + iy. It is thereby assumed that one series of operations is sufficient for
the simultaneous construction of u and v, instead of one series for u and
another series for v as in the general case of a complex function in 6.
If this assumption be justified by the same forms resulting from the two
different methods of construction, it follows that the two series of opera-
tions, which lead in the general case to u and to v, must be equivalent to
the single series and must therefore be connected by conditions ; that is,
u and v as functions of x and y must have their functional forms related.
We thus take
u + iv = w =f(z) =f(x + iy)
without any specification of the form of/. When this postulated equation
is valid, we have
dw _ dw dz _ v , dw
9# dz dx J dz '
dw _ dw dz _/// _ dw
dy-~dz dy-y (2) ~ l dz'
dw 1 dw dw
and therefore ^- = - ^- = -7- (1),
ox i dy dz ,
equations from which the functional form has disappeared. Inserting the
value of w, we have
. 9 , . , 3 . . .
I - - (U + IV) = =- (U + IV),
dx dy
whence, after equating real and imaginary parts,
dv _ du du _dv
dx dy ' dx dy"
These are necessary relations between the functional forms of u and v.
8 RIEMANN'S [7.
These relations are easily seen to be sufficient to ensure the required
functionality. For, on taking w = u + iv, the equations (2) at once lead to
dw _ 1 dw
dx i "by '
dw .dw
that is, to o- + * ^- = 0>
ax dy
a linear partial differential equation of the first order. To obtain the most
general solution, we form a subsidiary system
dx _\.dy _ dw
T = TT == 1T'
It possesses the integrals w, x + iy ; then from the known theory of such
equations we infer that every quantity w satisfying the equation can be
expressed as a function of x + iy, that is, of z. The conditions (2) are thus
proved to be sufficient, as well as necessary.
8. The preceding determination of the necessary and sufficient conditions
of functional dependence is based upon the existence of a functional form ;
and yet that form is not essential, for, as already remarked, it disappears
from the equations of condition. Now the postulation of such a form is
equivalent to an assumption that the function can be numerically calculated
for each particular value of the independent variable, though the immediate
expression of the assumption has disappeared in the present case. Experience
of functions of real variables shews that it is often more convenient to use
their properties than to possess their numerical values. This experience is
confirmed by what has preceded. The essential conditions of functional
dependence are the equations (1), and they express a property of the function w,
viz., that the value of the ratio , is the same as that of -* , or, in other
dz ox
words, it is independent of the manner in which dz ultimately vanishes In
the approach of the point z + dz to coincidence with the point z. We an-
thus led to an entirely different definition of functionality, viz. :
A complex quantity w is a function of another complex quantity z, when
they change together in such a manner that the value of -y- is independent of
the value "of the differential element dz.
This is Riemann's definition* ; we proceed to consider its significance.
We have
dw _du + idv
dz dx + idy
fdu .dv\ dx /du .dv\ dy
I !_ n . j I I I J /
\dx dxj dx + idy \dy dy, /<!.'+ id'/'
* Ges. ll'erke, p. 5 ; a modified definition is adopted by him, ib., p. 81.
8.] DEFINITION OF A FUNCTION
Let < be the argument of dz ; then
dsc cos <f>
dx+idy cos <j> + i sin </>
idy
dx + idy
and therefore
dw _ - J9w . 9v . 9w 9w| ^ _ 2 , f (du .dv .du dv\
dz " \dx dx dy dy} [das dx dy dy]
Since -?- is to be independent of the value of the differential element dz,
it must be independent of < which is the argument of dz ; hence the coefficient
of e~ 2<i>i in the preceding expression must vanish, which can happen only if
du _dv dv _ du (e> .
dx dy ' dx dy ' '
These are necessary conditions; they are evidently also sufficient to make
~y independent of the value of dz and therefore, by the definition, to secure
that w is a function of zjf'
By means of the conditions (2), we have
dw _ du .dv_dw
dz dx dx dx '
dw .du dv 1 dw
and also - = t + = _ _
dz dy dy ^ dy
agreeing with the former equations (1). They are immediately derivable from
the present definition by noticing that da;. and idy are possible forms of dz.
It should be remarked that equations (2) are the conditions necessary
arid sufficient to ensure that each of the expressions
uda- vdy and vdx + udy
is a perfect differential a result of great importance in many investigations
in the region of mathematical physics. Within that region, the quantities
u and v are frequently called conjugate functions. Sometimes they are
called harmonic functions ; but the latter term usually has a wider signi-
ficance associated with classes of functions that satisfy the equation of the
potential in ordinary three-dimensional space.
When the conditions (2) are expressed, as is sometimes convenient, in
terms of derivatives with regard to the modulus of z, say r, and the
argument of z, say 0, they take the new forms
du _ 1 dv dv _ I du ,,,
on = Z 5/a > 0,1 = 7, o2i \ ) '
10 CONFORM. \L [8.
We have so far assumed that the function has a differential coefficient
an assumption justified in the case of functions which ordinarily occur. But
functions do occur which have different values in different regions of the
2-plane, and there is then a difficulty in regard to the quantity --r- at the
boundaries of such regions ; and functions do occur which, though themselves
definite in value in a given region, do not possess a differential coefficient at
all points in that region. The consideration of such functions is not of
substantial importance at present : it belongs to another part of our subject.
du/
It must not be inferred that, lx cause . is independent of the direction
in which dz vanishes when w is a function of z. therefore - has only one
dz
value. The number of its values is dependent on the number of values of w ;
no one of its values is dependent on dz.
A quantity, defined as a function by Riemann on the basis of this
property, is sometimes* called an analytic function ; but it seems pre-
ferable to reserve the term analytic in order that it may be associated
hereafter ( 34) with an additional quality of the functions.
9. In the same way as' the complex variable z is represented upon
a plane, which is often called the 2-plane, so the complex variable w is
also represented upon a plane, which is often called the w-plane. The
two variables can obviously be represented upon different parts of the
z- plane. The relations of the two planes to one another, or of the
different parts of the same plane, when there is a functional connection
between z and w, will be the subject of later investigations ; one important
property will, however, be established at once.
Let P and p be two points in different planes, or in different parts of
the same plane, representing w and z respectively ; and suppose that P and
p are at a finite distance from the points (if any) which cause discontinuity
in the functional connection between the two variables. Let q and r be
any two other points, z -f dz and z + Bz, in the immediate vicinity of p ;
and let Q and R be the corresponding points, w + dw and w + ow, in the
immediate vicinity of P. Then
, dw , 5. dw ~
dw = -. dz, ow = -r- 02,
dz dz -
the value of -5- being the same for both equations, because, as lu is a function
of z, that quantity is independent of the differential element of z. Hence
ow _ Bz
dw dz'
on the ground that -=- is neither zero nor infinite at z, which is assumed not
* Harnack, 84.
9.] REPRESENTATION OF PLANES 11
to be a point of discontinuity in the functional connection. Expressing all
the differential elements in terms of their moduli and arguments, let
dz = a-e 81 , dw = rje^,
Bz = a'e ffi , Bw = T/'e* *,
and let these values be substituted in the foregoing relation"; then
i] _ <r'
rj <r
ii i /}/ /j
Q> u> ^ u ".
Hence the triangles QPR and qpr are similar to one another, though
not necessarily similarly situated. Moreover, the directions originally chosen
for pq and pr are quite arbitrary. Thus it appears that a functional connection
between two complex variables establishes the similarity of the corresponding
infinitesimal elements of those parts of two planes which are in the immediate
vicinity of the points representing the two variables.
The magnification of the w-plane relative to the ^r-plane at the corre-
sponding points P and p is the ratio of two corresponding infinitesimal
lengths, say of QP and qp. This is the modulus of -j- ', if it be denoted by
m, we have
_ dw 2 /3u\ 2 /9i;\ 2 _ /du\ 2 idv\"
m~ = j = I ^ I T I ^~ 1 I ^~ I + { zr~ I
dz \dx/ \dxj \dyJ \oy/
du dv du dv
dx dy dy dx '
Evidently the quantity m, in general, depends on the variables and
therefore it changes from one point to another ; hence a functional relation
between w and z does not, in general, establish similarity of finite parts of
the two planes corresponding to one another through the relation.
It is easy to prove that w = az + b, where a and b are constants, is the
only relation which establishes similarity of finite parts ; and that, with this
relation, a must be a real constant in order that the similar parts may be
similarly situated.
If u + iv = w = < (z), the curves u = constant and v = constant cut at
right angles ; a special case of the proposition that, if <f> (x + iy) = u + ve^ 1 ,
where X is a real constant and u, v are real, then u = constant and v = constant
cut at an angle X.
The process, which establishes the infinitesimal similarity of two planes
by means of a functional relation between the variables of the planes, may be
called the conformal representation of one plane on another*.
* By Gauss (Ges. Werke, t. iv, p. 262) it was styled conforme Albildung, the name
universally adopted by German mathematicians. The French title is representation conforme ;
and, in England, Cayley has used orthomorphosis and orthomorphic transformation.
12
CONDITIONS OF FUNCTIONAL DEPENDENCE
[9.
The discussion of detailed question connected with the conformal representation is
deferred until the later part of the treatise, principally in order to group all such
investigations together; but the first of the two chapters, devoted to it, need not be
deferred so late, and an immediate reading of some portion of Chapter XIX. will tend
to simplify many of the explanations relative to functional relations as they occur in
the early chapters of this treatise.
10. The analytical conditions of functionality, under either of the
adopted definitions, are the equations (2). From them it at once follows that
&u dhi =
dac z dy 2
so that neither the real nor the imaginary part of a complex function can be
arbitrarily assumed.
If either part be given, the other can be deduced. For example, let u be
given ; then we have
dv , dv ,
dv 5- dx + =- dy
ox dy
du
du
and therefore, except as to an additive constant, the value of v is
du , du
In particular, when u is an integral function, it can be resolved into the
sum of homogeneous parts
and then, again except as to an additive constant, v can similarly be
expressed as a sum of homogeneous parts
It is easy to prove that
du m du m
mv m = y - -- x -^ ,
J das dy
by means of which the value of v can be obtained.
The cast-, when u is homogeneous of zero dimensions, presents no
difficulty: tor then we have
u = I + a6,
v = c a log r,
where a, />. c are constants.
Similarly for other special cases ; and, in the most general case, onh
a quadrature is necessary.
10.]
EXAMPLE OF RIEMANN's DEFINITION
13
The tests of functional dependence of one complex variable on another are
of effective importance in the case when the supposed dependent variable
arises in the form u + iv, where u and v are real ; the tests are, of course,
superfluous when iv is explicitly given as a function of z. When w does
arise in the form u + iv and satisfies the conditions of functionality, perhaps
the simplest method (other than by inspection) of obtaining the explicit
expression in terms of z is to substitute z iy for x in u + iv; the simplified
result must be a function of z alone.
11. Conversely, when w is explicitly given as a function of z and it
is divided into its real and its imaginary parts, these parts individually
satisfy the foregoing conditions attaching to u and v. Thus log r, where r
is the distance of a point z from a point a, is the real part of log (z a) ;
it therefore satisfies the equation
Again, <, the angular coordinate of z relative to the same point a, is, ( !7TT
the real part of t log (z a) and satisfies the same equation: the morel:/
usual form of </> being tan" 1 {(y y^)l(x - UC Q }}, where a = x + iy . Again, iff
a point z be distant r from a and r' from b, then log(r/r'), being the real
part of log {(z - a)f(z- &)}, is a solution of the same equation.
The following example, the result of which will be useful subsequently*, uses the
property that the value of the derivative is independent of the differential element.
Consider a function
z -c
,
where c' is the inverse of c with regard to a circle, centre the origin and radius R.
Then
u=log
so the curves, u = constant, are circles. Let
(fig. 4)
then if
fit
= a, so that c=re a% , cf = -e
Fig. 4.
the values of X for points in the interior of the circle of radius R vary from zero, when
the circle u = constant is the point c, to unity, when the circle u = constant is the circle of
radius R. Let the point K( = 6e al ) be the centre of the circle determined by a value of X,
and let its radius be p ( = %MN}. Then since
cM _ r _ cN
~
we have
r+p 6 _ r 6 + p -r
* In 217, in connection with the investigations of Schwarz, by whom the result is stated,
Get. Werke, t. ii, p. 183.
14
whence
KXAMl'LES
-r)
[11.
Now if dn t>e an element of the normal drawn inwards at z to the circle NzJH, we have
dz = dx + idy = -dn. cos ^ idn . sin ^
where ^( = -
wt- have
But
so that
and
and therefore
M the argument of c relative to the centre of the circle. Hence, since
*?__L _L_
dz z c z - c"
du 'd^_dw_( 1 1 \ ^,
dn dn dn \z c' z c)
= A(/?-r") w _ ^
du ^.dv
, +i -j-
dn dn
1/1 -/2e a
Hence, equating the real parts, it follows that
du fP
_____
dn~ \R (W ~-'r*) {ft? - 2Rr\ cos (^ - a) + X 2 r 2 } '
the differential element dn being drawn inwards from the circumference of the circle.
The application of this method is evidently effective when the curves u= constant,
arising from a functional expression of w in terms of 2, are a family of non -intersecting
algebraical curves.
Ex. 1. Prove that, if 2 t and z z denote two complex variables,
i *1 ! + I *2 I , ! 1 ~2 I ^ I z l I ~ i 2 i
Ex. 2. Find the values of u and v when w is defined as a function of z in the following
cases:
(i) z=(w+i?- t
(ii) 3 = (1 + cos w) e ;
In each case, trace the curves u = a, v = c, regarded as loci in the plane of x, y.
Ex. 3. Shew that x" -y 2 lixy is not a function of z ; and that
is a function of z only when =0.
Ex. 4. Shew that a possible value of u is
and determine the associated value of w in terms of z.
Determine also the value of w in terms of z when the preceding expression is the value
of u v.
Ex. 5. Find the value of v, and of w in terms of 0, when
U= r .
cosh y cos x
11.] DEFINITIONS 15
Ex. 6. Prove that, when .< and y are regarded as functions of u and v (with the
foregoing notation), the relations
()x _ ay dx vy
du dv" 1 9 9'
are satisfied.
Ex. 7. Shew that, if A and Z? are any two fixed points in a plane, if P is any variable
point (x, y), and if 6 denotes the angle APB, then
<Jonstruct the function of s, =.v + iy, of which # is the real part, and also the function
of 3 of which id is the imaginary part.
Ex. 8. Given X, a function of x and y ; shew that </> (X) can be the real part of
a function of z if the quantity
is expressible in terms of X alone.
Verify that the condition is satisfied when \=x+(.v~+y 2 )^; and obtain the function
of z which has (X) for its real part.
12. As the tests which are sufficient and necessary to ensure that a
complex quantity is a function of z have been given, we shall assume that
all complex quantities dealt with are functions of the complex variable
( 6, 7). Their characteristic properties, their classification, and some of
the simpler applications will be considered in the succeeding chapters.
Some initial definitions and explanations will now be given.
(i). It has been assumed that the function considered has a differential
coefficient, that is, that the rate of variation of the function in any direction
is independent of that direction by being independent of the mode of change
of the variable. We have already decided ( 8) not to use the term analytic
for such a function. It is often called monogenic, when it is necessary to
assign a specific name ; but for the most part we shall omit the name, the
property being tacitly assumed*.
We can at once prove from the definition that, when the derivative
/ dw\ . . dw 1 dw .
w l \ = -7- ) exists, it is itself a function. For w 1 = -^- =--^- are equations
\ dzj dx % ciy
* This is in fact done by Eiemanu, who calls such a dependent complex simply a function.
Weierstrass, however, has proved (see 85, post) that the idea of a monogenic function of a complex
variable and the idea of dependence expressible by arithmetical operations are not coextensive.
The definition is thus necessary ; but the practice indicated in the text will be adopted, as non-
monogenic functions will be of relatively rare occurrence.
)EFINITIO1
[12.
which, when satisfied, ensure the existence of w t ; hence
I r/r, _ 1 f) fdw\
'/ i dy \dtcj
d 1 dw\
dx \i dy)
_dw }
~lte'
shewing, os in 8, that the derivative -~ is independent of the direction in
which dz vanishes. Hence w l is a function of z.
Similarly for all the derivatives in succession.
(ii). Since the functional dependence of a complex is ensured only if the
value of the derivative of that complex be independent of the manner in
which the point z-Vdz approaches to coincidence with z, a question naturally
suggests itself as to the effect on the character of the function that may be
caused by the manner in which the variable itself has come to the value of z.
If a function has only one value for each given value of the variable,
whatever be the manner in which the variable has come to that value, the
function is called uniform*. Hence two different paths from a point a to a
point z give at z the same value for any uniform function ; and a closed
curve, beginning at any point and completely described by the ^-variable
will lead to the initial value of w, the corresponding w-curve being closed, if z
has not passed through any point which makes w infinite.
The simplest class of uniform functions is constituted by rational
functions.
(iii). If a function has more than one value for any given value of the
variable, or if its value can be changed by modifying the path in which
the variable reaches that given value, the function is called multiform^.
Characteristics of curves, which are graphs of multiform functions corre-
sponding to a -curve, will hereafter be discussed.
One of the simplest classes of multiform functions is constituted by
algebraical irrational functions, that is, functions defined by an irresolublc
algebraic equation f(w, z) = Q, where f is a polynomial in w and z.
The rational functions in (ii) occur when f is of only the first degree in w.
(iv). A multiform function has a number of different values for the same
value of z, and these values vary with z': the aggregate of the variations of
any one of the values is called a branch of the function. Although the
function is multiform for unrestricted variation of the variable, it often
happens that a branch is uniform when the variable is restricted to
particular regions in the plane.
* Also monodromic, or monotropic ; with German writers the title is eindeutig, occasionally,
einSndrig.
t Also polytropic ; with German writers the title is mehrdeutig.
12.] DEFINITIONS 17
(v). A point in the plane, at which two or more branches of a multiform
function assume the same value, and near which those branches are inter-
changed ( 94, Note) by appropriate modification in the path of z, is called a
branch-point* of the function. The relations of the branches in the immediate
vicinity of a branch-point will be discussed hereafter.
(vi). A function, which is monogenic, uniform and continuous over any
part of the ^-plane, is called holomorphic^ over that part of the plane. When
a function is called holomorphic without any limitation, the usual implication
is that the character is preserved over the whole of the plane which is not at
infinity.
The simplest example of a holomorphic function is a polynomial in the
variable.
(vii). A root (or a zero) of a function is a value of the variable for which
the function vanishes.
The simplest case of occurrence of roots is in a rational integral
function, various theorems relating to which (e.g., the number of roots
included within a given contour) will be found in treatises on the theory
of equations.
(viii). The infinities of a function are the points at which the value of
the function is infinite. Among them, the simplest are the poles^ of the
function, a pole being an infinity such that in its immediate vicinity the
reciprocal of the function is holomorphic.
Infinities other than poles (and also the poles) are called the singular
points, or the singularities, of the function: their classification must be
deferred until after the discussion of properties of functions.
(ix). A function, which is monogenic, uniform and, except at poles,
continuous, is called a rneromorphic function. The simplest example is a
rational fraction.
^13. The following functions give illustrations of some of the preceding
definitions.
'
(a) In the case of a meromorphic function
F(z)
w = "77\>
/W
* Also critical point, which, however, is sometimes used to include all special points of a
function ; with German writers the title is Verzivngungspunkt, and sometimes Windungspunkt.
French writers use point de ramififatiort, and Italians punto di giramento and punto di
diramazione.
t Also synectic.
Also polar discontinuities ; also ( 32) accidental singularities.
Sometimes regular, but this term will be reserved for the description of another property of
functions.
F. P. 2
IS EXAMPLES ILLUSTRATING [!'!.
y. here F and / are polynomials in z without a common factor, the roots are
the roots of F (z) and the poles are the roots of / (z}. Moreover, according
as the degree of F is greater or is less than that of f,z = cc is a pole or a
zero of w.
(b) If w be a polynomial of order n, then each simple root of >/' is a
branch-point and a zero of w m , where m is a positive integer; 2=00 is
a pole of w ; and z=ao is a pole but not a branch-point or is an infinity
(though not a pole) and a branch-point of w*' according as n is even or odd.
(c) In the case of the function
1
w=
sn -
z
(the notation being that of Jacobian elliptic functions), the zeros are given by
1
- = ? A + 2mA + 2m'iK',
z
for all positive and negative integral values of m and of in. If we take
1
where may be restricted to values that are not large, then
so that, in the neighbourhood of a zero, w behaves like a holomorphic
function. There is evidently a doubly-infinite system of zeros ; they art-
distinct from one another except at the origin, where an infinite number
practically coincide.
The infinities of w are given by
-='2nK+2riiK',
z
for all positive and negative integral values of?? and of n '. If we take
1
2niK' +
1
then - =( l) n sn c,
w
so that, in the immediate vicinity of = 0,' is a holomorphic function.
Hence = is a pole of w. There is thus evidently a doubly-infinite system
of poles ; they are distinct from one another except at the origin, where an
infinite number practically coincide. But the origin is not a pole ; the
13.] THE DEFINITIONS 19
function, in fact, is there not determinate, for it has an infinite number of
zeros and an infinite number of infinities, and the variations of value are not
necessarily exhausted by zeros and infinities.
For the function - , the origin is a point which will hereafter be called
sn-
z
an essential singularity.
Ex. Obtain essential singularities of the functions
e\ sinh-, tanhz.
z
2 2
CHAPTER II.
INTEGRATION OF UNIFORM FUNCTIONS.
14. TIIE definition of an integral, that is adopted when the variables
are complex, is the natural generalisation of that definition for real variables
in which it is regarded as the limit of the sum of an infinite number of
infinitesimally small terms. It is as follows :
Let a and z be any two points in the plane ; and let them be connected
by a curve of specified form, which is to be the path of variation of the
independent variable. Let f(z) denote any function of z\ if any infinity
of f(z) lie in the vicinity of the curve, the line of the curve will be chosen
so as not to pass through that infinity. On the curve, let any number of
points Zi, z 3 z n in succession be taken between a and z\ then, if the sum
(*, - a)/(o) + (z, - z l )f(z l ) + ... + (*- *)/(*)
have a limit, when n is indefinitely increased so that the infinitely numerous
points are in indefinitely close succession along the whole of the curve from
a to z, that limit is called the integral off(z) between a and z. It is denoted,
as in the case of real variables, by
It is known* that the value of the integral of a function of a real variable
between limits a and 6 is independent of the manner in which, under the
customary definition, the interval between a and b is divided up. Assuming
this result, we infer at once that the same property holds for the complex
integral
for, if f(z) = u + iv, where u and v are real,
/ (z) dz = udx vdy +*iudy + ivdx,
and each of the integrals
fudx, $vdy, fudy, fvd.i;
* Harnack's Introduction to the Calculus, (Cathcart's translation), 103, 142.
14.] DEFINITIONS AS TO CONVERGENCE 21
taken between limits corresponding to the extremities of the curve, is inde-
pendent of the way in which the range is divided up.
The limit, as the value of the integral, is associated with a particular
curve : in order that the integral may have a definite value, the curve
(called the path of integration) must, in the first instance, be specified*.
The integral of any function whatever may not be assumed to depend in
general only upon the limits.
We have to deal with converging series ; it is therefore convenient to state the
definitions of the terms used. For proofs of the statements, developments, and appli-
cations in the theory of convergence, as well as the various tests of convergence, see
Bromwich's Theory of infinite series, Carslaw's Fourier's series and integrals, Hobson's
Functions of a real variable, and Pringsheim's article in the Encyclopadie der mathema-
tischen Wissenschaften, t. i, pp. 49 146, where full references are given.
A series, represented by
i, a 2 , ,), ...ad inf.,
is said to converge, when the limit of S n , where
S n = a l +az+. ..+,
as n increases indefinitely, is a unique finite quantity, say S. When, in the same circum-
stances, the limit of S n either is infinite or, if finite, is not unique (that is, may be one of
several quantities), the series is saidt to diverge.
The necessary and sufficient condition that the series
n 2> 3>---
should converge is that, corresponding to every finite positive quantity taken as small
as we please, an integer m can be found such that
for all integers n such that n ^ m, and for every positive integer r.
When the series
!i| !
converges, the series
i, 2 , a 3 , ...
converges ; and it is said to converge absolutely. When the series of moduli j i j , | 2 1 , | 3 1 , . . . , .
does not converge, though the series a l , 2 , :I , ... converges, the convergence of the latter
is said to be conditional. In a conditionally converging series, the order of the terms
must be kept : derangement of the order can lead to different limits ; and any assigned
sum, as a limit, can be obtained by appropriate derangement. In an absolutely converging
series, the order of the terms can be deranged without affecting the limit to which the
series converges ; the convergence is sometimes called unconditional.
These definitions apply to all infinite series, whatever be the source of their terms.
When the terms depend upon a variable quantity z, and the convergence of the series is
considered as z varies, wo have further classifications. Denote the series by
/i (4 / 2 (z), / 3 (*),.-. ad if.,
* This specification is tacitly supplied when the variables are real: the variable point moves
along the axis of x.
t Sometimes the series, such that the limit of ,S', ( when re is infinitely large is one of a
number of finite quantities (depending upon the way in which S n is formed), are called oscillating.
22 THEOREMS ON [14-.
and suppose that it converges for all values of z within a definite region. When any
quantity 8 has been chosen, and a positive integer m can be determined, such that
2
<8
v=n
for every value of n ^ m and for all values of z in the region, the convergence is said to lx;
uniform (sometimes continuous}.
Convergence may be uniform without being absolute ; it can be absolute without being
uniform.
When a series converges for all values of z such that j 2 | < >, but not for \s\> r, then
the circle, centre the origin of the variable z and radius equal to r, is called the circle of
rrjence : and the radius is sometimes called the radius of convergence. A series
such as
", 't t :, <i. >z-, ... ad inf.,
converges' absolutely within its circle of convergence, though not necessarily on its
circumference. It does not necessarily converge uniformly within its circle of convergence ;
but if r' is a positive quantity, less than the radius of convergence by a finite quantity
which can be taken small, the series converges uniformly within the circle of radius /'
concentric with its circle of convergence.
Again, when a uniformly converging series is integrated term by term over a finite
range, the resulting series also converges uniformly. But a uniformly converging series
can be differentiated term by term only if the series of derivatives converges.
15. Some inferences can be made from the definition of an integral.
(I.) The integral along any path from a to z passing through a point is
the sum of the integrals from a to % and from % to z along the same path.
Analytically, this is expressed by the equation
J
the paths on the right-hand side combining to form the path on the left.
(II.) When the path is described in the reverse direction, the sign of the
integral is changed : that is,
!
J
}dz = - f(z)dz,
J z
the curve of variation between a and z being the same.
(III.) The integral of the sum of a finite number of terms is equal to
the sum of the integrals of the separate terms, the path of integration being
the same for all.
(IV.) If a function f(z) be finite and continuous along any finite line
between two points a and z, the integral I f(z)dz is finite.
J a '
lo.J INTEGRATION 23
Let / denote the integral, so that we have / as the limit of
n
^ / \ f / \ .
i- =
H
hence 1 I\ = limit of 2 (z r+l z,.)f(z r }
Because f(z) is finite and continuous, its modulus is finite and therefore
must have a superior limit, say M, for points on the line. Thus
!/(*,) I <Jf,
so that | / } < limit of M^ \ z r+1 z f
<MS,
where S is the finite length of the path of integration. Hence the modulus
of the integral is finite ; the integral itself is therefore finite.
No limitation has been assigned to the path, except finiteness in length ;
the proposition is still true when the curve is a closed curve of finite length.
Hermite and Darboux have given an expression for the integral which
leads to the same result. We have as above
and / < * dz
/ < n
J a
where is a real positive quantity less than unity. The last integral involves
only real variables ; hence* for some point lying between a and z, we have
[ 3 '
J a
so that / =*0S\f(&\.
It therefore follows that there is some argument a such that, if X = 0e' a ,
This form proves the finiteness of the integral; and the result is the
generalisation t to complex variables of the theorem of mean value just
quoted for real variables.
* By what is usually called the " First theorem of mean value," in the integral calculus; for
a proof, see Carslaw's Fourier's series and integrals, 39.
t Hermite, Cours a la faculte des sciences de Paris (4 tnie ed., 1891), p. 59, where the reference
to Darboux is given.
\
24 FUNDAMENTAL THEOREM [15.
(V.) When a function is expressed as a uniformly converging series, the
integral of the function along any path of finite length is the sum of the
integrals of the terms of the series along the same path, provided that path
lies within the circle of convergence of the series: a result, which is an
extension of (III.) above.
Let MO + M, + ft* + be the converging series ; take
/(*) = MO + MI+ ... + i<n + R,
where j R \ can be made infinitesimally small with indefinite increase of n,
because the series converges uniformly. Then by (III.), or immediately from
the definition of the integral, we have
I f(z) dz = I u dz + I u^dz + ... + I u n dz + I Rdz,
J n J a J a J a J a
the path of integration being the same for all the integrals. Hence, if
@ = f(z) dz- u m dz,
J n m=QJ a
we have = I Rdz.
Let R' be the greatest value of j R \ for points in the path of integration
frem a to z, and let S be the length of this path, so that S is finite ;
then, by (IV.),
1 | <SR'.
Now S is finite ; and, as n is increased indefinitely, the quantity R' tends
towards zero as a limit for all points within the circle of convergence and
therefore for all points on the path of integration provided that the path lie
within the circle of convergence. When this proviso is satisfied, ! ! becomes
infinitesimally small and therefore also becomes infinitesimally small, with
indefinite increase of n. Hence, under the conditions stated in the enuncia-
tion, we have
r
J ti
which proves the proposition.
16. The following lemma* is of fundamental importance.
Let any region of the plane, on which the z- variable is represented, be
bounded by one or more simple f curves which do not meet one another :
each curve that lies entirely in the finite part of the plane will be considered
to be a closed curve.
* It is proved by Kiemann, Ges. Werke, p. 12, and is made by him (as also by Cauchy) the
basis of certain theorems relating to functions of complex variables.
t For the immediate purpose, a curve is called simple, if it have no multiple points. The
aim, in constituting the boundary from such curves, is to prevent the superfluous complexity that
arises from duplication of area on the plane. If, in any particular case, multiple points existed,
a method of meeting the difficulty would be to take each simple loop as a boundary.
16.]
IN INTEGRATION
25
If p and q be any two functions of x and y, which, for all points within the
region or along its boundary, are uniform, finite and continuous, then the
integral
extended over the whole area of the region, is equal to the integral
$(pdx + qdy),
taken in a positive direction round the whole boundary of the region.
(As the proof of the proposition does not depend on any special form of
region, we shall take the area to be (fig. 5) that which is included by the
curve QtP&'Pz and excluded by P 2 'Qo'P 3 4 and excluded by P/P 2 . The
positive directions of description of the curves are indicated by the arrows ;
and for integration in the area the positive directions are those of increas-
ing x and increasing y.)
First, suppose that both p and q are real. Then, integrating with regard
to x, we have*
fJs
where the brackets imply that the limits are to be introduced. When the
limits are introduced along a line CQ^'... parallel to the axis of x, then,
since C^Q/... gives the direction of integration, we have
[qdy] = - q.dy, + q.'dy,' - q^y, + q. 2 'dy 2 ' - q s dy s + q s 'dy s ',
where the various differential elements are the projections on the axis of y
of the various elements of the boundary at points along CQ&
* It is in this integration, and in the corresponding integration for p, that the properties of
the function q are assumed. Any deviation from uniformity, nniteness or continuity within the
region of integration would render necessary some equation different from the one given in
the text.
26 FUNDAMENTAL LEMMA [Ifi.
Now when integration is taken in the positive direction round the whole
boundary, the part of fqdy arising from the elements of the boundary at the
points on CQ&'... is the foregoing sum. For at Q 3 f it is q 3 'dy 3 ' because the
positive element dy~, which is equal to CD, is in the positive direction of
boundary integration; at Q :t it is (] 3 dy 3 because the positive element dy.>,
also equal to CD, is in the negative direction of boundary integration ;
at Q 2 ' it is q 2 'dy 2 ', for similar reasons ; at Q., it is q<dy, for similar reasons ;
and so on. Hence
corresponding to parallels through C and D to the axis of x, is equal to
the part of / qdy taken along the boundary in the positive direction for all
the elements of the boundary that lie between those parallels. Then when
we integrate for all the elements CD by forming $[qdy\, an equivalent is
given by the aggregate of all the parts of fqdy taken in the positive direction
round the whole boundary ; and therefore
on the suppositions stated in the enunciation.
Again, integrating with regard to y, we have
([ty
J J oy
when the limits are introduced along a line BP^P^... parallel to the axis
of y: the various differential elements are the projections on the axis of a; of
the various elements of the boundary at points along BP^PJ
It is proved, in the same way as before, that the part of j pdx arising
from the positively-described elements of the boundary at the points on
BPiPi ... is the foregoing sum. At P :{ ' the part of $pdx is p 3 'da; 3 ', because
the positive element dx 3 , which is equal to AB, is in the negative direction
of boundary integration ; at P s it is p 3 dx 3 , because the positive element
dx*, also equal to AB, is in the positive direction of boundary integration ;
and so on for the other terms. Consequently
- {pdx},
corresponding to parallels through A and B to the axis of y, is equal to
the part of jpdx taken along the boundary in the positive direction for all
the elements of the boundary that lie between those parallels. Hence
integrating for all the elements AB, we have as before
*- dxdy = jpdx ;
and therefore 1 1 ( ^ - \ dxdy = j(pdx + qdy).
16.] IN INTEGRATION 27
Secondly, suppose that p and q are complex. When they are resolved
into real and imaginary parts, in the forms // + ip" and q + iq" respectively,
then the conditions as to uniformity, finiteness and continuity, which apply
to p and q, apply also to p', q', p", q". Hence
~ y dxdy = -
!; - 10 ****
and therefore 1 1 (~ - J-) dxdy = j(pda- + qdy) :
aml
which proves the proposition.
No restriction on the properties of the functions p and q at points
that lie without the region is imposed by the proposition. They may have
infinities outside, they may cease to be continuous at outside points, or they
may have branch-points outside ; but so long as they are finite and continuous
everywhere inside, and in passing from any one point to any other point
always acquire at that other the same value whatever be the path of passage
in the region, that is, so long as they are uniform in the region, the lemma
is valid.
17. The following theorem due to Cauchy * can now be proved :
If a function f(z) be holomorphic throughout any region of the z-plane,
then the integral ff(z)dz, taken round the whole boundary of that region, is zero.
We apply the preceding result by assuming
p =f(z), q = ip = if(z) ;
owing to the character of f(z), these suppositions are consistent with the
conditions under which the lemma is valid. Since p is a function of z, we
have, at every point of the region,
dp _ 1 dp
dx i dy '
and therefore, in the present case,
dq _ .dp_dp
dx dx dy '
There is no discontinuity or infinity of p or q within the region ; hence
'dq _dp^
\dx dy;
* For an account of the gradual development of the theory and, in particular, for a
statement of Cauchy's contributions to the theory (with references), see Casorati, Teorica
delle funzioni di variabili complesse, pp. 64 90, 102106. The general theory ef functions,
as developed by Briot and Bouquet in their treatise Theorie dcs fonctions eUiptiqites, is based
upon Cauchy's method.
28 CAUCHY'S THEOREM [17.
the integral being extended over the region. Hence also
)(pdx + qdy) = 0,
when the integral is taken round the whole boundary of the region. But
pdx + qdy - pdx + ipdy
and therefore ff( z ) dz = 0,
the integral being taken round the whole boundary of the region within
which f(z) is holomorphic.
It should be noted that the theorem requires no limitation on the
character of f(z) for points z that are not included in the region.
The result can also be established by a slightly different use of the
original theorem. Writing
f(z) = u+ iv,
where, after the hypotheses concerning f(z), the real functions u and v
are uniform, finite, and continuous for all points within the region or along
the boundary, we have
ff(z) dz =f(w + iv) (dx + idy)
= j(udx vdy) + if(vdx + udy).
Owing to the character of u and v, we have
*** - vdy) =fj(- ^ - ? dxdy,
taken over the whole region ; but
du _ dv
dy dx'
and therefore
j(udac vdy} = 0.
Similarly
taken over the whole region ; but
du _ dv
doc dy '
and therefore
j(vdx + udy) = Q.
Hence, with the assumptions made as to f(z), we have
ff(z) dz = 0.
Some important propositions can be derived by means of the theorem, as
follows.
18.] INTEGRATION OF HOLOMORPHIC FUNCTIONS 29
18. When <i function f(z) is holomorphic over any continuous region
of the plane, the integral \ f(z)dz is a holomorphic function of z, provided the
J a
points z and a as well as the whole path of integration lie within that region.
The general definition ( 14) of an integral is associated with a specified
path of integration. In order to prove that the integral is a holomorphic
function of z, it will be necessary to prove (i) that the integral acquires the
same value in whatever way the point z is attained, that is, that the value is
independent of the path of integration, (ii) that it is finite, (iii) that it
is continuous, and (iv) that it is monogenic.
Let two paths ayz and aftz between a and z be drawn (fig. 6) in the
continuous region of the plane within which f(z) is
holomorphic. The line ayzffa is a contour over the area
of which f(z) is holomorphic ; and therefore ff(z) dz
vanishes when the integral is taken along ayz/Sa.
Dividing the integral into two parts and implying by
Zy, zp that the point z has been reached by the paths
ayz, af3z respectively, we have
f z y f a
and therefore I f(z) dz = I f(z) d
J d J Z
Thus the value of the integral is independent of the way in which z has
f*
acquired its value ; and therefore I f(z) dz is uniform in the region. Denote
it by F(z\
Secondly, f(z) is finite for all points in the region. After the result
of 17, we naturally consider only such paths between a and z as are finite in
length, the distance between a and z being finite. Hence ( 15, IV.) the
integral F(z) is finite for all points z in the region.
Thirdly, let z 1 '(=z + Bz) be a point infinitesimally near to z\ and consider
tt
f(z) dz. By what has just been proved, the path from a to / can be taken
afizz' ; therefore
t
I
J
rz+Sz rs rz+Sz
or f(z)dz-\ f(z}dz=\ f(z)dz,
J a J a J z
rz+Sz
so that F (z + Sz) -F(z}=\ f(z) dz.
J z
iRATIoX OF
Now at points in the infinitesimal line from z to /, the value of the
continuous function f(z) differs only by an infinitesimal quantity from its
value at z ; hunce the right-hand side is
where ] e ! is an infinitesimal quantity vanishing with Bz. It therefore follows
that
is an infinitesimal quantity with a modulus of the same order of small
quantities as \Bz\. Hence F(z) is continuous for points z in the region.
Lastly, we have
and therefore
F(z+Sz)-F(z)
has a limit when &z vanishes ; and this limit, f(z), is independent of the
way in which Bz vanishes. Hence F(z) has a differential coefficient ; the
integral is monogenic for points z in the region.
Thus F(z), which is equal to
J a
is uniform, finite, continuous, and monogenic ; it is therefore a holomorphic
function of z.
As in 16 for the functions p and q, so here for/(^), no restriction is
placed on properties of f(z} at points that do not lie within the region ;
so that elsewhere it may have infinities, or discontinuities, or branch-points.
The properties, essential to secure the validity of the proposition, are
(i) that no infinities or discontinuities lie within the region, and (ii) that the
same value of f(z) is acquired by whatever path in the continuous region
the variable reaches its position z.
COROLLARY. No change is caused in the value of the integral of a
holomorphic function between two points when the path of integration
between the points is deformed in any manner, provided only that, during the
deformation, no part of the path passes outside the boundary of the region
within which the function is holomorphic.
This result is of importance, because it permits the adoption of special
forms of the path of integration without affecting the value of the integral.
19. When a function f (z) is holomorphic aver a part of the plane
bounded by two simple curves (one lying within the other}, equal values of
ff(z)dz are obtained by integrating round each of the curves in a direction,
which relative to the whole area enclosed by each of them is positive.
19.] HOLOMOKPHK FUNCTIONS 31
The ring-formed portion of tin- plane (fig. 1, p. 3) which lies between
the two curves is a region over which f(z) is holomorphic; hence the integral
$f(z) dz taken in the positive sense round the whole of the boundary of
the included portion is zero. The integral consists of two parts : first, that
round the outer boundary the positive sense of which is DEF '; and second,
that round the inner boundary the positive sense of which for the portion of
area between ABC and DEF is AGE. Denoting the value of ff(z)dz round
DEF by (DEF), and similarly for the other, we have
The direction of an integral can be reversed if its sign be changed, so that
(ACS) = -(ABC): and therefore
(ABC) = (DEF).
But (ABC) is the integral $f(z)dz taken round ABC, that is, round the
curve in a direction which, relative to the area enclosed by it, is positive.
The proposition is therefore proved.
The remarks made in the preceding case as to the freedom from limitations
on the character of the function at places not within the bounded area are
valid also in this case.
COROLLARY I. When the integral of a function is taken round the whole
of any simple curve in the plane, no change is caused in its value by continuously
deforming the curve into any other simple curve provided the function is
holomorphic over the part of the plane in which the deformation is effected.
COROLLARY II. When a function f(z) is holomorphic over a continuous
portion of a plane bounded by any number of simple non-intersecting curves,
all but one of which are external to one another and the remaining one of
which encloses them all, the value of the integral ff(z) dz taken positively round
the single external curve is equal to the sum of the values taken round each
of the other curves in a direction which is positive relative to the area enclosed
by it.
These corollaries are of importance in many instances, as will be seen
later. The simplest instances arise in rinding the value of the integrals of
meromorphic functions round a curve which encloses one or more of the
poles ; the fundamental theorem, also due to Cauchy, for these integrals is
the following.
20. Let f(z) denote a function which is holomorphic over any region in
the z-plane, and let a denote any point within that region ; then
dz =,
z a
the integral being taken positively round the whole boundary of the region.
With a as centre and a very small radius p, describe a circle C, which
will be assumed to lie wholly within the region ; this assumption is justifiable
32 INTEGRATION OF [20.
because the point a lies within the region. Because /(z) is holomorphic over
the assigned region, the function f(z)/(z a) is holomorphic over the whole of
the region excluded by the small circle ('. Hence, by Corollary II. of 19, we
have
m*>
B Jz-a c .
'-'- dz,
z a
the notation implying that the integrations are taken positively round the
whole boundary B and round the circumference of C respectively.
For points on the circle C, let z a = pe?', so that 6 is the variable for
the circumference and its range is from to 2-n- ; then we have
dz .j a
z a
Along 'the circle f(z)=f(a + pe 9 *); the quantity p is very small and f(z) is
finite and continuous over the whole of the region, so that /(a 4- pe ei ) differs
from /(a) only by a quantity which vanishes with p. Let this difference
be e, which is a continuous small quantity ; thus j e j is a small quantity
which, for every point on the circumference of C, vanishes with p. Then
\f(a}+e\dd
= 1irif(a) + i I " edd.
J o
If E denote the value of the integral on the right-hand side, and rj the
greatest value of the modulus of e along the circle, we have, as in 15,
r2n
E\<\
J o
</:
d0
< 2-71-17.
Now let the radius of the circle diminish to zero. Then 77 also diminishes
to zero and therefore \E\, necessarily positive, becomes less than any finite
quantity however small, that is, E is itself zero ; and thus we have
^' z a
which proves the theorem.
When a is not a zero off(z), this result is the simplest case of the integral
f (z)
of a meromorphic function. The subject of integration is , a function
Z """ Cb
which is monogenic and uniform throughout the region and which, every-
where except at z a, is finite and continuous ; moreover, z a is a pole.
20.] MEROMORPHIC FUNCTIONS 33
because in the immediate vicinity of a the reciprocal of the subject of
integration, viz. (z a)/f(z), is holomorphic.
The theorem may therefore be expressed as follows :
If g (z) be a meromorphic function, which in the vicinity of a can be
expressed in the form f(z)/(z a) where /(a) is not zero, and which at all
other points in a region enclosing a is holomorphic, then
- .Jg (z) dz limit of (z a) g (z) when z = a,
the integral being taken round a curve in the region enclosing the point a.
The pole a of the function g (z) is said to be simple, or of the first order,
or of multiplicity unity.
Corollary. The more general case of a meromorphic function with a
finite number of poles can easily be deduced. Let these be a ly . .., a n , each
assumed to be simple ; and let
G (z) (z- a,) (z - a. 2 ). . .(z - a n ).
Let f(z) be a holomorphic function within a region of the z-plane bounded
by a simple contour enclosing the n points a ls a 2 , ..., a n , no one of which is a
zero off(z). Then since
1 I _J. 1_
#(*)~ r =i G'(a r )z-a r '
f(z} 1 f(z)
we have -~- = 2 ^r7\ jLLJ '.
G (z) r =i G (a r ) z a r
We therefore have
r f(*\
dz
$ 1 [f(z) ,
= 2 n , . . - - dz,
r= i G (a r ) J z- a r
G(z)
each integral being taken round the boundary. But the preceding proposition
gives
J Z ~~"" CLf
because f(z) is holomorphic over the whole region included in the contour;
and therefore
the integral on the left-hand side being taken in the positive direction*.
The result just obtained expresses the integral of the meromorphic
function round a contour which includes a finite number of its simple poles.
It can be obtained otherwise from Corollary II. of 19, by adopting
* "We shall for the future assume that, if no direction for a complete integral be specified, the
positive direction is taken.
F. F. 3
34 INTEGRATION OF
a process similar to that adopted above, viz., by making each of the curves in
that Corollary circles round the points a,, ..., a n with radii sufficiently small
to secure that each circle is outside all the others.
Ex. 1. A function f(z) is holomorphic over an area bounded by a simple closed
ourve; and a, b, c are three points within the area. Find the value of the integral
J_ ( /W d:
2iri ](z-a)(z-b)(z-c)
taken round the curve; and shew what it becomes
(i) when a and b coincide,
(ii) when a, 6, c coincide.
Ex, 2. Let S(-\ denote the sum of any set of selected terms of the series
!+ + +..., ICKI'I,
and let /(f)=ao+aiC+a 2 f 2 +---
where /(f) is a holomorphic function of f within the range ; shew that the sum of the
same set of terms selected from /() can be expressed in the form
21. The preceding theorems have sufficed to evaluate the integral of
a function with a number of simple poles. We now proceed to obtain
further theorems, which can be used among other purposes to evaluate
the integral of a function with poles of order higher than the first.
We still consider a function f(z) which is holomorphic within a given
region. Let a be a point within the region which is not a zero of f(z) :
we have
. a
Let a + Ba be any other point within the region, so that, if a be near the
boundary, \Ba\ is to be chosen less than the shortest distance from a to
the boundary ; then
dz>
_
2-TTi .' z a Sa
and therefore
z a z-a
^- -2^2 \f(z)dz
-ay**(z-ay(z-a-8a)l j(
the integral being in every case taken round the boundary.
Since / (z) is monogenic, the definition of f (a), the first derivative of
/(a), gives /'(a) as the limit of
21.] MEROMORPHIC FUNCTIONS 35
when Sa ultimately vanishes ; hence we may take
/(a + Sa)-/(a) >/"> A .
gj - =/ (a) + o-,
where o- is a quantity which vanishes with So, and is therefore such that | <r \
also vanishes with Sa. Hence
I/' (a) + -}..
dividing out by Sa and transposing, we have
/(*) - ^
_ /T
/(
t_L-i
- a) 2 (5 - a - Sa)
(z - of
As yet, there is no limitation on the value of Sa; we now proceed to a
limit by making a + Sa approach to coincidence with a, viz., by making Sa
ultimately vanish. Taking moduli of each of the members of the last
equation, we have
Sa
2-rri (z - a)
(z a) 2 (z a - Sa)
dz
< o- +
Saj
~t^
Let the greatest modulus of
a) 2 (z a Sa)
for points z along the
-. -. .
(z ay(z a Sa)
boundary be M, which is a finite quantity on account of the conditions
applying to f(z) and of the fact that the points a and a + Sa lie within
the region and are not on the boundary. Then, by 15,
/(*)
dz
<MS,
J (z af(z a Sa)
where S is the whole length of the boundary, a finite quantity. Hence
f (a) - g,
-
dz
<\ a
When we proceed to the limit in which Sa vanishes, we have Sa | =
and | cr | = 0, ultimately ; hence the modulus on the left-hand side ultimately
vanishes, and therefore the quantity to which that modulus belongs is itself
zero, that is,
_ * J \*) d
^ i \ *
so that , ,^, . ,
(z - a) 2
This theorem evidently corresponds in complex variables to the well-
known theorem of differentiation with respect to a constant under the
integral sign when all the quantities concerned are real.
32
36 PROPERTIES OF [21.
Proceeding in the same way, we can prove that
-/_ 2! f /(*) , , ff
where is a small quantity which vanishes with Ba. Moreover the integral
on the right-hand side is finite, for the subject of integration is everywhere
finite along the path of integration which itself is of finite length. Hence,
first, a small change in the independent variable leads to a change of the
same order of small quantities in the value of the function /' (a), which
shews that f (a) is a continuous function. Secondly, denoting
/'(a + Sa)-/'(a)
t // / \
by Sf (a), we have the limiting value of -*- equal to the integral on
the right-hand side when 8a vanishes, that is, the derivative of f (a) has
a value independent of the form of Ba and therefore f (a) is monogenic.
Denoting this derivative by /" (a), we have
Thirdly, the function f (a) is uniform : for it is the limit of the value
of /v ^ ~/ W . an( j D0 th /(a) and/(a + Sa) are uniform. Lastly, it
is finite; for (S 15) it is the value of the integral ~ . f! 1 -r*dz, in which
2-m (z a) 2
the length of the path is finite and the subject of integration is finite at
every point of the path.
Hence f (a) is continuous, monogenic, uniform, and finite, throughout
the whole of the region in which f(z) has these properties : it is a holo-
morphic function. Hence :
When a function is holomorphic in any region of the plane bounded by
a simple curve, its derivative is also holomorphic within that region.
And, by repeated application of this theorem :
When a function is holomorphic in any region of the plane bounded by
a simple curve, it has an unlimited number of successive derivatives each of
which is holomorphic within the region.
All these properties have been shewn to depend solely upon the holo-
morphic character of the fundamental function ; but the inferences relating
to the derivatives have been proved only for points within the region and
not for points on the boundary. If the foregoing methods be used to prove
them for points on the boundary, they require that a consecutive point shall
be taken in any direction ; in the absence of knowledge concerning the
fundamental function for points outside (even though just outside), no
inferences can be drawn justifiably.
21.] HOLOMORPHIC FUNCTIONS 37
An illustration of this statement is furnished by the hypergeometric
series which, together with all its derivatives, is holomorphic within a circle
of radius unity and centre the origin. The series converges everywhere
on the circumference, provided y > a + $. But the corresponding condition
for convergence on the circumference ceases to be satisfied for some one of
the derivatives and for all which succeed it : as such functions dd not then
converge, the circumference of the circle must be excluded from the region
within which the derivatives are holomorphic.
Ex. Let F(z) and G (z) denote two functions of z, holomorphic in a region enclosing
the point a, which is a zero of G (z) and a non-zero of F(z) ; prove that
& ,_F'(a}G'(a)-F(a}G"(a)
when a is a simple root of &(z)=0, and that
when a is a double root of G(z) = 0, both integrals being taken round a small contour
which encloses a but no other zero of G(z).
22. Expressions for the first and the second derivatives have been
obtained.
By a process similar to that which gives the value of f (a), the derivative
of order n is obtainable in the form
(z - a) n+
the integral being taken round the whole boundary of the region or round
any curves which arise from deformation of the boundary, provided that no
point of the curves in the final form of the boundary or in any intermediate
form of the boundary is indefinitely near to a.
In the case when the curve of integration is a circle, no point of which
circle may lie outside the boundary of the region, we have a modified form
for /<"> (a).
For points along the circumference of the circle with centre a and radius
r, let z a = re ei , so that, as before,
z a
then and 2-rr being taken as the limits of 0, we have
f ST
I e ~ n6i f( a + re ^ d6 '
Let M be the greatest value of the modulus of f(z) for points on the
38 PROPERTIES OF [22.
circumference (or, as it may be convenient to consider, for points on or within
the circumference) : then
i fw < a \ i < JLL I ' I e -noi I l/(a + re 9 *) j d0
2irr n J o
<s?D
M
Now, let a function < (z) be defined by the equation
M
r
evidently it can be expanded in a series of ascending powers of z a which
converges within the circle. The series is
z a (z a) z
i i V * \
d n <f)(z) , M ( , , ,z - a
so that _Z = n ! ji + ( n + i)_ - + ...}
dz n r n ( r j
retool ,^
Hence = n ,
L dz n ] z=a r"'
so that, if the value of the nth derivative of <j>(z), when z = a, be denoted
by 0W (a), we have j/< n > (a) j < < (n > (a).
These results can be extended to functions of more than one variable :
the proof is similar to the foregoing proof. When there are two variables,
say z and /, the results may be stated as follows :
For all points z within a given simple curve C in the 2-plane and all
points z' within a given simple curve C' in the ^'-plane, let f(z, z) be a
holomorphic function ; then, if a be any point within C and a' any point
within C",
n\ri \ [[ f(z, z'} d n+n 'f(a, ftp
(27n') 2 J J (z - a) n+l (z - a') n ' +l da n da' n '
where n and n' are any integers and the integral is taken positively round the
two curves C and C'.
If M be the greatest value of \f(z, z') \ for points z and / within their
respective regions when the curves C and C" are circles of radii r, r' and
centres a, a', then
n I n
and if $ (z, z') =
da n da' n ' ' r n r'
M
22.]
then
HOLOMORPHIC FUNCTIONS
39
d n+n 'f(a, a')
da n da' n '
d n+n> $ (z, z')
dz n dz" n>
when z = a and z' = a in the derivative of <f> (z, z').
A function (f>, related in this manner to a function f in association with
which it is constructed, is sometimes called* a dominant function.
23. All the integrals of meromorphic functions that have been considered
have been taken along complete curves : it is necessary to refer to integrals
along curves which are lines only from one point to another. A single
illustration will suffice at present.
rz f(z\
Consider the integral I dz ; the function f(z) is
supposed holomorphic in the given region : z and z are
any two points in that region. Let some curves joining
z to z be drawn as in the figure (fig. 7).
r (*)
Then
is holomorphic over the whole area en-
dz
taken round the boundary of that area. Hence, as in the earlier case, we have
closed by z ftzSz : and therefore we have I -dz = Q, the integral being
J z a
z-a
z a
The point a lies within the area enclosed by z Q ryz/3z , and the function
\
is holomorphic, except in the immediate vicinity of z = a ; hence
the integral on the left-hand side being taken round z yz^z . Accordingly
f(z)
We denote - by g (z), so that g (z) is a function which has one pole a
Z ~~ CL
in the region considered.
The preceding results are connected only with the simplest form of
meromorphic functions ; other simple results can be derived by means of the
other theorems proved in 17 21. Those which have been obtained are
sufficient however to shew that: The integral of a meromorphic function
fg (z) dz, from one point to another of the region of the function, is not in
general a uniform function. The value of the integral is not altered by
any deformation of the path which does not meet or cross a pole of the
* Poincar^ uses the term majorante.
40 GENERAL PROPOSITIONS [23.
function ; but the value is altered when the path of integration is so
deformed as to pass over one or more poles. Therefore it is necessary to
specify the path of integration when the subject of integration is a mero-
morphic function ; only partial deformations of the path of integration are
possible without modifying the value of the integral.
24. The following additional propositions* are deduced from limiting
cases of integration round complete curves. In the first, the curve becomes
indefinitely small ; in the second, it becomes infinitely large. And in neither,
are the properties of the functions to be integrated limited as in the preceding
propositions, so that the results are of wider application.
I. If f(z) be a function which, whatever be its character at a, has no
infinities and no branch-points in the immediate vicinity of a, the value of
/f(z) dz taken round a small circle with its centre at a tends towards zero
when the circle diminishes in magnitude so as ultimately to be merely the
point a, provided that, as \z a\ diminishes indefinitely, the limit of(z a)f(z)
tend uniformly to zero.
Along the small circle, initially taken to be of radius r, let
z a = re 6 *,
dz
z a
r2ir
= idd,
ff(z)dz = ir(z-a)f(z)d0.
Jo
Jo (Z ~ a
<
.'0
r
Jo
Mdd
< 27T JT ,
where M' is the greatest value, of M, the modulus of (z a)f(z), for points
on the circumference. Since (z-a)f(z) tends uniformly to the limit zero
as | z a | diminishes indefinitely, j Jf(z) dz \ is ultimately zero. Hence the
integral itself ff(z) dz is zero, under the assigned conditions.
Note. If the integral be extended over only part of the circumference of
the circle, it is easy to see that, under the conditions of the proposition,
the value of //(z) dz still tends towards zero.
* The form of the first two propositions, which is adopted here, is due to Jordan, Cours
d'Analyse, t. ii, 256.
24.] IN INTEGRATION 41
COROLLARY. If (z a)f(z} tend uniformly to a limit k as z a\
diminishes indefinitely, the value offf(z)dz taken round a small circle, centre
a, tends towards 2mk in the limit.
Thus the value of / -, taken round a very small circle centre a, where a is
J (a 2 - 2 2 )2
not the origin, is zero: the value of / - round the same circle is -. ( - ) .
J (a z) (a + z) 2 * \ a /
Neither the theorem nor the corollary will apply to a function, such as sn : ,
Z " Ct
which has the point a for an essential singularity: the value of (z a) sn , as
Z CL
| z- a, | diminishes indefinitely, does not tend ( 13) to a uniform limit. As a matter of
fact, the function sn has an infinite number of poles in the immediate vicinity of a
z a
as the limit z=a is being reached.
II. Whatever be the character of a function f (z) for infinitely large values
of z, the value of ff(z) dz, taken round a circle with the origin for centre, tends
towards zero as the circle becomes infinitely large, provided that, as \z\
increases indefinitely, the limit of zf(z) tend uniformly to zero.
Along a circle, centre the origin and radius R, we have z = Re ei , so that
dz ...
- = idv,
z
and therefore If( 2 ) dz = i \ zf(z) dd.
J o
Hence | // (*) dz \ = I f *"zf(z) d6
zf(z} | M
2ir
< I MdO
where M' is the greatest value of M, the modulus of zf(z), for points on
the circumference. When R increases indefinitely, the value of M' is zero
on the hypothesis in the proposition ; hence | ff(z) dz \ is ultimately zero.
Therefore the value of ff(z) dz tends towards zero, under the assigned con-
ditions.
Note. If the integral be extended along only a portion of the circumfer-
ence, the value of ff(z) dz still tends towards zero.
COROLLARY. If zf(z) tend uniformly to a limit k as \z
ncreases
indefinitely, the value of $f(z) dz, taken round a very large circle, centre the
origin, tends towards %Trik.
Thus the value of J (1 - z n }~^dz round an infinitely large circle, centre the origin, is zero
if n > 2, and is 2ir if n = 2.
42 GENERAL PROPOSITIONS [24.
III. If all the infinities and the branch-points of a function lie in a finite
region of the z-plane, then the value of jf(z) dz round any simple curve, which
includes all those points, is zero, provided the value of zf(z), as \ z increases
indefinitely, tends uniformly to zero.
The simple curve can be deformed continuously into the infinite circle
of the preceding proposition, without passing over any infinity or any
branch-point ; hence, if we assume that the function exists all over the plane,
the value of ff(z)dz is, by Cor. I. of 19, equal to the value of the integral
round the infinite circle, that is, by the preceding proposition, to zero.
Another method of stating the proof of the theorem is to consider
the corresponding simple curve on Neumann's sphere ( 4). The surface
of the sphere is divided into two portions by the curve*: in one portion lie
all the singularities and the branch-points, and in the other portion there is
no critical point whatever. Hence in this second portion the function is holo-
morphic : since the area is bounded by the curve we see that, on passing back
to the plane, the excluded area is one over which the function is holomorphic.
Hence, by 19, the integral round the curve is equal to the integral round
an infinite circle having its centre at the origin and is therefore zero, as
before.
COROLLARY. If, under the same circumstances, the value of zf(z}, as
z | increases indefinitely, tend uniformly to k, then the value of ff(z) dz round
the simple curve is %irik.
Thus the value of / - along any simple curve, which encloses the two points
a and a, is 2ir ; the value of
f dz
J{(i- 2 2)n
round any simple curve enclosing the four points 1, 1, -r, T, is zero, k being a non-
K K
vanishing constant; and the value of J(l z 2n )~*dz, taken round a circle, centre the
origin and radius greater than unity, is zero when n is an integer greater than 1.
But the value of
/
J {(*-
round any circle, which has the origin for centre and includes the three distinct points
i e 2> e 3i is n t zero - The subject of integration has 2=00 for a branch-point, so that the
condition in the proposition is not satisfied ; and the reason that the result is no longer
valid is that the deformation into an infinite circle, as described in Cor. I. of 19,
is not possible because the infinite circle would meet the branch-point at infinity.
* The fact that a single path of integration is the boundary of two portions of the surface
of the sphere, within which the function may have different characteristic properties, will be
used hereafter ( 104) to obtain a relation between the two integrals that arise according as. the
path is deformed within one portion or within the other.
25.] EXAMPLES 43
25. The further consideration of integrals of functions, that do not possess
the character of uniformity over the whole area included by the curve of in-
tegration, will be deferred until Chap. IX. Some examples of the theorems
proved in the present chapter will now be given.
Ex. 1. It is sufficient merely to mention the indefinite integrals (that is, integrals from
an arbitrary point to a point z) of rational integral functions of the variable. After the
preceding explanations it is evident that they follow the same laws as integrals of similar
functions of real variables.
Ex. 2. Consider the integral I ^^ , taken round a simple curve.
j v* **/
When n is 0, the value of the integral is zero if the curve do not include the point a,
and it is 2iri if the curve include the point a.
When n is a positive integer, the value of the integral is zero if the curve do not
include the point a (by 17); and the value of the integral is still zero if the curve do
include the point a (by 22, for the function f(z} of the text is 1 and all its derivatives
are zero). Hence the value of the integral round any curve, which does not pass through
a, is zero.
We can now at once deduce, by 20, the result that, if a holomorphic function be
constant along any simple closed curve within its region, it is constant over the whole
area within the curve. For let t be any point within the curve, z any point on it, and C
the constant value of the function for all the points z ; then
the integral being taken round the curve, so that
*<>-/*
= 0,
since the point t lies within the curve.
Ex. 3. The integral - . I /' (z) log - - dz is taken round a circle, centre the origin
ZTT'l J ZL
and radius greater than unity; and the function f(z) is holomorphic everywhere within
the circle. Prove that the value of the integral is
Ex. 4. Consider the integral \e~*dz.
In any finite part of the plane, the function e~^ is holomorphic; therefore ( 17) the
integral round the boundary of a rectangle
(fig. 8), bounded by the lines x a, y=0,
y b, is zero : and this boundary can be
extended, provided the deformation remain
in the region where the function is holo-
morphic. Now as a tends towards infinity,
the modulus of e~^, being e-* 2 + 2 ) tends
towards zero when y remains finite ; and p.
therefore the preceding rectangle can be
extended towards infinity in the direction of the axis of x, the side b of the rectangle
remaining unaltered.
44 EXAMPLES IN [25.
Along A' A, we have z=x : so that the value of the integral along the part A' A of the
fa
boundary is I e'^dx.
J -a
Along AS, we have z=a+iy, so that the value of the integral along the part AB
Along BB', we have z=x + ib, so that the value of the integral along the part BE'
is t %-<* + ) 2 cfo.
J a
Along B'A\ we have z=-a + iy, so that the value of the integral along the part
f
B'A' is i I e-(- a+i Wdy.
J b
fb
The second of these portions of the integral is e~ a * , i . I e^-^dy, which is easily seen
J o
to be zero when the (real) quantity a is infinite.
Similarly the fourth of these portions is zero.
Hence as the complete integral is zero, we have, on passing to the limit,
f
J
whence d>'- \ e-* f - 2aac dx= I e-**dx=Tr$,
J -oc J -oc
I e~ z "(cos2&# i sin %bx)dx=ir*e~ vt ;
or
and therefore, on equating real parts, we obtain the well-known result
/
This is only one of numerous examples* in which the theorems in the text can be
applied to obtain the values of definite integrals with real limits and real variables.
Ex. 5. By taking the integral \e~*dz along the perimeter of a sector of a circle
between the radii of a circle given = 0, # = TT, and the intercepted part of the circum-
ference of radius r which is ultimately increased without limit, establish the value
(JTT)& for each of Fresnel's integrals
cos v? du. I sin 2 dii.
I cos v? du, I si
Jo Jo
Ex. 6. Prove that, when a 2 + 6 2 < 1, the value of the integral
/ 2ir a cos x + ft sin x + y
a cos #+6 sin x + l
for real values of x within the range, is
27T
* See Briot and Bouquet, Theorie des fonctions elliptiques,\ (2nd ed.), pp. 141 et sqq., from
which examples 4 and 8 are taken.
f
J
25.] CONTOUR INTEGRATION 45
A'.c. 7. Evaluate the following integrals by the process of contour integration :
cos ax
(*+ :i)(^+4) ^ where a 1S real ;
cos ax cos 6.r ,
o ~^~~ (m)
where a and b are real and lie between and 1 ;
f<*> gOX
(iv) / - -<&F, where < a < 1.
/ 2 n-l
ofe, where TO is a real positive quantity less than
1 + 2
unity.
The only infinities of the subject of integration are the origin and the point - 1 ;
the branch-points are the origin and 2 = 00. Everywhere else in the plane the function
behaves like a holomorphic function ; and, therefore, when we take any simple closed
curve enclosing neither the origin nor the point 1, the integral of the function round
that curve is zero.
Choose the curve, so that it lies on the positive side of the axis of x and that it is
made up of :
(i) a semicircle C 3 (fig. 9), centre the origin and radius R which is made to increase
indefinitely :
(ii) two semicircles, Cj and c 2 , with their centres at and 1 Respectively, and with
radii r and r', which ultimately are made infinitesimally small :
(iii) the diameter of C 3 along the axis of x excepting those ultimately infinitesimal
portions which are the diameters of Cj and of c 2 .
The subject of integration is uniform within the area thus enclosed although it
is not uniform over the whole plane. We shall take that value of z n ~ l which has its
argument equal to (n 1) 0, where 6 is the argument of z.
Fig. 9.
The integral round the boundary is made up of four parts.
z n-\
(a) The integral round C 3 . The value of z . , as | z \ increases indefinitely, tends
1+2
uniformly to the limit zero ; hence, as the radius of the semicircle is increased indefinitely,
the integral round (7 S vanishes ( 24, n., Note).
2 n-l
(6) The integral round Ci. The value of z. , as ] z \ diminishes indefinitely,
L -\-z
tends uniformly to the limit zero ; hence as the radius of the semicircle is diminished
indefinitely, the integral round c t vanishes ( 24, I., Note).
46 EXAMPLES IN [25.
2~1
(c) The integral round c 2 . The value of (1 + z) , as j 1 + z \ diminishes indefinitely
for points in thcs area, tends uniformly to the limit ( I)"" 1 , i.e., to the limit e^ H ~ 1 ^'.
Hence this part of the integral is
being taken in the direction indicated by the arrow round c 2 , the infinitesimal semicircle.
Evidently =idO and the limits are IT to 0, so that this part of the whole integral is
(d) The integral along the axis of x. The parts at 1 and at which form the
diameters of the small semicircles are to be omitted ; so that the value is
( f-l-r' f-r /) #n-l
\ + + - dx.
(J -oo J -i + r' J r) l+#
This is what Cauchy calls the principal value of the integral
dx.
-oo 1+X
Since the whole integral is zero, we have
/ fM-l TO r n-l
-dx, P'= f dx,
I+X J -oc 1+^
/ x n 1
, -- dx,
l-X
principal values being taken in each case. Then, taking account of the arguments, we have
/ ( 3-\ n ~ l r x r n -ls7-r
x > dx=(-iy-i\
o 1-* Jo l-x
=e (-l)T'' q
Since i,re niri +P + P' = 0, we have
P-e^Q^-irre,
so that
V Q cos nir IT sin nw, sin nir = IT cos nir.
f" x n ~ l
Hence = - dx=P=
Jo 1+Jc
ir cosec nir.
1-*
Ex. 9. In the same way it may be proved that
cos ax , .IT n
where n is an integer, a is positive and o> is e 2n .
25.] CONTOUR INTEGRATION 47
Ex. 10. By considering the integral \e~*z n ~ l dz round the contour of the sector of a
circle of radius r, bounded by the radii = 0, 6 = a, where a is less than ?r and n is positive,
it may be proved that
J
on proceeding to the limit when r is made infinite. (Briot and Bouquet.)
Ex. 11. By considering the integral l(z^-\) m z~ ai ~ m ~ l dz, taken round a semicircle,
prove that
~ 2" H (* TO + $ai)
provided the real part of wi is greater than - 1.
Similarly deduce the value of
r *r
sin 6 cos" d e ae dd,
where the real parts of m and n are each greater than - 1, from a consideration of the
integral .
J(2 2 - l) m (2 2 + l) n z-ai-m-n-1 ^
taken round a semicircle.
(Many of the results stated in de Haan, Nouvelles tables d'integrales definies, can be
obtained in a similar manner.)
/dz
, where n is an integer. The subject of integration
z" 1
is meromorphic: it has for its poles (each of which is simple) the n points <a r for r=0,
1, ..., n 1, where a> is a primitive ?ith root of unity ; and it has no other infinities and no
branch-points. Moreover the value of , as | z \ increases indefinitely, tends uniformly
Z J.
to the limit zero ; hence ( 24, m.) the value of the integral, taken round a circle centre
the origin and radius > 1, is zero.
This result can be derived by means of Corollary II. in 19. Surround each of the
poles with an infinitesimal circle having the pole for centre ; then the integral round
the circle of radius > 1 is equal to the sum of the values of the integral round the
infinitesimal circles. The value round the circle having w r for its centre is, by 20,
z (limit of , when z=<B r |
\ 21 /
n
Hence the integral round the large circle
2
n r =o
=0.
/ e an
-3 - dz, taken round a semicircle, prove that
2^+1
provided a is positive.
48 EXAMPLES IN [25.
Ex. 14. Taking as the definition of Bernoulli's numbers that they are the coefficients
in the expansion
i i 7?
- y I _ 1 \rn-l m r 2m-l
+ - ( ~ "
prove (by contour integration) that
_
m ~ (2nf m n =in
In the same way, obtain expressions for the coefficients, in the expansion in powers of z,
of the quantity
(Hermite.)
Ex. 15. In all the preceding examples, the poles that have occurred have been
simple : but the results proved in 21 enable us to obtain the integrals of functions
which have multiple poles within an area. As an instance, consider the integral
/ - - -- round any curve which includes the point i but not the point -i, these
7( 2n +
points being the two poles of the subject of integration, each of multiplicity + l.
We have seen that /(") (a)= . I _ ^ + 1 efe,
where f(z) is holomorphic throughout the region bounded by the curve round which the
integral is taken.
In the present case a is i, and /() = -; r- r ; so that
(Z + t)
n\
f
J
dz _ 2iri ... . Zn\ it
i~ TO! f (l) ~n\n\2 2n '
In the case of the integral of a function round a simple curve which contains several
of its poles, we first ( 20) resolve the integral into the sum of the integrals round simple
curves each containing only one of the points, and then determine each of the latter
integrals as above.
Another method, that is sometimes possible, makes use of the expression of the uniform
function in partial fractions. After Ex. 2, we need retain only those fractions which are of
the form Aj(z a}: the integral of such a fraction is 2iriA, and the value of the whole
integral is therefore ZiriSA. It is thus sufficient to obtain the coefficients of the inverse
first powers which arise when the function is expressed in partial fractions corresponding
to each pole. Such a coefficient A, being the coefficient of - - in the expansion of the
Z "^ CL
function, is called by Cauchy the residue of the function relative to the point.
For example,
25.] CONTOUR INTEGRATION 49
so that the residues relative to the points - 1, -o>, -<a 2 are , $, $o> 2 respectively.
Hence if we take a semicircle, of radius >1 and centre the origin with its diameter
along the axis of y, so as to lie on the positive side of the axis of y, the area between the
semi-circumference and the diameter includes the two points - o> and o> 2 ; and therefore
the value of
dz
taken along the semi-circumference and the diameter, is
that is, the value is - $ tri.
Ex. 16. Let u denote I I * y / dzdz", f being a rational integral function
J(C')J(C) zz'-\
2A mn z m z'* of the complex variables z, z 1 , the integrations being taken in the positive sense
round the closed contours C, C", of which C is a circle of unit radius with its centre
at the origin. Shew that u = if C' lies wholly inside C, or if C and C' lie wholly outside
one another, and that u=47T 2 2A mm (m=0, 1, 2, ...) if C'' completely surrounds C.
Discuss also the value of u if C' is a circle passing through the points i but not
coinciding with G, and f(z, z')=f( z, z'}.
(Math. Trip., Part II., 1898.)
NOTE. For further applications of Cauchy's theory of residues, together with many
references to Cauchy's own results, Lindelof's monograph Le calcul des residus (Gauthier-
Villars, 1905) may be consulted.
P. P. 4
CHAPTER III.
EXPANSION OF FUNCTIONS IN SERIES OF POWERS.
26. WE are now in a position to obtain the two fundamental theorems
relating to the expansion of functions in series of powers of the variable :
they are due to Cauchy and Laurent respectively.
Cauchy's theorem is as follows* :
When a function is holomorphic over the area of a circle of centre a, it can
be expanded as a series of positive integral powers of z a, converging for all
points within the circle.
Let z be any point within the circle ; describe a concentric circle of
radius r such that
\z a\ = p <r< R,
i r '
where R is the radius of the given circle. If t
denote a current point on the circumference of the
new circle, we have
, w
27TI jt
Stri] t a z a
t a
Fig. 10.
the integral extending along the whole circumference of radius r. Now
1-
z a
z
T^<
z
-a\ n \t-a)
t a
so that, by 15 (III.), we have
t a) z a
t a
* Exercices d' Analyse et de Physique Mathematique, t. ii, pp. 50 et seq.; the memoir was first
made public at Turin in 1832.
26.] CAUCHY'S THEOREM ON THE EXPANSION OF A FUNCTION 51
Now/() is holomorphic over the whole area of the circle; hence, if t be
not actually on the boundary of the region ( 21, 22), a condition secured by
the hypothesis r < R, we have
and therefore
Let the last term be denoted by L. Since | z a = p and t a \ = r \
it is at once evident that t - z \ ^ r p. Let M be the greatest value of
f(t) \ for points along the circle of radius r ; then M must be finite, owing to
the initial hypothesis relating tof(z). Taking
a=
so that
we have
dt = i(t a) dO,
L
2-7T |J t-z(t-a) n
2?r r n (r-
n n+i
r n (r - p)
< (efirii -
\rj \
Now r was chosen to be greater than p ; as M becomes infinitely large,
(-) becomes infinitesimally small. Also M ( 1 -j is finite. Hence as
n increases indefinitely, the limit of \L\, necessarily not negative, is in-
finitesimally small and therefore, in the same case, L tends towards zero.
It thus appears, exactly as in 15 (V.), that, when n is made to increase
without limit, the difference between the quantity /(.z) and the first n + 1
terms of the series is ultimately zero ; hence the series is a converging series
having f(z) as the limit of the sum, so that
which proves the proposition under the assigned conditions. It is the form
of Taylor's expansion for complex variables.
Note. A. series, such as that on the right-hand side and not necessarily
arising through the expansion of a given function/^), is frequently denoted
by P (z a), where Pisa general symbol for a converging series of positive
integral powers of z a : it is also sometimes* denoted by P (z \ a). Con-
formably with this notation, a series of negative integral powers of z a
Weierstrass, Ges. Werke, t. ii, p. 77.
42
52 CAUCHY'S THEOREM ON THE EXPANSION OF A FUNCTION [26.
would be denoted by P (- )' a series of negative integral powers of z
either by Pf-j or by P(z\ao\ the latter implying a series proceeding in
\f I
positive integral powers of a quantity which vanishes when z is infinite,
that is, in positive integral powers of z~*.
If, however, the circle can be made of infinitely great radius so that the
function f(z) is holomorphic over the finite part of the plane, the equivalent
series is denoted by G (z a), and it converges over the whole plane*.
Conformably with this notation, a series of negative integral powers of z a
which converges over the whole plane is denoted by G ( - - J .
Ex. If the expansion, taken in the form Oo + ai2 + a 2 2 2 + ..., be valid over the whole of
the finite part of the plane, then the limit of
as m increases indefinitely, is zero. More generally, if the circle of convergence of the
series be of radius r, then the limit of the preceding quantity is 1/r. (Cauchy.)
27. The following remarks on the proof and on inferences from it should
be noted.
(i) In order that (t z)~ l may be expanded in the required form, the
point z must be taken actually within the area of the circle of radius R',
and therefore the convergence of the series P {z a) is not established for
points on the circumference.
(ii) The coefficients of the powers of z a in the series are the
values of the function and its derivatives at the centre of the circle ; and the
character of the derivatives is sufficiently ensured ( 21) by the holomorphic
character of the function for all points within the region. It therefore
follows that, if a function be holomorphic within a region bounded by a
circle of centre a, its expansion in a series of ascending powers of z a,
which converges for all points within the circle, depends only upon the values
of the function and its derivatives at the centre.
Conversely, a converging power-series in z a, having assigned
coefficients /(a), /' (a), ..., defines a uniform function within the radius
of convergence of the series.
But instead of having the values of the function and of all its derivatives
at the centre of the circle, it will suffice to have the values of the holomorphic
function itself over any region at a or along any line through a, the region
or the line being not merely a point. The values of the derivatives at a can
be found in either case; for/'(&) is the limit of {/(b + Bb) f(b)}/8b, so that
the value of the first derivative can be found for any point in the region or
on the line, as the case may be ; and so for all the derivatives in succession.
* It then, is often called an integral function.
27.] DARBOUX'S EXPRESSION 53
(iii) The form of Maclaurin's series for complex variables is at once
derivable by supposing the centre of the circle at the origin. We then
infer that, if a function be kolomorphic over a circle, centre the origin, it can be
represented in the form of a series of ascending, positive, integral powers of the
variable given by
where the coefficients of the various powers of z are the values of the derivatives
of f(z} at the origin ; and the series converges for all points within the circle.
Thus, the function e? is holomorphic over the finite part of the plane ;
therefore its expansion is of the form G(z). The function log(l +2) has a
singularity at 1 ; hence within a circle, centre the origin and radius unity,
it can be expanded in the form of an ascending series of positive integral
powers of z, it being convenient to choose that one of the values of the
function which is zero at the origin. Again, tan" 1 z 2 has singularities at the
four points z* = 1, which lie on the same circumference : choosing the value
at the origin which is zero there, we have a similar expansion in a series,
converging for points within the circle.
Similarly for the function (1 + z) n , which has 1 for a singularity unless
n is a positive integer.
(iv) Darboux's method* of derivation of the expansion of f(z) in
positive powers of z a depends upon the expression, obtained in 15 (IV.),
for the value of an integral. When applied to the general term
/ 2 _ Q\ n+i
= L say, it gives L = \r ( ^- )
\i a>
where is some point on the circumference of the circle of radius r, and X is
Z ~ CL
a complex quantity of modulus not greater than unity. The modulus of -z
? ~~ a
is less than a quantity which is less than unity ; the terms of the series of
moduli are therefore less than the terms of a converging geometric progress-
ion, so that they form a converging series ; the limit of [ L \ , and therefore
of L, can, with indefinite increase of n, be made zero and Taylor's expansion
can be derived as before.
00
Ex. 1. Prove that the arithmetic mean of all values of z~ n 2 a v z v , for points lying along
v=0
a circle j z \ =r entirely contained in the region of continuity, is a n . (Rouche, Gutzmer.)
Prove also that the arithmetic mean of the squares of the moduli of all values of
00
2 a v z v , for points lying along a circle \ z \=r entirely contained in the region of continuity,
-=0
is equal to the sum of the squares of the moduli of the terms of the series for a point on
the circle. (Gutzmer.)
* Liouville, 3* me S6r., t. ii, (1876), pp. 291312.
LAURENT'S EXPANSION OF
[27.
Ex. 2. Prove that the function 2 a's" 2 ,
is finite and continuous, as well as all its derivatives, within and on the boundary of the
circle | z | = 1, provided | a j < 1. (Fredholm.)
Ex. 3. The radii of convergence of the series
are p and p ; prove that pp' is the radius of convergence of the series
Denoting the singularities of f(z) by 1} 2 ,..., and those of g (z 1 ) by /, s 2 ',..., prove
that the singularities of h (z") are given by s m s n ', for all values of m and . (Hadamard.)
Ex. 4. (See also Ex. 2, 20.) It is possible to express the sum of selected terms
in the form of a definite integral. Thus, writing
for i=l, 2,..., consider the finite series
t a
or
Ex. 5 Establish the following results in a similar manner :
f(t)
f(t}(t-a)
(i)
(ii)
(iii)
28. Laurent's theorem is as follows* :
A function, which is holomorphic in a part of the plane bounded by two
concentric circles with centre a and finite radii, can be expanded in the form
of a double series of integral powers, positive and negative, of z a ; and the
series converges in the part of the plane bettveen the circles.
* Comptes Rendus, t. xvii, (1843), p. 939.
.JL (_
2iri J (t-.
28.]
A FUNCTION IN SERIES
55
Let z be any point within the region bounded by the two circles of radii
R and R'\ describe two concentric circles of
radii r and r', such that
R > r > | z - a | > r > R'.
Denoting by t and by s current points on the
circumference of the outer and of the inner
circles respectively, and considering the space
which lies between them and includes the point
z, we have, by 20,
/w = *
2-Tn ] s z Fig. 11.
a negative sign being prefixed to the second integral because the direction
indicated in the figure is the negative direction for the description of the
inner circle regarded as a portion of the boundary.
Now we have
t a
t-z
z a
t a
z-a\- (z - aV* \t-aJ
t a) \t a) z-
J. ~
t-a) _
a
t-a
this expansion being adopted with a view to an infinite converging series,
because
t-z
z a
t a
dt =
is less than unity for all points t; and hence, by 15,
t a
1+1
dt
t z\t
Now each of the integrals, which are the respective coefficients of. powers of
z a, is finite, because the subject of integration is everywhere finite along
the circle of finite radius, by 15 (IV.). Let the value of
"*%**
be %7riu r : the quantity u r is not necessarily equal to f (r) (a) 4- r !, because no
knowledge of the function or of its derivatives is given for a point within
the innermost circle of radius R'. Thus
1
2tri jt z\t a
The modulus of the last term is less than
M (^ n+l
~v
r
+(z-a) n u n
dt.
56 LAURENT'S EXPANSION OF [28.
where p is \z a\ and M is the greatest value of \f(t)\ for points along
the circle. Because p < r, this quantity diminishes to zero with indefinite
increase of n ; and therefore the modulus of the expression
27rt J t z
becomes indefinitely small with unlimited increase of n. The quantity itself
therefore vanishes in the same circumstances ; and hence
1 rf(t) _
( [_ _ fit __ <\i I I iy __ ft\ it I I / ty .^ ft \m -jt I
C> * I * " ^> ' ^ N ' "* '
so that the first of the integrals is equal to a series of positive powers. This
series converges within the outer circle, for the modulus of the (m + l) th term
is less than
which is the (m + l) th term of a converging series.
As in 27, the equivalence of the integral and the series can be affirmed
only for points which lie within the outermost circle of radius R.
Again, we have
fs a\ n
1 - I
\z a/
z a
r ..... + 1 - I +
s z z a \z a/ s a
z a
this expansion being adopted with a view to an infinite converging series,
because
s a
z a
is less than unity for all points s. Hence
+ ^-. If -) J-^ds.
a/ z s
The modulus of the last term is less than
M' / r '\ n+ '
P
where M' is the greatest value of \f(s) \ for points along the circle of radius
r. With unlimited increase of n, the modulus of this last term is ultimately
zero ; and thus, by an argument similar to the one which was applied to the
former integral, we have
z-a
where v m denotes the integral J (s - a) in - 1 f(s) ds taken round the circle.
28.] A FUNCTION IN SERIES 57
As in the former case, the series is one which converges, its converg-
ence being without the inner circle ; the equivalence of the integral and
the series is valid only for points z that lie without the innermost circle of
radius R '.
[fo o:
The coefficients of the various negative powers of z a are of the form
(s - a)
a form that suggests values of the derivatives of f(s) at the point given by
= 0, that is, at infinity. But the outermost circle is of finite radius ;
s a
and no knowledge of the function at infinity, lying without the circle, is
given, so that the coefficients of the negative powers may not be assumed
to be the values of the derivatives at infinity, just as, in the former case, the
coefficients u r could not be assumed to be the values of the derivatives at the
common centre of the circles.
Combining the expressions obtained for the two integrals, we have
f(z) = u + (z-a) w, + (z - a) 2 u 2 + , . .
Both parts of the double series converge for all points in the region between
the two circles, though not necessarily for points on the boundary of the
region. The whole series therefore converges for all those points : and we
infer the theorem as enunciated.
Conformably with the notation ( 26, Note) adopted to represent Taylor's
expansion, a function f (z) 'of the character required by Laurent's Theorem
can be represented in the form
the series P 1 converging within the outer circle and the series P 2 converging
without the inner circle ; their sum converges for the ring-space between the
circles.
29. The coefficient u in the foregoing expansion is
the integral being taken round the circle of radius r. We have
JL-M
t a
58 LAURENT'S THEOREM [29.
for points on the circle ; and therefore
so that j w 1 < I M t < M',
M' being the greatest value of M t , the modulus of f(f), for points along the
circle. If M be the greatest value of |/(^)i for any point in the whole
region in which f(z) is defined, so that M' ^ M, then we have
| tt 1 < M,
that is, the modulus of the term independent of z a in the expansion of
f(z) by Laurent's Theorem is less than the greatest value of \f(z) \ at points
in the region in which it is defined.
Again, (z a)~ m f(z) is a double series in positive and negative powers of
z a. the term independent of z a being n m ; hence, by what has just been
proved, \u m \ is less than p~ m M, where p is z a j. But the coefficient u m
does not involve z, and for any point z we can therefore choose a limit. The
lowest limit will evidently be given by taking z on the outer circle of radius
R, so that | u m | < MR". Similarly for each coefficient v m ; and therefore we
have the result :
If f(z) be expanded as by Laurent's Theorem in the form
00 00
MO+ 2 (z-a) m u m + 2 (z-a)- m v m ,
Mt = l 7tt=l
then | u m < MR~ m , \v m \< MR' m ,
where M is the greatest value of \f(z) \ at points within the region in which
f(z) is defined, and R and R are the radii of the outer and the inner circles
respectively.
COROLLARY. If M(r) denote the greatest value of \f(z) \ for values of z
on the circumference of the circle \z a =r, then
I Mm I < r~ m M (r), | v m \ < r m M (r) :
which may be lower limits than the preceding. As above, we have
W =
taken round the circle | z a \ = r ; so that
Similarly, as u m is the term independent of z a in the Laurent expansion
of (z a)~ m f(z\ we have
| m I < greatest value of | (z a)~ m f(z) \ along | z a \ = r
^r" M (r);
and so for v m .
80.] EXPANSION IN NEGATIVE POWERS 59
30. The following proposition is practically a corollary from Laurent's
Theorem :
When a function is holomorphic over all the plane which lies outside a
circle of centre a, it can be expanded in the form of a series of negative integral
powers of z a, the series converging everywhere in that part of the plane.
It can be deduced as the limiting case of Laurent's Theorem when the
radius of the outer circle is made infinite. We then take r infinitely large,
and substitute for t by the relation
t a = re ei ,
so that the first integral in the expression (i), p. 55, forf(z) is
2?rJo t z
t a
Since the function is holomorphic over the whole of the plane which lies
outside the assigned circle, f(t) cannot be infinite at the circle of radius r
when that radius increases indefinitely. If f(t) tend towards a (finite)
limit k, which must be uniform owing to the hypothesis as to the functional
character of/ (z), then, since the limit of (t - z)/(t a) is unity, the preceding
integral is equal to k.
The second integral in the same expression (i), p. 55, for f(z) is
unaltered by the conditions of the present proposition; hence we have
the series converging without the circle, though it does not necessarily
converge on the circumference.
The series can be represented in the form
conformably with the notation of 26.
Of the three theorems in expansion which have been obtained, Cauchy's
is the most definite, because the coefficients of the powers are explicitly
obtained as values of the function and of its derivatives at an assigned
point. In Laurent's theorem, the coefficients are not evaluated into simple
expressions. In the corollary from Laurent's theorem the coefficients are,
as is easily proved, the values of the function and of its derivatives for infinite
values of the variable. The essentially important feature of all the theorems
is the expansibility of the function in converging series under assigned
conditions.
31. It was proved (21) that, when a function is holomorphic in any
region of the plane bounded by a simple curve, it has an unlimited number
of successive derivatives each of which is holomorphic in the region. Hence,
60 EXPANSION OF FUNCTIONS [31.
by the preceding propositions, each such derivative can be expanded in
converging series of integral powers, the series themselves being deducible
by differentiation from the series which represents the function in the region.
In particular, when the region is a finite circle of centre a, within which
f(z) and consequently all the derivatives off(z) are expansible in converging
series of positive integral powers of z a, the coefficients of the various
powers of z a are save as to numerical factors the values of the
derivatives at the centre of the circle. Hence it appears that, when a function
is holomorphic over the area of a given circle, the values of the function and all
its derivatives at any point z within the circle depend only upon the variable
of the point and upon the values of the function and its derivatives at the
centre.
32. Some of the classes of points in a plane that usually arise in
connection with uniform functions may now be considered.
(i) A point a in the plane may be such that a function of the variable
has a determinate finite value there, always independent of the path by
which the variable reaches a; the point a is called an ordinary point* of
the function. The function, supposed continuous in the vicinity of a, is
continuous at a: and it is said to behave regularly in the vicinity of an
ordinary point.
Let such an ordinary point a be at a distance d, not infinitesimal, from
the nearest of the singular points (if any) of the function ; and let a circle of
centre a and radius just less than d be drawn. The part of the .z-plane lying
within this circle is called f the domain of a ; and the function, holomorphic
within this circle, is said to behave regularly (or to be regular) in the domain
of a. From the preceding section, we infer that a function and its derivatives
can be expanded in a converging series of positive integral powers of z a
for all points z in the domain of a, an ordinary point of the function : and
the coefficients in the series are the values of the function and of its derivatives
at a.
The property possessed by the series that it contains only positive
integral powers of z a at once gives a test which is both necessary and
sufficient to determine whether a point is an ordinary point. If the point a
be ordinary, the limit of (z a}f(z) necessarily is zero when z becomes equal
to a. This necessary condition is also sufficient to ensure that the point is
an ordinary point of the function f(z), supposed to oe uniform ; for, since
f(z) is holomorphic, the function (z-a)f(z) is also holomorphic and can be
expanded in a series
MO + u l (z a) + u 2 (z a) 2 + . . . ,
* Sometimes a regular point.
t The German title is Umgebung, the French is domaine.
32.] CLASSES OF POINTS DEFINED 61
converging in the domain of a. The quantity u is zero, being the value
of (z a)f(z) at a and this vanishes by hypothesis ; hence
(z - a)/ (2) = (z - a) { uj, + u 2 (z - a) + . . . },
shewing that f(z) is expressible as a series of positive integral powers of
z a converging within the domain of a, or, in other words, that/(^) certainly
has a for an ordinary point in consequence of the condition being satisfied.
(ii) A point a in the plane may be such that a function f(z) of the
variable has a determinate infinite value there, always independent of the
path by which ^the variable reaches a, the function behaving regularly for
points in the vicinity of a ; then -^-^ has a determinate zero value there, so
J\ z )
that a is an ordinary point of ^^r- The point a is called a pole ( 12)
or an accidental singularity 1 * of the function.
A test, necessary and sufficient to settle whether a point is a pole of
a function, will subsequently ( 42) be given.
(iii) A point a in the plane may be such that f(z) has not a determinate
value there, either finite or infinite, though the function is definite in value
at all points in the immediate vicinity of a other than a itself.
Such a point is called j~ an essential singularity of the function. No
hypothesis is postulated as to the character of the function for points
at infinitesimal distances from the essential singularity, while the relation
of the singularity to the function naturally depends upon this character at
points near it. There may thus be various kinds of essential singularities
all included under the foregoing definition, even for uniform functions ;
one classification is effected through the consideration of the character of
the function at points in their immediate vicinity. (See 88.)
One sufficient test of discrimination between an accidental singularity
and an essential singularity is furnished by the determinateness of the value
at the point. If the reciprocal of the function have the point for an ordinary
point, the point is an accidental singularity it is, indeed, a zero for the
reciprocal. But when the point is an essential singularity, the value of the
reciprocal of the function is not determinate there ; and then the reciprocal,
as well as the function, has the point for an essential singularity.
In these statements and explanations, it is assumed that the essential
singularity is an isolated point. It will hereafter be seen that uniform
functions can be constructed for which this is not the case ; thus there are
uniform functions which have lines of essential singularity. For the present,
we shall deal only with essential singularities that are isolated points.
* Weierstrass, Ges. Werke, t. ii, p, 78, to whom the name is due, calls it aussenvesentliche
singuldre Stelle ; the term non-essential is suggested by Mr Cathcart, Harnack, p. 148.
t Weierstrass calls it wesentliche singuldre Stelle.
62 EXAMPLES [32.
Ex. 1. Consider the function cos - in the vicinity of the origin.
The value at s=0 clearly is indeterminate; but it tends to limits that depend upon
the mode by which z approaches the origin.
Thus suppose that z approaches the origin along the axis of imaginary quantities ; and
let 2= ai, where a is real and can be made as small as we please. Then
1 l - - l -
COS - = +<? ;
if a be positive then the first term, and if a be negative then the second term, can be
made larger than any assigned finite quantity by sufficiently diminishing a : that is,
by these methods of approach of z to its origin, the function cos - ultimately acquires an
infinite value.
Next suppose that z approaches the origin along the axis of real quantities, and
assume it to have positive values, (the same reasoning applies if it has negative values) ;
in particular, consider real values of z, such that ^ z ^ , where /3 is a quantity that may
be assigned as small as we please. When /3 is assigned, take any positive integer m, such
that
ftir
so that m will be any integer lying between some one integer (that will be large, in
dependence upon the value of /3) and infinity. Let
j=(2m + l)| + f,
where f is a positive quantity such that ^ ^ it ; then
1 1
1*1-
and so < z < /3. For such values we have
cos - = ( l) m-1 sin ,
and therefore with the range of f from to rr, the function ranges continuously in
numerical value between and 1. In particular, when f=0, the function has a zero
value; (also when f=7r, but this in effect gives the next greater value of m); and this
holds for each of the integers tn so assumed. Hence it follows that within the range
^ z ^ for real values of 2, no matter how small the real quantity /3 may be assigned,
the function cos - has an unlimited number of zeros ; also that, within the same range,
Z
the function cos < (where * is a real quantity not greater than unity) has an unlimited
number of zeros.
Ex. 2. Consider the function cos - in the vicinity of the origin, when the variable z is
Z
made to approach the origin along the spiral 0=pr, where z re e *, and p is a parametric
quantity ; and shew that, in the immediate vicinity of the origin along this path,
sinh p. ^
cosh 2/x.
s
Discuss the possibility of so choosing the approach of z to the origin as, for values of z
such that | z \ < y where y is a quantity that may be made as small as we please, to
make cos - acquire a value A + iB.
32.] EXAMPLES 63
Ex. 3. Shew that the function cosec - has an unlimited number of poles in the
z
immediate vicinity of its essential singularity z=0.
1
Ex. 4. Consider the variations in value of the function e* for values of z, such that
| z | is not greater than some assigned small quantity K.
l
In particular, consider the possibility of e* either acquiring, or tending to, any assigned
l
value A. The values of z for which e z A are given by
1
- = 2&7Ti+lOg A,
z
where k is any integer, positive or negative. Let A = ae al , where a and a are real ;
so that
1
- = (2/for + a) i + log a.
z
If z=x + iy as usual, then
_
and therefore all the points, for which e acquires the value A, lie upon the circle
Accordingly, we consider an arc of this circle which lies within the circle
Not every point on the arc leads to the value A of e z for taking any point (, 77 )
on it, let
Moga
where m is an integer, and ^ Q < 2ir ; thus
l
so that the value of ez is l <W<-*(*> Mr +*> 1 = a<T w , which is only the same as ae ai for
l_
particular points. It is however clear that e* is the same for all points on the circular
arc.
1
The values of z for which e z = A are given by
'~
where k is an integer. It is manifest that a value of k (say k^) can be chosen for which
\Z\<K,
I
this inequality holding for all values of k greater than k^ : so that the function e" acquires
the value A at an unlimited number of points in the region | z \ < K. Further, by
sufficiently increasing k, we can make | z \ smaller than any assigned quantity however
l^
small; and therefore A is one of the (unlimited number of) values of e" as z ultimately
becomes zero.
64 UNIFORM FUNCTIONS AT AND NEAR [32.
It may be remarked at once that there must be at least one infinite-
value among the values which a uniform function can assume at an essential
singularity. For iff(z) cannot be infinite at a, then the limit of (z d)f(z)
would be zero when z = a, no matter what the non-infinite values of f(z)
may be, and no matter by what path z acquires the value a ; that is, the
limit would be a determinate zero. The function (z a)f(z) is regular in
the vicinity of a : hence by the foregoing test for an ordinary point, the point
a would be ordinary and the value of the uniform function f(z) would be
determinate, contrary to hypothesis. Hence the function must have at least
one infinite value at an essential singularity.
Further, a uniform function must be capable of assuming any value C
at an essential singularity. For an .essential singularity of f(z) is also an
essential singularity of f(z) C and therefore also of ^/ r ^ . The last
J \ )
function must have at least one infinite value among the values that it
can assume at the point; and, for this infinite value, we have f(z) = C
at the point, so that f(z} assumes the assigned value C at the essential
singularity.
Note. This result, that a uniform function can acquire any assigned
value at an isolated essential singularity, is so contrary to the general idea of
the one-valuedness of the function, that the function is often regarded as not
existing at the point ; and the point then is regarded as not belonging to the
region of significance of the function. The difference between the two views
is largely a matter of definition, and depends upon the difference between
two modes of considering the variable z. If no account is allowed to be
taken of the mode by which z approaches its value at an essential singularity
a, the function does not tend uniformly to any one value there. If such
account is allowed, then it can happen (as in Ex. 4, above) that z may
approach the value a along a particular path through a limiting series of
values in such a way that the function can acquire any assigned value in the
limit when z coincides with a after the specified mode of approach.
33. There is one important property possessed by every uniform funct-
ion in the immediate vicinity of any of its isolated essential singularities ;
it was first stated by Weierstrass*, as follows : In the immediate vicinity of
an isolated essential singularity of a uniform function, there are positions a,t
which the function differs from an assigned value by a quantity not greater
than a non-vanishing magnitude that can be made as small as we please.
* Weierstrass, Ges. Werke, t. ii, pp. 122 124; Durege, Elemente der Theorie der Funktionen,
p. 119 ;: Holder, Math. Ann., t. xx, (1882), pp. 188143; Picard, " Memoire sur les fonctions
entieres," Annales de I'Ecole Norm. Sup., 2""= Ser., t. ix, (188U), pp. 145 166, which, in this
repaid, shouM be consulted in connection with the developments in Chapter V. See also 62.
Picard's proof is followed in the text.
33.]
AN ESSENTIAL SINGULARITY
Let a be the singularity, G an assigned value, and e a non-vanishing
magnitude which can be chosen arbitrarily small at our own disposal ; and in
the vicinity of a, represented by
\z a\< p,
consider the function ^y ~. For values of z in the range
j (z) Li
0< \z a | < p,
this function may have poles, or it may not.
If it has poles, then at each of them f(z) = 0: that is, the function
f(z) actually attains the value C, so that the difference between f(z) and C
for such positions is not merely less than e, it actually is zero.
If it has no poles, then the function
1
is regular everywhere through the domain
< | z a < p,
because no point in that domain is either a pole or an essential singularity.
Accordingly, by Laurent's theorem, it can be expanded in that domain in a
converging series of positive and negative powers, in the form
77-T r, = u o + ( z ~ a ) u i + + (z-a) n u n +
z - a
'
i
T
,
T
(z - a) 2 (*.-)"
Choose a quantity p such that < p' < p. The series of positive powers
converges everywhere within and on a circle, centre a and radius p' : let 8(2)
denote its value at z. The series of negative powers converges everywhere
in the plane outside the point a ; and therefore the series
i ~ - T~-^ ......
z a (z a-) 2
converges everywhere outside the point a : let T(z) denote its value, so that
z-a
Accordingly, as \S(z}\ is finite and |T(^)| not zero it may be a rapidly
increasing quantity as | z a \ decreases choose \za\ so that, while not
being zero, it gives the modulus of the right-hand side as greater than - .
As z a occurs in a denominator, this can be done. Then, for such a value
of z,
and therefore
which proves the theorem.
F. F.
/(*).-
|/(*)- (7 1
e,
66 CONTINUATIONS OF A FUNCTION [33.
It may happen that the function attains the value C only at the essential singularity,
where C is one of its unlimited number of values. Thus to find the zeros of the function
cosec in the vicinity of the origin, we must have sin - infinite at them ; this can only
z z
occur when z becomes zero along the axis of imaginaries, and cannot occur for any value
of 2 such that \z\ > 0. Such a value is called an exceptional value ; the discussion of
exceptional values is effected by Picard in his memoir quoted.
Ex. Discuss the character of the functions cos (1/2), tan (1/2) for values of j z \ which
]_
are very small ; and the character of the functions tan 2, e?, z~*e*, e z , z log 2, for values
of 1 2 | which are very large.
34. Let f(z} denote the function represented by a series of powers
PI(Z a), the circle of convergence of which is the domain of the ordinary
point a, and the coefficients in which are the values of the derivatives of
f(z) at a. The region over which the function f(z) is holomorphic may
extend beyond the domain of a. although the circumference bounding that
domain is the greatest of centre a that can be drawn within the region.
The region evidently cannot extend beyond the domain of a in all directions.
Take an ordinary point b in the domain of a. The value at b of the
function f(z) is given by the series P x (b a), and the values at b of all its
derivatives are given by the derived series. All these series converge within
the domain of a and they are therefore finite at b ; and their expressions
involve the values at a of the function and its derivatives.
Let the domain of b be formed. The domain of b may be included in
that of a, and then its bounding circle will touch the bounding circle of the
domain of a internally. If the domain of b be not entirely included in that
of a, part of it will lie outside the domain of a ; but it cannot include the
whole of the domain of a unless its bounding circumference touch that of
the domain of a externally, for otherwise it would extend beyond a in all
directions, a result inconsistent with the construction of the domain of a.
Hence there must be points excluded from the domain of a which are also
excluded from the domain of b.
For all points z in the domain of b, the function can be represented by
a series, say P 2 (zb), the coefficients of which are the values at b of the
function and its derivatives. Since these values are partially dependent
upon the corresponding values at a, the series representing the function may
be denoted by P 2 (z b, a).
At a point z in the domain of b lying also in the domain of a, the two
series P! (z a) and P 2 (? b, a) must furnish the same value for the
function f(z) ; and therefore no new value is derived from the new series P 2
which cannot be derived from the old series P 1 . For all such points the new
series is of no advantage ; and hence, if the domain of b be included in that
of a, the construction of the series P 2 (z b,a) is superfluous. Thus, in
choosing the ordinary point b in the domain of a we choose a point, if
possible, that will not have its domain included in that of a.
34.]
OVER ITS REGION OF CONTINUITY
67
At a point z in the domain of 6, which does not lie in the domain of a,
the series P z (z b,a) gives a value for f(z) which cannot be given by
PI (2- a). The new series P 2 then gives an additional representation of the
function ; it is called * a continuation of the series which represents the function
in the domain of a. The derivatives of P 2 give the values of the derivatives
of f(z) for points in the domain of b.
It thus appears that, if the whole of the domain of b be not included in
that of a, the function can, by the series which is valid over the whole
of the new domain, be continued into that part of the new domain excluded
from the domain of a.
Now take a point c within the region occupied by the combined domains
of a and b ; and construct the domain of c. In the new domain, the
function can be represented by a new series, say P 3 (z c), or, since the
coefficients (being the values at c of the function and of its derivatives)
involve the values at a and possibly also the values at b of the function*
and of its derivatives, the series representing the function may be denoted
by P 3 (z c, a, b). Unless the domain of c include points, which are not
included in the combined domains of a and b, the series P 3 does not give
a value of the function which cannot be given by Pj or P 2 : we therefore
choose c, if possible, so that its domain will include points not included in
the earlier domains. At such points z in the domain of c as are excluded
from the combined domains of a and b, the series P 3 (z c, a, b) gives a value
for f(z) which cannot be derived from P x or P 2 ; and thus the new series
is a continuation of the earlier series.
Proceeding in this manner by taking successive points and constructing
their domains, we can reach all parts of the plane connected with one
another where the function preserves its holomorphic character; their
combined aggregate is called -f- the region of continuity of the function.
With each domain, constructed so as to include some portion of the region of
continuity not included in the earlier domains, a series is associated, which is
a continuation of the earlier series and, as such, gives a value of the function
not deducible from those earlier series ; and all the associated series are
ultimately deduced from the first.
Each of the continuations is called an Element of the function. The
aggregate of all the distinct elements is called a monogenic analytic function :
it is evidently the complete analytical expression of the function in its region
of continuity.
Let z be any point in the region of continuity, not necessarily in the
circle of convergence of the initial element of the function ; a value of the
* Biermann, Theorie der analytischen Functionen, p. 170, which may be consulted in
connection with the whole of 34; the German word is Fortsetzung.
t Weierstrass, Ges. Werke, t. ii, p. 77.
52
68 EEGION OF CONTINUITY OF [34.
function at z can be obtained through the continuations of that initial
element. In the formation of each new domain (and therefore of each new
element) a certain amount of arbitrary choice is possible ; and there may,
moreover, be different sets of domains which, taken together in a set, each
lead to z from the initial point. When the analytic function is uniform, as
before defined ( 12), the same value at z for the function is obtained,
whatever be the set of domains. If there be two sets of elements, differently
obtained, which give at z different values for the function, then the ana-
lytic function is multiform, as before defined ( 12); but not every change
in a set of elements leads to a change in the value at z of a multiform
function, and the analytic function is uniform within such a region of the
plane as admits only equivalent changes of elements.
Th6 whole process is reversible when the function is uniform. We can
pass back from any point to any earlier point by the use, if necessary, of
intermediate points. Thus, if the point a in the foregoing explanation
be not included in the domain of 6 (there supposed to contribute a continu-
ation of the first series), an intermediate point on a line, drawn in the
region of continuity so as to join a and b, would be taken ; and so on,
until a domain is formed which does include a. The continuation, associated
with this domain, must give at a the proper value for the function and its
derivatives, and therefore for the domain of a the original series Pj (z a)
will be obtained, that is, P l {z a) can be deduced from P 2 (z b, a) the
series in the domain of b. This result is general, so that any one of the
continuations of a uniform function, represented by a power-series, can be
deduced from any other ; and therefore the expression of such a function in
its region of continuity is potentially given by one element, for all the
distinct elements can be deduced from any one element.
35. It has been assumed that the property, characteristic of some of the
uniform functions adduced as examples, of possessing either accidental or
essential singularities, is characteristic of all such functions; it will be proved
( 40) to hold for every uniform function which is not a mere constant.
The singularities limit the region of continuity ; for each of the separate
domains is, from its construction, limited by the nearest singularity, and the
combined aggregate of the domains constitutes the region of continuity when
they form a continuous space*. Hence the complete boundary of the region
of continuity is the aggregate of the singularities of the function f.
* Cases occur in which the region of continuity of a function is composed of isolated spaces,
each continuous in itself, but not continuous into one another. The consideration of such cases
will be dealt with briefly hereafter, an I they are assumed excluded for the present : meanwhile,
it is sufficient to note that each continuous apace could be deduced from an element belonging to
some domain of that space and that a new element would be needed for a new space.
t See Weierstrass, Ges. Werke, t. ii, pp. 7779; Mittag-Leffler, "Sur la representation analy-
tique des fonctions monogenes uniformes d'une variable independante, " Acta Math., t. iv, (1884),
pp. 1 et seq., especially pp. 1 8.
35.] AN ANALYTIC FUNCTION 69
It may happen that a function has no singularity except at infinity ; the
region of continuity then extends over the whole finite part of the plane but
it does not include the point at infinity.
It follows from the foregoing explanations that, in order to know a
uniform analytic function, it is necessary to know some element of the
function, which has been shewn to be potentially sufficient for the derivation
of the full expression of the function and for the construction of its region of
continuity. But the process of continuation is mainly descriptive of the
analytic function, and in actual practice it can prove too elaborate to be
effected*.
To avoid the continuation process, Mittag-Leffler has de vised f another
method of representing a uniform function. Let a be an ordinary point of
the function, and let a line, terminated at a, rotate round it. In the vicinity
of a, let the element of the function be denoted by P (z a) ; and imagine
the continuation of this element to be effected along the vector as far as
possible. It may happen that the continuation can be effected to infinity
along the vector; if not, there is some point a' on the vector beyond which
the continuation is impossible. In the latter case, the part of the vector J
from a to infinity is excluded from the range of variation of the variable.
Let this be done for every position of the vector ; then the part of the plane,
which remains after these various ranges have been excluded, gives a star-
shaped figure, which is a region of continuity of the uniform function of
which P(z d) is the initial element. The function manifestly can be
continued over the whole of this star, by means of appropriate elements ; but
there is no indication as to the necessary number of elements. Instead of
using the elements to express the function, Mittag-Leffler constructs a single
expression, which is the valid representation of the function over the whole
star; the expression is an infinite series of polynomials, and not merely a
power-series.
Thus let there be a power-series
&o + &i ( z ~ + si ^2 ( z a ) + ;n "3 \ z ~ a ) + >
2 ! 6 \
which converges uniformly in a region round the point a ; the radius of convergence of the
series is r, where l/r is the upper limit of the quantities (b n /n !). Let the star-shaped
figure be constructed; the following is the simplest form of expression as obtained by
Mittag-Leffler to represent, over the whole star, the function of which the foregoing series
is an element. Let the quantity
2
A.,=O A 2 =o" A P =O AI! X 2 !...A P ! \ p
* Some examples have been constructed by Prof. M. J. M. Hill, Proc. Land. Math. Soc.,
vol. xxxv, (1903), pp. 388416.
t Exact references are given at the beginning of Chapter VII.
In effect, this is a section, in the sense used in 1Q3.
70 CONTINUATION OF ANALYTIC FUNCTION
which is a polynomial, be denoted by g v (z) ; and take
C/" n ( Zj := g n (z) o n _ j v^/j lor z = 1 , AJ
Mittag-Leffler's expression is
[35.
and it converges everywhere within the star.
Again, an element representing a function is effective only within its own
circle of convergence, while it may be known that the function is holomorphic
over some closed domain which touches the circle of convergence externally.
The process of continuation would make it possible to obtain the analytical
representation over the whole domain by means of appropriate elements :
but again there is no indication as to the necessary number of elements.
Painleve* has shewn how to construct a single expression, which is the valid
representation of the function over the whole domain ; this expression also is
an infinite series of polynomials, and not merely a power-series.
For the establishment of these results, we refer to the memoirs quoted.
36. The method of continuation of a function, by means of successive
elements, is quite general; there is one particular continuation, which is
important in investigations on conformal representation. It is contained in
the following proposition, due to Schwarzf:
If an analytic function w of z be defined only for a region S' in the
positive half of the z-plane, and if continuous real values of w correspond to
continuous real values of z, then w can be continued across the axis of real
quantities.
Consider a region 8", symmetrical with 8' relative to the axis of real
quantities (fig. 12). Then a function is
defined for the region 8" by associating
a value w , the conjugate of w, with z ,
the conjugate of z.
Let the two regions be combined
along the portion of the axis of ac which
is their common boundary; they then
form a single region S' + S".
Consider the integrals
Fig. 12.
I [ w 1 [ W Q
~ . / T, dz and ^ : ..
Z-mJB'Z-t 2mJ S ',z -
,
az
taken round the boundaries of S' and of S" respectively. Since w is
continuous over the whole area of S' as well as along its boundary, and
* Comptes Rendus, t. cxxvi, (1898), pp. 320, 321; see also the references to Painlev at
the beginning of Chapter VIL
t Crelle, t. Ixx, (1869), pp. 106, 107, and Ges. Math. Abh., t. ii, pp. 6668. See also Darboux,
Theorie generate des surfaces, t. i, 130.
36.] DUE TO SCHWABZ 71
likewise w relative to S", it follows that, if the point f be in S', the value of
the first integral is w () and that of the second is zero ; while, if f be in S",
the value of the first integral is zero and that of the second is W Q (). Hence
the sum of the two integrals represents a unique function of a point in either
S' or S". But the value of the first integral is
1 f A wdz 1 [ B w (x) dx
B z $ 2-rnJ
x ~ b
the first being taken along the curve EGA and the second along the axis
AxB ; and the value of the second integral is
w dz
the first being taken along the axis BxA and the second along the curve
ADB. But
w (x) = w (x\
because conjugate values w and w correspond to conjugate values of the
argument by definition of w , and because w (and therefore also w ) is real
and continuous when the argument is real and continuous. Hence when the
sum of the four integrals is taken, the two integrals corresponding to the
two descriptions of the axis of x cancel ; we have as the sum
If A /J/l/y ^ r Z* /i/| /y iy
Wi.l& J. M/Q \JV&Q
and this sum represents a unique function of a point in S' + S". These two
integrals, taken together, are
1 fw'dz
27Ttl Z-V
taken round the whole contour of S' + S", where w' is equal to w() in the
positive half of the plane and to w () in the negative half.
For all points in the whole region S' + S", this integral represents a
single uniform, finite, continuous function of ; its value is w() in the
positive half of the plane and is w () in the negative half; and therefore
w () is the continuation, into the negative half of the plane, of the function
which is defined by w() for the positive half.
For a point c on the axis of x, we have
and all the coefficients A, B, C, ... are real. If, in addition, w be such
.a function of z that the inverse functional relation makes z a uniform
analytic function of w, obviously A must not vanish. Thus the functional
relation may be expressed in the form
w(z)-w (c) = (z c)P(z c),
where P (z c) does not vanish when z = c.
CHAPTER IV.
GENERAL PROPERTIES OF UNIFORM FUNCTIONS, PARTICULARLY OF THOSE
WITHOUT ESSENTIAL SINGULARITIES.
37. IN the derivation of the general properties of functions, which will be
deduced in the present and the next three chapters from the results already
obtained, it is to be supposed, in the absence of any express statement to
other effect, that the functions are uniform, monogenic and, except at -either
accidental or essential singularities, continuous*.
THEOREM I. A function, which is constant throughout any region of the
plane however small, or which is constant along any line however short, is
constant throughout its region of continuity.
For the first part of the theorem, we take any point a in the region of the
plane where the function is constant ; and we draw a circle of centre a and
of any radius, taking care that the circle remains within the region of
continuity of the function. At any point z within this circle, we have
/(*) =f(a) + (z- a)/' (a) + ( -f" () + .. .,
a converging series the coefficients of which are the values of the function
and its derivatives at a. Let a point a + Sa be taken in the region ; then
>//; T ., f f(
f (a) = Limit of -
K- - - ,
which is zero because f(a + Ba) is the same constant as f(a}: so that the
first derivative is zero at a. Similarly, all the derivatives can be shewn to
be zero at a; hence the above series after its first term is evanescent,
and we have
/(*)-/(),
that is, the function preserves its constant value throughout its region of
continuity.
The second result follows in the same way, when once the derivatives are
proved zero. Since the function is monogenic, the value of the first and
* It will be assumed, as in 35 (note, p. 68), that the region of continuity consists of a single
space. Functions, which exist in regions of continuity consisting of a number of separated
spaces, will be discussed in Chap. VII.
37.] ZEROS OF A UNIFORM FUNCTION 73
of each of the successive derivatives will be obtained, if we make the
differential element of the independent variable vanish along the line.
Now, if a be a point on the line and a + Sa a consecutive point, we have
f(a + Sa) =f(a) ; hence/' (a) is zero. Similarly the first derivative at any
other point on the line is zero. Therefore we have /' (a 4- 8a) =/' (a), for
each has just been proved to be zero : hence /" (a) is zero. Similarly the
value of the second derivative at any other point on the line is zero. So on
for all the derivatives : the value of each of them at a is zero.
Using the same expansion as before and inserting again the zero values
of all the derivatives at a, we find that
/<*)-/<*),
so that under the assigned condition the function preserves its constant value
throughout its region of continuity.
It should be noted that, if in the first case the area and in the second the
line reduce to a point, then consecutive points cannot be taken ; the values
at a of the derivatives cannot be proved to be zero and the theorem cannot
then be inferred.
COROLLARY I. If two functions have the same value over any area of
their common region of continuity however small or along any line in that
region however short, then they have the same values at all points in their
common region of continuity.
This is at once evident : for their difference is zero over that area or along
that line and therefore, by the preceding theorem, their difference has a
constant zero value, that is, the functions have the same values, everywhere
in their common region of continuity.
But two functions can have the same values at a succession of isolated
points, without having the same values everywhere in their common region
of continuity ; in such a case the theorem does not apply, the reason being
that the fundamental condition of equality over a continuous area or along
a continuous line is not satisfied.
COROLLARY II. A function cannot be zero over any area of its region
of continuity however small, or along any line in that region however short,
without being zero everywhere in its region of continuity.
It is deduced in the same manner as the preceding corollary.
If, then, there be a function which is evidently not zero everywhere, we
conclude that its zeros are isolated points though such points may be multiple
zeros.
Further, in any finite area of the region of continuity of a function that is
subject to variation, there can be at most only a finite number of its zeros, when
74 ZEROS OF A [37.
no point of the boundary of the area is an essential singularity. For if there
were an infinite number of such points in any such region, there must be a
cluster in at least one area or a succession along at least one line, infinite in
number. Either they must then constitute a continuous area or a continuous
line where the function is everywhere zero : which would require that the
function should be zero everywhere in its region of continuity, a condition
excluded by the hypothesis. Or they must be so close to some point, say c,
that the function has an unlimited number of zeros within a, region
| Z C | < 6,
where e can be made as small as we please : and so for non-zero values of the
function. After the general properties which have been established, and
the proposition of 33, it is clear that c is an essential singularity of the
function, contrary to the hypothesis as to the region of continuity of the
function.
It immediately follows that the points within a region of continuity,
at which a function assumes any the same value, are isolated points ; and
that only a finite number of such points occur in any finite area.
This result may be established in another way.
Let f(z) be a uniform monogenic function ; we proceed to shew that,
when /(a) is not zero, we can choose a region round a in which f(z) nowhere
vanishes. We have
/ (z) = a + a l (z - a) + a 2 (z - a) 2 + . . . ,
where a is not zero, the series for f(z) converging absolutely and uniformly
for values of z such that
| z a | ^ r < R.
Within or on the circle r, let M be the greatest value of
\a l + a?(z-a} + ... \,
so that M is, of course, finite. Let
J a 1 = Ms,
so that s is finite ; and take values of z such that
z a
Then
|/(.z)i^|ao \z a
^ a <rM
a- < s.
so that, at no place within this region can/ (z) vanish.
Now let c be a zero off(z) of order n, so that
37.] UNIFORM FUNCTION 75
where g (c) is not zero and g (z) is uniform and monogenic. By what has just
been proved, we can choose a region round c such that g (z) has no zero within
it. Then obviously f(z) has no zero within that region except at the place c ;
in other words, the zero off(z) is an isolated point.
38. THEOREM II. The multiplicity m of any zero a of a, function is
finite provided the zero be an ordinary point of the function, supposed not to be
zero throughout its region of continuity ; and the function can be expressed in
the form
(*-o)0(*),
where (f> (z) is holomorphic in the vicinity of a, and a is not a zero of <f> (z).
Let f(z) denote the function ; since a is a zero, we have f(a) = 0.
Suppose that /' (a), f" (a), ....".. vanish : in the succession of the derivatives
of f, one of finite order must be reached which does not have a zero value.
Otherwise, if all vanish, then the function and all its derivatives would
vanish at a; the expansion of f(z) in powers of z a would lead to zero as
the value of f(z), that is, the function would everywhere be zero in the
region of continuity, if all the derivatives vanish at a.
Let, then, the rath derivative be the first in the natural succession which
does not vanish at a, so that m is finite. Using Cauchy's expansion, we have
m ,. ni
f(z) =
ml J (m+ 1)1
where <f> (z) is a function that does not vanish with a and, being the quotient
of a converging series by a monomial factor, is holomorphic in the immediate
vicinity of a.
COROLLARY I. If infinity be a zero of a function of multiplicity m and
at the same time be an ordinary point of the function, then the function can be
expressed in the form
where </> ( -) is a function that is continuous and different from zero for infinitely
\zJ
large values of z.
The result can be derived from the expansion in 30 in the same way as
the foregoing theorem from Cauchy's expansion.
COROLLARY II. The number of zeros of a function, account being taken of
their multiplicity, which occur within a finite area of the region of continuity
of the function, is finite, when no point of the boundary of the area is an
essential singularity.
By Corollary II. of 37, the number of distinct zeros in the limited area
is finite, and, by the foregoing theorem, the multiplicity of each is finite ;
'
76 ZEROS OF A [38.
hence, when account is taken of their respective multiplicities, the total
number of zeros is still finite.
The result is, of course, a known result for a polynomial in the variables ;
but the functions in the enunciation are not restricted to be of the type of
polynomials.
Note. It is important to notice, both for Theorem II. and for its Corol-
lary I., that the zero is an ordinary point of the function under consideration ;
the implication therefore is that the zero is a definite zero and that in the
immediate vicinity of the point the function can be represented in the form
P(z - a) or P f - j , the function P(a a) or P f j being always a definite
zero.
Instances do occur for which this condition is not satisfied. The point
may not be an ordinary point, and the zero value may be an indeterminate
zero ; or zero may be only one of a set of distinct values though everywhere
in the vicinity the function is regular. Thus the analysis of 13 shews that
z = a is a point where the function sn - has any number of zero values and
any number of infinite values, and there is no indication that there are not
also other values at the point. In such a case the preceding proposition does
not apply ; there may be no limit to the order of multiplicity of the zero, and
we certainly cannot infer that any finite integer ra can be obtained such that
is finite at the point. Such a point is ( 32, 33) an essential singularity of
the function.
39. THEOREM III. A multiple zero of a function is a zero of its
derivative; and the multiplicity for the derivative is less or is greater by
unity according as the zero is not or is at infinity.
If a be a point in the finite part of the plane which is a zero of f(z)
of multiplicity n, we have
/<)*(*~r+<')t
and therefore /' (z) = (z- a) n ~ l {n<j> (z) + (z-a) <f>' (z)}.
The coefficient of (z a) n ~ l is holomorphic in the immediate vicinity of a and
does not vanish for a ; hence a is a zero for f (z) of decreased multiplicity
n-1.
If z = oo be a zero of f(z) of multiplicity r, then
39.] UNIFORM FUNCTION 77
where <j> ( - ) is holomorphic for very large values of z and does not vanish at
\*/
infinity. Therefore
The coefficient of s~ r ~ 1 is holomorphic for very large values of z, and does
not vanish at infinity ; hence z oo is a zero of /' (z) of increased multiplicity
r + 1.
Corollary I. If a function be finite at infinity, then z = oo is a zero of the
first derivative of multiplicity at least two.
Corollary II. If a be a finite zero of f(z} of multiplicity n, we have
z = n ( #( 8 )
'
f(z) z-a
Now a is not a zero of <f> (z); and therefore . . is finite, continuous, uniform
and monogenic in the immediate vicinity of a. Hence, taking the integral
of both members of the equation round a circle of centre a and of radius
so small as to include no infinity and no zero, other than a, of f (z) and
therefore no zero of < (z} we have, by former propositions,
1 f/'O) ,
^. J -..~ dz = n.
2-JTl ! f (z}
40. THEOREM IV. A function must have an infinite value for some finite
or infinite value of the variable.
If M be a finite maximum value of the modulus for points in the plane,
then ( 22) we have
v'V, n ! M
l/ (n) ()l<-^r-'
where r is the radius of an arbitrary circle of centre a, provided the whole of
the circle is in the region of continuity of the function. But as the function
is uniform, monogenic, finite and continuous everywhere, this radius can be
increased indefinitely ; when this increase takes place, the limit of
!/<>() I
is zero, and therefore f (n) (a) vanishes. This is true for all the indices 1, 2,...
of the derivatives.
Now the function can be represented at any point z in the vicinity of a
by the series
/(a) + (* - a)/' (a) 4 ( -? f" ()+...,
78 INFINITIES OF A [40.
which degenerates, under the present hypothesis, toy (a), so that the function
is everywhere constant. Hence, if a function has not an infinity somewhere
in the plane, it must be a constant.
The given function is not a constant ; and therefore there is no finite
limit to the maximum value of its modulus, that is, the function acquires
an infinite value somewhere in the plane.
COROLLARY I. A function must have a zero value for some finite or
infinite value of the variable.
For the reciprocal of a uniform monogenic analytic function is itself a
uniform monogenic analytic function ; and the foregoing proposition shews
that this reciprocal must have an infinite value for some value of the
variable, which therefore is a zero of the original function.
COROLLARY II. A function must assume any assigned value at least once.
COROLLARY III. Every function which is not a mere constant must have
at least one singularity, either accidental or essential. For it must have
an infinite value : if this be a determinate infinity, the point is an accidental
singularity ( 32) ; if it be an infinity among a set of values at the point, the
point is an essential singularity ( 32, 33).
41. Among the infinities of a function, the simplest class is that con-
stituted by its poles or accidental singularities, already defined ( 32) by the
property that, in the immediate vicinity of such a point, the reciprocal of
the function is regular, the point being an ordinary (zero) point for that
reciprocal.
It follows from this property that, because ( 37) an ordinary zero of a
uniform function is an isolated point, every pole of a uniform function is also
an isolated point : that is to say, in some non-infinitesimal region round a
pole a, no other pole of the function can occur.
THEOREM V. A function, which has a point c for an accidental singularity,
can be expressed in the form
(z-c)- n <f>(z),
where n is a finite positive integer and <f> (z) is a continuous function in the
vicinity of c.
. 1
Since c is an accidental singularity of the function f(z), the function
is regular in the vicinity of c and is zero there ( 32). Hence, by 38, there
is a finite limit to the multiplicity of the zero, say n (which is a positive
integer), and we have
yV) = ^~ c)n%( ^
41.] UNIFORM FUNCTION 79
where % (2) is uniform, monogenic and continuous in the vicinity of c and is
not zero there. The reciprocal of ^ (z), say < (z), is also uniform, monogenic
and continuous in the vicinity of c, which is an ordinary point for </> (z) ;
hence we have
/<*)->-o)r*'f&X
which proves the theorem.
The finite positive integer n measures the multiplicity of the accidental
singularity at c, which is sometimes said to be of multiplicity n or of
order n.
Another analytical expression for f(z) can be derived from that which
has just been obtained. Since c is an ordinary point for $ {z) and not a zero,
this function can be expanded in a series of ascending, positive, integral
powers of z c, converging in the vicinity of c, in the form
= + w a - c) + ...'+ !*_! (z - c) 71 " 1 + u n (z - c) n + ...
= U Q + Mj (Z - C) + ... + U n _, (Z - G) n ~ l + (Z- C} n Q(z - C),
where Q (z c), a series of positive, integral, powers of z c converging in the
vicinity of c, is a monogenic analytic function of z. Hence we have
, .
" "
the indicated expression for /"(.?), valid in the immediate vicinity of c, where
Q (z - c) is uniform, finite, continuous and monogenic.
COROLLARY. A function, which has z = co for an accidental singularity .of
multiplicity n, can be expressed in the form
where < ( - ) is a continuous function for very large values of \z , and is not
\zj
zero when z = oo . It can also be expressed in the form
a z n + a^' 1 + ...+ a n _, z + Q ( - } ,
\zj
where Q ( ) is uniform, finite, continuous and monogenic for very large values
o/!4
The derivation of the form of the function in the vicinity of an accidental
singularity has been made to depend upon the form of the reciprocal of the
function.
As the accidental singularities of a function are isolated points, there is
only a finite number of them in any limited portion of the plane.
80 INFINITIES OF A [4*2.
42. We can deduce a criterion which determines whether a given
singularity of a uniform function f(z) is accidental or essential.
When ohe point is in the finite part of the plane, say at c, and a finite
positive integer n can be found such that
is not infinite at c, then c is an accidental singularity.
When the point is at infinity and a finite positive integer n can be found
such that
is not infinite when z= oo , then z = oo is an accidental singularity.
If the condition be not satisfied in the respective cases, the singularity
at the point is essential. But it must not be assumed that the failure of the
limitation to finiteness in the multiplicity of the accidental singularity is
the only source or the complete cause of essential singularity.
Since the association of a single factor with the function is effective in
preventing an infinite value at the point when the condition is satisfied,
it is justifiable to regard the discontinuity of the function at the point
as not essential, and to call the singularity either non-essential or accidental
< 32).
43. THEOREM VI. The poles of a function, that lie in the finite part
of the plane, are all the poles (of increased multiplicity) of the derivatives of
the function that lie in the finite part of the plane.
Let c be a pole of the function f(z) of multiplicity p : then, for any point
z in the vicinity of c,
where < (z) is holomorphic in the vicinity of c, and does not vanish for z = c.
We have
/' C 2 ) = ( z ~ c)" 9 <j>' (z) p(z c)"^" 1 <f> (z)
= (z c)~ p ~ l {(z c) <f>' (z) p<f> (z)}
where % (z) is holomorphic in the vicinity of c, and does not vanish for z = c.
Hence c is a pole of/' (z) of multiplicity p + l. Similarly it can be shewn
to be a pole of f (r} (z) of multiplicity p+ r.
This proves that all the poles of f(z) in the finite part of the plane are
poles of its derivatives. It remains to prove that a derivative cannot have
a pole which the original function does not also possess.
Let a be a pole of /' (z) of multiplicity m : then, in the vicinity of a,
f'(z) can be expressed in the form
(z - a)- 77 * i/r (z),
UNIFOEM FUNCTION 81
where ^r(z) is holomorphic in the vicinity of a and does not vanish for z^a.
Thus
and therefore /' (z) = P~^ + J_ ^ ^ + . .. ,
so that, integrating, we have
f/\ ty (g) ty' (a)
~ (m -l)(z- a)- 1 ~ (m - 2) (z - a)" 1 " 2 ~ ""
When there is no terra in log(z a) in this expression, f(z) is uniform:
that is, a is a pole of f(z). When there is a term in log (z - a), then f(z) is
not uniform.
An exception occurs in the case when m is unity: for then
the integral of which leads to
/(*) = ^(
so ihatf(z) is no longer uniform, contrary to hypothesis. Hence a derivative
cannot have a simple pole in the finite part of the plane ; and so this exception
is excluded.
The theorem is thus proved.
COROLLARY I. The r th derivative of a function cannot have a pole in the
finite part of the plane of multiplicity less than r + 1.
COROLLARY II. If c be a pole off(z) of any order of multiplicity /*, and
v f (^) be expressed in the form
(z- c y+ r (z-
there are no terms in this expression with the indices 1, 2, ..., r.
COROLLARY III. If c be a pole of f(z) of multiplicity p, we have
f'(z) -P , <ft'(3)
f(?\ r r rh (?}'
J\ z ) z V 9 \Z)
where 9 (z} is a holomorphic function that does not vanish for z = c, so that
is a holomorphic function in the vicinity of c. Taking the integral of
9 (z)
f ( 7\ *
^.-~r round a circle, with c for centre, with radius so small as to exclude all
other poles or zeros of the function f(z}, we have
;rx dz = -p.
COROLLARY IV. If a simple closed curve include a number N of zeros
of a uniform function f(z) and a number P of its poles, in both of which
F. F. 6
82 INFINITIES OF A [43.
numbers account is taken of possible multiplicity, and if the curve contain
no essential singularity of the function, then
the integral being taken round the curve.
f (z)
The only infinities of the function J , within the curve are the zeros
J \ z )
and the poles of f(z). Round each of these draw a circle of radius so small
as to include it but no other infinity; then, by Cor. II. 19, the integral
round the closed curve is the sum of the values when taken round these
circles. By the Corollary II. 39 and by the preceding Corollary III., the
sum of these values is
= N-P.
It is easy to infer the known theorem that the number of roots of a
polynomial of order n is n, as well as the further result that STT (N P)
is the variation of the argument of f(z), when z describes the closed curve
in a positive sense.
Ex. 1. A function f(z) is uniform over an area bounded by a contour ; it has no
essential singularity within that area; and it has no zero and no pole on the contour.
Prove that the change in the argument of f(z\ as z makes a complete description of the
A contour, is 2?r (n ~p\ where n is the number of zeros and p is the number of poles within
the area. (Cauchy.)
Ex. 2. Prove that, if F(z) be holomorphic over an area of simple contour, which con-
tains roots aj, a 2 ,...of multiplicity m l ^ 7n 2 ,...and poles q, c 2 ,... of multiplicity p l ,p%,...
respectively of a function f(z) which has no other singularities within the contour, then
9W l F ^ J f^ dz= 2 m r F(a r -)- 2 p r F(c r \
Zrt J f(z) r=l r=l
the integral being taken round the contour.
In particular, if the contour contains a single simple root a and no singularity, then
that root is given by
the integral being taken as before. (Laurent.)
Ex. 3. Discuss the integral in the preceding example when F(z) = log z, and the origin
is excluded by a small circle of radius p, less than the smallest of the quantities | a r \ and
I CT \ ' (Goursat.)
44. THEOREM VII. If infinity be a pole of f(z}, it is also a pole of
f (z) only when it is a multiple pole of f(z).
Let the multiplicity of the pole for f(z) be n : then for very large values
of z we have
44.]
UNIFORM FUNCTION
83
where <f> is holomorphic for very large values of z and does not vanish at
infinity ; hence
The coefficient of z n ~ 1 is holomorphic for very large values of z and does not
vanish at infinity ; hence infinity is a pole off (z) of multiplicity n 1.
If n be unity, so that infinity is a simple pole of f(z), then it is not a
pole of f'(z}', the derivative is then finite at infinity.
45. THEOREM VIII. A function, which has no singularity in a finite
part of the plane, and has 2=00 for a pole, is a polynomial in z.
Let n, necessarily a finite integer, be the order of multiplicity of the pole
at infinity : then the function f(z) can be expressed in the form
where Q ( - j is a holomorphic function for very large values of z, and is finite
(or zero) when z is infinite.
Now the first n terms of the series constitute a function which has no
singularities in the finite part of the plane : and f(z) has no singularities
in that part of the plane. Hence Q ( - j has no singularities in the finite part
of the plane : it is finite for infinite values of z. It thus can never have an
infinite value : and it is therefore merely a constant, say a n . Then
f(z) = a z n + ct 1 z n ~ l + + a n _iZ + a n ,
a polynomial of degree equal to the multiplicity of the pole at infinity,
supposed to be the only pole of the function.
The above result may be obtained also in the following manner.
Since z = oo is a pole of multiplicity n, the limit of z~ n f(z) is not infinite
when z = oo .
Now in any finite part of the plane the function is everywhere finite, so
that we can use the expansion
where
2-irt
the integral being taken round a circle of any radius r enclosing the point z
and having its centre at the origin. As the subject of integration is finite
everywhere along the circumference, we have, by Darboux's expression in
(IV.) 15,
r n+i _ _
62
84 TRANSCENDENTAL AND [45.
where r is some point on the circumference and X is a quantity of modulus
not greater than unity.
Let T = re** ; then .
^^-^/(i) I
T " i_V-<
r
f( T \
By definition, the limit of -^ as T (and therefore r) becomes infinitely
/ 2 \ l
large is not infinite; in the same case, the limit of II e~ ai } is unity.
Since | X | is not greater than unity, the limit of \/r in the same case is zero ;
hence with indefinite increase of r, the limit of R is zero, and so
shewing as before that f(z) is a polynomial in z.
46. As the quantity n is necessarily a positive integer*, there are two
distinct classes of functions discriminated by the magnitude of n.
The first (and the simpler) is that for which n has a finite value. The
function then contains only a finite number of terms, each with a positive
integral index; it is a polynomial or a rational integral function of z, of
degree n.
The second (and the more extensive, as significant functions) is that
for which n has an infinite value. The point z x is not a pole, for then
the function does not satisfy the test of 42 : it is an essential singularity
of the function, which is expansible in an infinite converging series
of positive integral powers. To functions of this class the general term
transcendental is applied.
The number of zeros of a function of the former class is known : it is
equal to the degree of the function. It has been proved that the zeros of a
transcendental function are isolated points, occurring necessarily in finite
number in any finite part of the region of continuity of the function, no
point on the boundary of the part being an essential singularity; but no
test has been assigned for the determination of the total number of zeros of
a function in an infinite part of the region of continuity f.
Again, when the zeros of a polynomial are given, a product-expression can
at once be obtained that will represent its analytical value. Also we know
* It is unnecessary to consider the zero value of n, for the function is then a polynomial of
order zero, that is, it is a constant.
t In connection with the zeros of a transcendental function, as expressed in a Taylor's series ,
a paper by Hadamard, Liouville, 4 me Ser., t. viii, (1892), pp. 101 186, may be consulted with
advantage.
RATIONAL UNIFORM FUNCTIONS
85
that, if a be a zero of any uniform analytic function of multiplicity n, the
function can be represented in the vicinity of a by the expression
(at - a) n <f> (z),
where <f> (z} is holomorphic in the vicinity of a. The other zeros of the
function are zeros of <b (z) ; this process of modification in the expression
can be continued for successive zeros so long as the number of zeros taken
account of is limited. But when the number of zeros is unlimited, then the
inferred product-expression for the original function is not necessarily a
converging product; and thus the question of the formal factorisation of a
transcendental function arises.
47. THEOREM IX. A function, all the singularities of which are accid-
ental, is a rational meromorphic function.
Since all the singularities are accidental, each must be of finite
multiplicity ; and therefore infinity, if an accidental singularity, is of finite
multiplicity. All the other poles are in the finite part of the plane; they
are isolated points and therefore only finite in number, so that the total
number of distinct poles is finite and each is of finite order. Let them be
!, a 2 , , a M of orders m 1 , w 2 , , ra M respectively: let m be the order
of the pole at infinity : and let the poles be arranged in the sequence of
decreasing moduli such that | a^ > | a^^ \> > | a x .
Then, since infinity is a pole of order m, we have
f(z) = a m z m + a m ^z m -i + + a,z +/ 0),
where / (z) is not infinite for infinite values of z. Now the polynomial
m
^. a i z i is not infinite for any finite value of z ; hence f (z) is infinite for all
the finite infinities of f(z) and in the same way, that is, the function f (z)
has !, , a M for its poles and it has no other singularities.
Again, since a M is a finite pole of multiplicity m M , we have
Om,.. h.
z a L
(z a^ 1 *
where / x (z) is not infinite for z = a^ and, as / (z) is not infinite for 2 = oo ,
evidently /i (z) is not infinite for z = oo . Hence the singularities of fi(z) are
merely the poles a x , , a M _i; and these are all its singularities.
Proceeding in this manner for the singularities in succession, we ultimately
reach a function / M (z) which has only one pole a x and no other singularity,
so that
a,
where g(z) is not infinite for z = a*. But the function /^(z) is infinite only
86 UNIFORM [47.
for z = a l} and therefore g (z) has no infinity. Hence g (z) is only a constant,
say k : thus
g (z) = & .
Combining all these results we have a finite number of finite series to add
together : and the result is that
'
where 0, (z) is the series k Q + a 1 z+ ...... + a m z m , and ^-^ is the sum of the
ffs( z )
finite number of fractions. Evidently g 3 (z) is the product
(z Oj)" 1 ' (z aa)" 1 * ...... (z a M ) m i* ;
and cf 2 (z) is at most of degree
w^ + Wa + ...... + m M 1.
If F(z) denote gi(z)g 3 (z)+g 2 (z\ the form of f(z) is
?.<*)'
that is, f(z) is a rational meromorphic function.
It is evident that, when the function is thus expressed as a rational
fraction, the degree of F(z) is the sum of the multiplicities of all the poles
when infinity is a pole.
COROLLARY I. A function, all the singularities of which are accidental,
has as many zeros as it has accidental singularities in the plane.
When z = oo is a pole, it follows that, because f(z) can be expressed in
the form
the function has as many zeros as F(z), unless one such should be also a zero of
g 3 (z}. But the zeros of g 3 (z) are known, and no one of them is a zero of F(z\ on
account of the form of f(z) when it is expressed in partial fractions. Hence
the number of zeros off(z) is equal to the degree of F(z), that is, it is equal
to the number of poles of f (z).
When z = oo is not a pole, two cases are possible ; (i) the function f(z) may
be finite for z = oo , or (ii) it may be zero for z= oo . In the former case, the
number of zeros is, as before, equal to the degree of F(z), that is, it is equal
to the number of infinities.
In the latter case, if the degree of the numerator F (z} be K less than
that of the denominator g, (z), then z = so is a zero of multiplicity K ; and it
follows that the number of zeros is. .equal to the degree of the numerator
together with K, so that their number is the same as the number of accidental
singularities.
47.]
87
COROLLARY II. At the beginning of the proof of the theorem of the
present section, it is proved that a function, all the singularities of which are
accidental, has only a finite number of such singularities.
Hence, by the preceding Corollary, such a function can have only a finite
number of zeros.
If, therefore, the number of zeros of a function be infinite, the function
must have at least one essential singularity.
COROLLARY III. When a uniform function has no essential singularity,
if the (finite) number of its poles, say c 1} ..., c m , be m, no one of them being
at z= oo , and if the number of its zeros, say a 1} ..., a m , be also m, no one of
them being at 2 = oo , then the function is
z a,
c r
except possibly as to a constant factor.
When z = oo is a zero of order n, so that the function has m n zeros, say
!, 02, ..., in the finite part of the plane, the form of the function is
r=l
- Or)
(Z - C r )
r=l
and, when z = oo is a pole of order p, so that the function has m p poles,
say GU c a , ..., in the finite part of the plane, the form of the function is
m
II (z a r )
r=l _
m-p
r=l
(z-C r )
COROLLARY IV. All the singularities of rational meromorphic functions
are accidental.
48. Some properties of the simplest functions thus defined may con-
veniently be given here*. We shall begin with polynomials.
(i) Let P(z) denote
az
a ,
where the coefficients a are constants which may be complex; it is con-
tinuous., for every one of the finite number of terms is continuous ; it
is finite for all finite values of z ; and \P(z)\ tends to become infinite as
z | tends to become infinite.
* For these and other properties, reference may be made to Jordan's Cours d'Analy$e, t. 5,
p. 198.
SOME PROPERTIES OF
[48.
Further, a finite value of | z \ can be determined which will make | P (z)
greater than ?ny assigned finite value, say A. For we have
a \
|m i \m
a,
Now take
then \P(z)\>A+\. a m \ (\z - c).
Hence if | z , already supposed greater than unity, is also greater than c
should c be greater than unity, we have
P(z)\>A,
for values of z such that | z \ > 1, | z \ > c.
(ii) Next, the equation P(z) = Q always has a root. The quantity
j P (z) | is continuous, is never negative, and tends to become infinite as
| z\ tends to become infinite. Hence, if it cannot be zero, there must be
at least one minimum value greater than zero below which it cannot fall.
Denote this value by //, ; and suppose it acquired for the value c of z, so that
Construct a circle of radius greater than [ c j , and take a place c + h lying
within that circle. Then
where the coefficient of h m is a m , a quantity different from zero. As
(hypothetically) P(c) is not zero, the first term and the last term in
P (c + h) do not disappear ; but intervening terms may disappear, and
so we write
P (c + h) = P (c) + b r h r + b r+1 h r+1 + ...... + a m h m ,
where r is the lowest index of the powers of h that survive. Now choose h
in such a way that h \ is small enough to secure the inequality
while at the same time
r {arg. h} + (arg. B r ] = {arg. P (c)} + (In + 1) IT,
48.]
RATIONAL FUNCTIONS
89
so that the arguments of B r h r and P(c) differ by an odd multiple of TT.
Hence, if
then
so that
and therefore
Now
consequently
B r h r = -\B r h r \e ei ,
P(c) + B r h r ={\P(c)\-\B r h*\}<F,
j P (c) + B r h r \ = | P (c) | - | B r h r \ .
P (c + h) = P (c) + h r B r + h r+l B r+1 + ...;
P(c)\-\h\r{\B r \-\h\\B r+1 \-...}.
As B r differs from zero, the coefficient of - | h r on the right-hand side is
positive when h is quite small ; consequently, for such values of h,
that is, the modulus of P(z) in the immediate vicinity of c can be made less
than | P (c) , contrary to the hypothesis that j P (c) \ is a minimum different
from zero. Thus there cannot be a minimum different from zero, and P (z)
can always be diminished so long as it is different, from zero. Hence there
must be a value of z which makes P (z) zero.
It now follows, by the customary argument, that there are m such values.
(iii) Any rational function of z, say w, is of the form
where Q (z) and P (z} are polynomials in z of degrees m and n respectively.
Every zero of Q (z) is a zero of w. Every zero of P (z) is a pole of w.
The place z= oo is a pole of w if m > n, and it is of order m n; it is a zero
of w if m < n, and it is of order m n ; it is neither if m = n. The number
of poles is equal to the number of zeros, being the greater of the two
integers m and n.
Two results, which are of use in one method of establishing some of the special
cases of Abel's theorem concerning integrals of algebraic functions, may be noted.
Let the roots of P(z) be simple, say a ls ..., a n . Let A be the coefficient of z n in P(z).
Then
(a) when i, the order of Q (z), is less than n I,
I ejr)
'
(j8) when m = n-l, and B l is the coefficient of z n ~ l in Q (z),
Q(a r ) B l
CHAPTER V.
TRANSCENDENTAL INTEGRAL FUNCTIONS.
49. WE now proceed to consider the properties of uniform functions
which have essential singularities.
The simplest instance of the occurrence of such a function has already
been referred to in 42 ; the function has no singularity except at z= cc ,
and that value is an essentialsingularity solely through the failure of the
limitation to finiteness that would render the singularity accidental. The
function is then an integral function of transcendental character ; and it is
analytically represented ( 26) by G (z\ an infinite series in positive powers of
z, which converges everywhere in the finite part of the plane and acquires
an infinite value at infinity alone.
The preceding investigations shew that uniform functions, all the singu-
larities of which are accidental, are rational functions of the variable their
character being completely determined by their uniformity and the accidental
nature of their singularities, and that among such functions having the same
accidental singularities the discrimination is made, save as to a constant
factor, by means of their zeros.
Hence the zeros and the accidental singularities of a rational function
determine, save as to a constant factor, an expression of the function which
is valid for the whole plane. A question therefore arises how far the zeros
and the singularities of a transcendental function determine the analytical
expression of the function for the whole plane.
We have to deal with converging products ; it is therefore convenient to state, as for
converging series, the definitions of the terms used. For proofs of the statements,
developments, and applications, as well as the various tests of convergence, the references
which were given at the beginning (p. 21) of Chapter II. may be consulted.
When a series of quantities
ttiy u%, 11,3, ... ad inf.
is given, the infinite product
49.] DEFINITIONS AS TO CONVERGENCE 91
is said to converge when the limit of n n , where
n n =U(l+u a \
8=0
as increases indefinitely, is a unique finite quantity P different from zero. (The last
condition, that P should not be zero, is omitted by some writers : as our products arise
through quantities involving z and do not vanish for every value of z, no difficulty
is caused. See also Pringsheim, Math. Ann., t. xxxiii, p. 125.) When, in the same
circumstances, the limit of II* either is infinite, or is zero, or if finite is not unique
(that is, may be one of several quantities), the infinite product is said to diverge.
The necessary and sufficient conditions that the product should converge are : that U n
is finite and different from zero, however large n may be ; and that, corresponding to
every finite positive quantity e taken as small as we please, an integer m can be found
such that
n n+r
for all integers n such that n ^ m. and for every integer r.
When the product
n (1 + 1 u s | )
=0
converges, the product
also converges ; and it is said to converge absolutely. In an absolutely converging product,
the factors may be arranged in any order without affecting the convergence or the value
of the product. The convergence is sometimes called unconditional. The necessary and
sufficient condition for the absolute convergence of the product is that the series
should converge absolutely.
00
When the series u lt u 2 , u 3 , ... does not converge absolutely, while the product n (1 + u,)
s=o
converges, the convergence of the infinite product is called conditional. The tests differ
according as the quantities u are real or complex : we shall not be concerned with
conditionally converging infinite products.
The instances, which we shall have to consider, are those where the quantities u
depend upon a variable (complex) quantity z. The convergence is required as z varies, the
quantities u being regular functions throughout' the region in which z varies. When any
small quantity 8 has been chosen, and a positive integer m can be determined, such
that
n n+r
for every value of n ^ m, for all positive integers r, and for all values of z within the
region, the convergence of the infinite product is said to be uniform within the region.
Convergence of an infinite product may be uniform without being unconditional;
it may be unconditional without being uniform.
When an infinite product converges uniformly and unconditionally within a given
region, then every partial product, which is formed by taking any number of factors
in the original product, also converges uniformly and unconditionally within that region.
When an infinite product converges uniformly and unconditionally within a region,
the series constituted by the logarithms of the factors (that is, taking the principal
92 CONVERGING [49.
logarithms, whose imaginary part is to, where IT ^ a ^ rr) also converges uniformly
and unconditionally at all points within the region except the zeros of the factors : and
the logarithmic series can be differentiated, if the series of the derivatives of the terms in
this -logarithmic series itself converges uniformly. In other words, we can (under the
condition stated) take logarithmic derivatives of an infinite product, which converges
uniformly and unconditionally within a region ; and the infinite series is equal to the
logarithmic derivative of the value of the product.
50. We shall consider first how far the discrimination of transcendental
integral functions, which have no infinite value except for z = oo , is effected
by means of their zeros*.
Let the zeros a 1} a 2 , a*, ... be arranged in order of increasing moduli; a
finite number of terms in the series may have the same value so as to allow
for the existence of a multiple zero at any point. After the results stated
in 46, it will be assumed that the number of zeros is infinite ; that,
subject to limited repetition, they are isolated points; and, in the present
chapter, that, as n increases indefinitely, the limit of | a n \ is infinity. And it
will be assumed that | a x > 0, so that the origin is temporarily excluded from
the set of zeros.
Let z be any point in the finite part of the plane. Then only a limited
number of the zeros can lie within and on a circle centre the origin and
radius equal to j z \ ; let these be a 1( a^, ... , a^-i, and let a r denote any one of
the other zeros. We proceed to form the infinite product of quantities u r ,
where u r denotes
and g r is a rational integral function of z which, being subject to choice, will
be chosen so as to make the infinite product converge everywhere in the
plane. We have
1 / z\ n
\ogu r = g r - 2 -f-J ,
n =i n \a r /
a series which converges because z | < | a r \ . Now let
oo \ f Z \ n
then log u r = 2 -1 1 ,
*=,- W
and therefore
u r = e
* The following investigations are based upon the famous memoir by Weierstrass, " Zur
Theorie der eindeutigen analytischen Functionen," published in 1876: see his Ges. Werke, t. ii,
pp. 77124.
In connection with the product-expression of a transcendental function, Cay ley, "Memoire sur
les fonctions doublement periodiques," Liouville, t. x, (1845), pp. 385 420, or Collected Mathe-
matical Papers, voL i, pp. 156 182, should be consulted.
50.]
INFINITE PRODUCTS
93
Hence
n Ur = e
- z
-
if the expression on the right-hand side is finite, that is, if the series
oo oo \ f Z \ n
rlk n=a n W/
converges. Denoting the modulus of this series by M, we have
oo oo I
M< 2 2 -
r=k n=s n
sM< 2 2
r=fc n=s
< 2
a r
whence, since 1
sum, we have
is the smallest of the denominators in terms of the last
sM
Sri.
< 2
7 \n I s
r=k I "T \
If, as is not infrequently the case, there be any finite integer s for which
(and therefore for all greater indices) the series
2 1
and therefore the series 2 | a r ~*, converges, we choose s to be that least
r=k
integer. The value of M then is -finite for all finite values of z ; the series
.I / _ \n
oo oo ..I / _ \
2 2 -(-)
r =jfc n = s n \r/
converges unconditionally, and therefore
n u r
r=k
is a product, which converges unconditionally, when
u r = 1 --- I e
a*
94 WEIERSTRASS'S CONVERGING
Moreover, it converges uniformly. We have
[50.
]l + F
n u r
r=k -.
-TfS^Y 1
= 6 r-i..n\a r / _1
I
n v r
r=k
l+V oo 1
2 2 i
<e r=ln=tn
z n
^ -1
CO 1
- y.
11}
-1.
Now the series 2 ; - ^ converged; hence when any finite quantity e is
r=l
assigned, we can choose an integer I such that, for all integers I" ^ I,
rmVn a r
< 6.
Denoting by p any positive quantity which is less than | ai j , consider a region
in the 2-plane given by | z \ ^ p. Let 8 denote any assigned finite quantity,
however small ; and, after 8 is assigned, choose a quantity e so that
I-
Oil
e< Log (1 + 8),
taking the principal logarithm. Then
i+i'
n
r=k
n
r=k
p
<e s \ l ~ KJf -
shewing that the product converges uniformly for all values of z such that
\z\^.p. But I can be taken as large as we please : so that the product
converges uniformly for all finite values of z.
Let the finite product
-i i
be associated as a factor with the foregoing infinite converging product. Then
the expression
i* a?i infinite product, converging uniformly and unconditionally for all finite
00
values of z, provided the finite integer s be such as to make the series 2 | a r |~*
r=l
converge.
51.]
INFINITE PRODUCT
95
51. But it may happen that no finite integer s can be found which will
make the series
!-*
converge*. We then proceed as follows.
Instead of having the same index s throughout the series, we associate
with every zero a r an integer m r , chosen so as to make the series
M-T
a n \aj
n=l
converge. To obtain these integers, we take any series of decreasing real
positive quantities e, e 1 , e 2 , . . . , such that (i) e is less than unity and (ii) they
form a converging series ; and we choose integers m r such that
e r .
These integers make the foregoing series of moduli converge. For,
neglecting the limited number of terms for which z | ^ | a e, and taking the
first term for a fc such that
z
a k
we have for all succeeding terms (r = k+ 1, k+ 2, ...)
z
a r
and therefore
m r +i
Hence, except for the first k 1 terms, the sum of which is finite, we have
2
n=k
m n
which is finite because the series e + e x + e 2 + . . . converges. Hence the series
00
2
n=l
is a converging series.
Just as in the preceding case a special expression was formed to serve as
a typical factor in the infinite product, we now form a similar expression
for the same purpose. Evidently
_ I " r
For instance, there is no finite integer s that can make the infinite series
converge. This series is given in illustration by Hermite, Cours a lafaculte des Sciences, (4 m e"d.,
1891), p. 86.
WEIERSTRASS S CONVERGING
if | x | < 1. Forming a function E(x, m) defined by the equation
[51.
E(x, m)
S -
we have E (x, m)-=e r-1 m r r .
In the preceding case it was possible to choose the integer m so that
it should be the same for all the factors of the infinite product, which was
&
ultimately proved to converge. Now, we take x = and associate m n as
the corresponding value of m. Hence, if
n=k
1 ( z \
m n \a n j
r+m n
where | a t _! | < | z \ < \ a k \, we have
The infinite product represented by f(z) will converge, if the double series in
the exponential be a converging series.
Denoting the double series by S, we have
\8\< 2
n=k r=l
<
=&
a-
on effecting the summation for r. Let J. be the value of 1
all the remaining values of n, we have
; then for
1-
and so
V
n=k
Z l+"*n
This series converges ; hence for finite values of | z , the value of | 8 j i
finite, so that S is an unconditionally converging series. Hence it follow
that f(z) is an unconditionally converging product. We now associate wit)
f(z) as factors the k 1 functions
51.] INFINITE PRODUCT 97
for i= 1, 2, ..., k 1 ; their number being finite, their product is finite and
therefore the modified infinite product still converges. We thus have
G(z)= II E ( , m n ] ;
n=l \Q"n '
it is an unconditionally converging product.
In the same way as for the simpler case, we prove that the infinite
product converges uniformly for finite values of z.
Denoting the series in the exponential by g n (z), so that
we have E [ , m n } = 1 1 - ] ev ;
\a n J \ aj
and therefore the function obtained is
G(z)=Il ffl -
. . . i
The series g n usually contains only a limited number of terms ; when the
number of terms increases without limit, it is only with indefinite increase
of j a n |, and the series is then a converging series.
Since the product G(z) converges uniformly and unconditionally, no
product constructed from its factors E, say from all but one of them, can
be infinite. The factor
vanishes for the value z = a n and only for this value ; hence G (z) vanishes for
2 = a n . It therefore appears that G(z) has the assigned points Oj, a z , a s , ...
for its zeros.
Further, take any finite quantity, say p ; and let a m be such that
p<\a m \<
m+1
< ....
mv r* ( -\ TT 35* I ,\ TT
1 nen (JT (z) = 11 & I , m n I 11
=i \a Tl
m f- 1
o ( / v \
But n
m.,+s
The double sum in the index is a series, which converges unconditionally for
values of z such that z\<p\ and therefore it is expressible in the form
P(z, ra + 1), which is a power-series converging absolutely for those values.
Hence e~ p{z ' m) can be expressed in the form
p. P.
98 TRANSCENDENTAL INTEGRAL FUNCTION [51.
converging absolutely for values of z such that z < p. Also each of
the finite number of factors E( , m n ), for n = l, ..., w, is expressible in a
\&tt /
series of the form
1 + n 1 z + n 2 z*+ ...,
which converges absolutely for finite values of z and therefore for values of z
such that j z \ < p. The product of all these n + 1 series is also an absolutely
converging series, of the form
which is an expression for G(z) representing it as a holomorphic uniform
function. Clearly we can take p as large as we please without affecting the
foregoing argument.
In the first place, since G (z) is a uniform analytic function which has no
singularity in any finite part of the plane and which clearly is transcendental,
the value z=<x> is an essential singularity of G (z}.
In the second place, G (z) has no zero other than the assigned zeros. For
let a be a value of z ; and choose ra sufficiently large to secure that a lies
within the region of convergence of P(z, ra + 1); hence e~ p(z>m+1) is finite for
z a. No one of the factors
z \
-, tn n ) O=l, ..., m)
a n
can vanish, if a is not included in the set a^, a 2 , ..., a m . Therefore G could
not vanish for a, proving the statement.
It should be noted that the factors of the infinite product G (z) are the
expressions E. No one of these expressions, for the purposes of the product,
is resoluble into factors that can be distributed and recombined with similarly
obtained factors from other expressions E\ for there is no guarantee that
the product of the factors, when so modified, would converge uniformly and
unconditionally. It is to secure such convergence that the expressions
E have been constructed.
It was assumed, merely for temporary convenience, that the origin was
not a zero of the required function ; there obviously could not be a factor of
exactly the same form as the factors E, if a were the origin.
If, however, the origin were a zero of order X, we should have merely
to associate a factor Z K with the function already constructed.
We thus obtain Weierstrass's theorem :
It is possible to construct a transcendental integral function such that it
shall have infinity as its only essential singularity and have the origin (of
multiplicity A.), Oi, a^, a 3 , ... as zeros; and such a function is
fi \(l-L\ e g m i\
n=l 1\ n/ j
51.] AS AN INFINITE PRODUCT 99
where g n (z) is a rational, integral function of z, the form of which is dependent
upon the law of succession of the zeros.
52. But, unlike uniform functions with only accidental singularities, the
function is not unique : there are an unlimited number of transcendental
integral functions with the same series of zeros and infinity as the sole essential
singularity, a theorem also due to Weierstrass.
For, if G! (z) and (z) be two transcendental, integral functions with the
same series of zeros in the same multiplicity, and z = oo as their only essential
singularity, then
G(z)
is a function with no zeros and no infinities in the finite part of the plane.
Denoting it by G 2 , then
^
G, dz
is a function which, in the finite part of the plane, has no infinities ; and
therefore it can be expanded in the form
^ + 2(73* + 30,*'+...,
a series converging everywhere in the finite part of the plane. Choosing a
constant G so that (r 2 (0) = e c '% we have on integration
where g (z) = C + <V + <V 2 + ,
and g (z) is finite everywhere in the finite part of the plane. Hence it follows
that, ifg (z) denote any integral function of z which is finite everywhere in the
finite part of the plane, and if G(z) be some transcendental integral function
with a given series of zeros and z = oo as its sole essential singularity, all
transcendental integral functions with that series of zeros and z = oo as the
sole essential singularity are included in the form
COROLLARY I. A function which has no zeros in the finite part of the
plane, no accidental singularities, and z=<x> for its sole essential singularity,
is necessarily of the form
e g (z) t
where g (z) is an integral function of z finite everywhere in the finite part
of the plane.
COROLLARY II. Every transcendental function, which has the same zeros
in the same multiplicity as a polynomial A (z) the number, therefore, being
necessarily finite , which has no accidental singularities, and has = oo for its
sole essential singularity, can be expressed in the form
72
100 INFINITE PRODUCTS [52.
COROLLARY III. Every function, which has an assigned set of zeros
and an assigned set of poles, and has z = oo for its sole essential singularity,
is of the form
where the zeros of G (z) are the assigned zeros and the zeros of G p (z) are the
assigned poles.
For if G p (z) be any transcendental integral function, constructed as in
the proposition, which has as its zeros the poles of the required function in
the assigned multiplicity, the most general form of that function is
where h (z) is integral. Hence, if the most general form of function which
has those zeros for its poles be denoted by /(.?),
is a function with no poles, with infinity as its sole essential singularity, and
with the assigned series of zeros. But if G (z) be any transcendental integral
function with the assigned zeros as its zeros, the most general form of function
with those zeros is
and so /(*) O p (z) e h & = G (z) e^ *>,
whence /(*) = L {^ (Z) >
in which g (z) denotes g(z) h (z).
If the number of zeros be finite, we evidently may take G (z) as the
polynomial in z with those zeros as its only zeros.
If the number of poles be finite, we evidently may take G p (z) as the
polynomial in z with those poles as its only zeros.
And, lastly, if a function has a finite number of zeros, a finite number
of accidental singularities, and z = oo as its sole essential singularity, it can
be expressed in the form
where P and Q are polynomials. This is valid, even though the number of
assigned zeros be not the same as the number of assigned poles; the sole
effect of the inequality of these numbers is to complicate the character of the
essential singularity at infinity.
53. It follows from what has been proved that any uniform function,
having z=<x> for its sole essential singularity and any number of assigned
53.] PRIMARY FACTORS 101
zeros, can be expressed as a product of expressions of the form
Such a quantity is called* a primary factor of the function.
It has also been proved that :
(i) If there be no zero a n , the primary factor has the form
e*w.
(ii) The exponential index g n (z) may be zero for individual primary
factors, though the number of such factors must, at the utmost,
be finite f.
(iii) The factor takes the form z when the origin is a zero.
Hence we have the theorem, due to Weierstrass :
Every uniform integral function of z can be expressed as a product of
primary factors, each of the form
where g (z) is an appropriate polynomial in z vanishing with z, and where k, I
are constants. In particular factors, g (z) may vanish ; and either k or I, but
not both k and I, may vanish with or without a non-vanishing exponential
index g (z).
54. It thus appears that an essential distinction between transcendental
integral functions is constituted by the aggregate of their zeros : and we may
conveniently consider that all such functions are substantially the same when
they have the same zeros.
There are a few very simple sets of functions, thus discriminated by their
zeros : of each set only one member will be given, and the factor e^, which
makes the variation among the members of the same set, will be neglected
for the present. Moreover, it will be assumed that the zeros are isolated
points.
I. There may be a finite number of zeros ; the simplest function is then
a polynomial.
II. There may be a singly-infinite set of zeros. Various functions will
be obtained, according to the law of distribution of the zeros.
Thus let them be distributed according to a law of simple arithmetic
progression along a given line. If a be a zero, o> a quantity such that j &> |
is the distance between two zeros and arg. to is the inclination of the line,
we have
a + meo,
* Weierstrass's term is Prim/unction ; see Ges. Werke, t. ii, p. 91.
t Unless the class ( 59) be zero, when the index is zero for all the factors.
102 PRIMARY [54.
for integer values of m from oo to + oo , as the expression of the set of
the zeros. Without loss of generality, we may take a at the origin this
is merely a change of origin of coordinates and the origin is then a
simple zero: the zeros are given by mco, for integer values of m from
oo to + oo .
Now 2 = - 2 is a diverging series ; but an integer s the lowest
f 1 V
- -
raw w m
value is s = 2 can be found for which the series 2 ( ^ - ) converges uncon-
ditionally. Taking 5 = 2, we have
- V 1 (_L\ n - __
~*=iw \<W "raw'
so that the primary factor of the present function is
(>-
M^~.
raw/
and therefore, by 52, the product
00 (f z \
= zU \(l ) e mu>
converges uniformly and unconditionally for all finite values of z.
The term corresponding to m = is to be omitted from the product ; and
it is unnecessary to assume that the numerical value of the positive infinity
for ra is the same as that of the negative infinity for m. If, however, the
latter assumption be adopted, the expression can be changed into the ordinary
product-expression for a sine, by combining the primary factors due to values
of m that are equal and opposite. In any case, we have
-, , w . TTZ
f(z) = sin .
7T ft)
This example is sufficient to shew the importance of the exponential term in the
primary factor. If the product be formed exactly as for a polynomial, then the function is
m=p /, z \
z n (l )
m=-q\ ma>/
in the limit when both p and q are infinite. But this is known* to be
(a \ ~ at . irz
sin .
p) IT u>
Another illustration is afforded by Gauss's n-function, which is the limit when k is
infinite of
1-2-3 * j.
(*+l)(+2) (z+k) '
* Hobson's Trigonometry, 287.
54.] FACTORS 103
This is transformed by Gauss* into the reciprocal of the expression
(!+,) /(i+iv y-i
m =2 IV mj \m-lj y
that is, of (1 + 2) 5
the primary factors of which have the same characteristic form as in the preceding
investigation, though not the same literal form. This is associated with the Gamma
Function t.
It is chiefly for convenience that the index of the exponential part of the primary
S-l 1 /2 \
factor is taken, in 50, in the form 2 -71, With equal effectiveness it may be
n=i n \a r j
*-l 1
taken in the form 2 -b r<n z n , provided the series
r=k n=l
converges uniformly and unconditionally.
Ex. 1. Prove that each of the products
n i(i - *L\ e l (i + *!} n u fll - 2Z } e^~]
a lV 1 mj 6 /' V T/JLLl (an-l)./ 6 J
for m=l, 3, 5, ...... to infinity, the term for =0 being excluded from the latter
product, converges uniformly and unconditionally, and that each of them is equal to
cos z. (Hermite and Weyr.)
Ex. 2. Prove that, if the zeros of a transcendental integral function be given by the
series
0? w, 4<a, +9o>, ...... to infinity,
the simplest of the set of functions thereby determined can be expressed in the form
f Mil f- Mil
sm \ TT - V\ sm {ITT (- L
I W J I \/ J
^r. 3. Construct the set of transcendental integral functions which have in common
the series of zeros determined by the law m?a>i + 2m<* 2 + o> 3 for all integral values of m
between - oo and +00 ; and express the simplest of the set in terms of circular functions.
Ex. 4. A one-valued analytical function satisfies the equation
where |a-|^l; it has a simple zero at each of the points #=a w (m = 0, l,...)and no
other zero, and it is finite for all values of x which are neither zero nor infinite. Shew
that it has essential singularities at #=0, x= oo ; and resolve it into primary factors.
(Math. Trip., Part II., 1898.)
* Ges. Werke, t. iii, p. 145; the example is quoted in this connection by Weierstrass, Ges.
Werke, t. ii, p. 15.
t On the theory of the Gamma Function, a paper by Barnes, Messenger of Mathematics, t. xxix,
(1900), pp. 64 128, may be consulted. Keferences to later memoirs on the subject are to be found
in Whittaker and Watson's Modern Analysis (2nd ed.).
104 PRIMARY [54.
Ex. 5. Three straight lines are drawn through a point 'equally inclined to one
another ; and by means of three infinite series of lines, respectively parallel to these three
lines, the plane is divided into an infinite number of equilateral triangles. Construct an
integral uniform function which vanishes at the centre of each of the triangles.
(Math. Trip., Part II., 1894.)
Ex. 6. Take a series of concentric circles
n (n=l, 2, 3,...).
in the plane ; and four common radii
0=0, -*, d = *r, 6 = %ir.
Construct a function which shall vanish at every one of these radial points on the
circumferences : and express it by means of circular functions.
55. The law of distribution of the zeros, next in importance and sub-
stantially next in point of simplicity, is tjjiat in which the zeros form a
doubly-infinite double arithmetic progression, the points being the oo 2
intersections of one infinite system of equidistant parallel straight lines
with another infinite system of equidistant parallel straight lines.
The origin may, without loss of generality, be taken as one of the zeros.
If &) be the coordinate of the nearest zero along the line of one system
passing through the origin, and aj be the coordinate of the nearest zero along
the line of the other system passing through the origin, then the complete
series of zeros is given by
ft = rao> + m'u) ',
for all integral values of m and all integral values of m between oo and
+ oo . The system of points may be regarded as doubly-periodic, having o>
and o>' for periods.
It must be assumed that the two systems of lines intersect. Other-
wise, w and o>' would have 'the same argument, and their ratio would be a
real quantity, say a ; and then
n
= m + m .
o>
If o be commensurable, let - denote its value, where p and q are positive
integers having no common factor ; also let - be expressed as a continued
in'
fraction, and let f denote the convergent next before the last (which, of
= > pq'-p'q=l;
course, is - 1 . Then
fl
and therefore = - = + (q'w ato) = CD",
P 9
55.] FACTORS 105
that is, o>' and w are 'integral multiples of a single period &>" ; and the
apparently double system of points would be singly-periodic.
When a is incommensurable, the number of pairs of integers for which
in + m'a. may be made less than any assigned small quantity 8 is infinite ;
and then the function would have an unlimited number of zeros in any
assigned small region round the origin. This would make the origin an
essential singularity instead of, as required, an ordinary point of the tran-
scendental integral function. Hence the ratio of the quantities co and QJ is
not real.
56. For the construction of the primary factor, it is necessary to render
the series
2fl-."
converging, by appropriate choice of integers s m , m '. It is found to be
possible to choose an integer s to be the same for every term of the series,
corresponding to the simpler case of the general investigation, given in 50.
As a matter of fact, the series
2n-*
diverges for * = 1 (we haVe not made any assumption that the positive and
the negative infinities for ra are numerically equal, nor similarly as to ra') ;
the series tends to a finite value for s = 2, but the value depends upon the
relative values of the infinities for m and m' ; and s = 3 is the lowest integral
value for which, as for all greater values, the series converges uncon-
ditionally.
There are various ways of proving the unconditional convergence of the
series SH"* 4 when //, > 2 : the following proof is based upon a general method
due to Eisenstein*.
j = oo n = oo
First, the series 2 S (m 2 + w 2 )"* 4 converges unconditionally, if /*> 1.
m= oo TJ= co
Let the whole series be arranged in partial series : for this purpose, we
choose integers k and I, and include in each such partial series all the terms
which satisfy the inequalities
< n
so that the number of values of m is 2 fc and the number of values of n is 2*.
Then, if k + I = 2x, we have
2 2(C < 2 2K+1 < 2 2 * + 2^ < ra 2 + n\
so that each term in the partial series ^ S^T.. The number of terms in the
& ^
* Crelle, t. xxxv, (1847), p. 161. A geometrical exposition is given by Halphen, Traite des
fonctions elliptiques, t. i, pp. 358 362 ; and another by Goursat, Cours d' Analyse MathGmatique,
t. ii, 324.
106 WEIERSTRASS'S FUNCTION AS [56.
partial series is 2* . I 1 , that is, 2 2<c : so that the sum of the terras in the
partial series is
"^ )2(| 1) *
Expressing the latter in the form
*-i) * 2* d*- 1 ) '
and taking the upper limit of k and I to be p, ultimately to be made infinite,
we have the sum of all the partial series
which, when p oo , is a finite quantity if yu, > 1.
Next, let CD = a + fii, &>' = 7 + Si, so that
fl = mo + w<u' = wet + ny + i (m/3 + nS) ;
hence, if = wo + 717, < = m/3 + nS,
we have | fl 2 = I? 2 + < 2 .
Now take integers r and s such that
r<0<r+l, sfxs+l.
The number of terms fl satisfying these conditions is definitely finite and is
independent of m and n. For since
m (aS - 7) = 0S - <f>y,
n(aS-/3y)=- 0/3 + <,
and a8 7 does not vanish because ta' fat is not purely real, the number of
values of m is the integral part of
(r+l)B-sy
aS /3y
less the integral part of
rS (s 4- 1) 7
a&-/3y
that is, it is the integral part of (7 + S)/(otS $7), or is greater than it by
unity. Similarly, the number of values of n is the integral part of
or is greater than it by unity. Let the product of the two numbers be q ;
then the number of terms fl satisfying the inequalities is q.
56.] A DOUBLY-INFINITE PRODUCT 107
Then 22 | H |-* = 22 (8* + <f>*)-"
which, by the preceding result, is finite when yu, > 1. Hence
22 (mo) + m'a)')-^
converges unconditionally when /LI > 1 ; and therefore the least integer s, for
which
22 (mco + m'(a')~ s
converges unconditionally, is 3. But this series converges unconditionally for
any real value of s which is definitely greater than 2.
The series 22 (mo> + m'o>')~ 2 has a finite sum, the value of which depends* upon
the infinite limits for the summation with regard to m and mf. This dependence is
inconvenient, and it is therefore excluded in view of the present purpose.
Ex. Prove in the same manner that the series
22 ...... 2(m 1 2 +m 2 :i + ...... +m n *)-,
the multiple summation extending over all integers wij, m 2 , ...... , m n between - oo and
+ 00, converges unconditionally if 2/i>%. (Eisenstein.)
57. Returning now to the construction of the transcendental integral
function the zeros of which are the various points O, we use the preceding
result in connection with 50 to form the general primary factor. Since
s = 3, we have
s-l
and therefore the primary factor is
Moreover, the origin is a simple zero. Hence, denoting the required function
by a (z), we have
OC 00
(z} = zu n
- 00 00
as a transcendental integral function which, since the product converges uni-
formly and unconditionally for all finite values of z, exists and has a finite
value everywhere in the finite part of the plane; the quantity O denotes
mo) + niQ)', and the double product is taken for all values of m and of mf
between - oo and + oo , simultaneous zero values alone being excluded.
This function will be called Weierstrass's <r-function ; it is of import-
ance in the theory of doubly-periodic functions which will be discussed in
Chapter XI.
* See a paper by the author, Quart. Journ. of Math., vol. xxi, (1886), pp. 261 280.
108 PRIMARY' FACTORS [57.
Ex. If the doubly-infinite series of zeros be the poiuts given by
<ai, 6> 2 , W 3 being complex constants such that Q does not vanish for real values of m and n,
then the series
00 00
2 2 Q~
_00 00
converges for 5 = 2 but not for 5 = 1, The primary factor is thus
. .. .............
and the simplest transcendental integral function having the assigned zeros is
z n n
The actual points that are the zeros are the intersections of two infinite systems of
parabolas.
58. One other result of a negative character will be adduced in this
connection. We have dealt with the case in which the system of zeros is a
singly-infinite arithmetical progression of points along one straight line, and
with the case in which the system of zeros is a doubly-infinite arithmetical
progression of points along two different straight lines. We proceed to prove
that a uniform transcendental integral function cannot exist with a triply-
infinite arithmetical progression of points for zeros.
A triply-infinite arithmetical progression of points would be represented
by all the possible values of
for all possible integer values for p 1 , p<2, p 3 between oc and 4- oc , where no
two of the arguments of the complex constants f! 1} Q 2 ^3 are equal. Let
n r =o> r + tV, (r=l, 2, 3);
then, as will be proved ( 107) in connection with a later proposition, it is
possible* and possible in an unlimited number of ways to determine
integers p^, p 2 , p 3 so that, save as to infinitesimal quantities,
Pi J>2 _ P*
all the denominators in which equations differ from zero on account of the
fact that no two arguments of the three quantities fi lt H 2 , fi 3 are equal. For
each such set of determined integers, the quantity
is zero or infinitesimal. If it is zero, then (as in 107 for periods) the triple
infinitude is really only a double infinitude. If it is infinitesimal, then (as
at the end of 55) the origin is an essential singularity, contrary to the
* Jacobi, Get. Werke, t. ii, p. 27.
58.] CLASS OF A FUNCTION 109
hypothesis that the only essential singularity is for z = oo . Hence a uniform
transcendental function cannot exist having a triply-infinite arithmetical
succession of zeros.
59. In effecting the formation of a transcendental integral function by
means of its primary factors, it has been proved that the expression of the
primary factor depends upon the values of the integers which make
S | a n I""- 1 \z\ m *
w=l
a converging series. Moreover, the primary factors are not unique in form,
because any finite number of terms of the proper form can be added to the
exponential index in
he
TOra-l 1 z r
s -
the added terms will only the more effectively secure the convergence of the
infinite product. But there is aiower limit to the removal of terms with the
highest exponents from the index of the exponential ; for there are, in general,
least values for the integers m 1 , m 2 , ..., below which these integers cannot be
reduced, if the convergence of the product is to be secured.
The simplest case, in which the exponential must be retained in the
primary factor in order to secure the convergence of the infinite product, is
that discussed in 50, viz., when the integers m lt m 2 , ... are equal to one
another. Let m denote this common value for a given function, and let
ra be the least integer effective for the purpose : the function is then said*
to be of class m, and the condition that it should be of class m is, that the
integer m be the least integer to make the series
converge, the constants a n being the zeros of the function.
Thus algebraical polynomials are of class ; the circular functions sin z
and cos z are of class 1 ; Weierstrass's o--function and the Jacobian elliptic
function sn z are of class 2, and so on : but for no one of these classes do the
functions mentioned constitute the whole of the functions of that class.
60. One or two of the simpler properties of an aggregate of transcendental
integral functions of the same class can easily be obtained.
Let a function f(z\ of class n, have a zero of order r at the origin and
have a,, a 2 , ... for its other zeros, arranged in order of increasing moduli.
Then, by 50, the function f(z) can be expressed in the form
/(,)-e^n
i = \
* The French word is genre ; the Italian is genere. Laguerre (see references on p. 113)
appears to have been the first to discuss the class of transcendental integral functions.
110
CLASS-PROPERTIES OF
[60.
n 1 / z \
where g^z} denotes the series 2 - ( ) and G(z) must be properly determined
=1 8 \&t'
to secure the equality.
Now consider the series
for all values of z that lie outside circles round the points a, taken as small
as we please. The sum of the series of the moduli of its terms is
f-1
. \n+i
Let d be the least of the quantities
-i
1 --
1 i , necessarily non-evanescent
because z lies outside the specified circles; then the sum of the series
which is a converging series since the function is of class n. Hence the
series of moduli converges, and therefore the original series converges.
00
Moreover, the series 2 ja^)"" 7 *" 1 converges. Denoting by e any real positive
quantity, as small as we please, we can choose an integer m such that
21 n . n i ^ c
\ "t ^ >
p=p.
for all integers /* ^ m and for all positive integers r. Accordingly, for the
values of z considered, we have
i r 1 1
-i
a,-
for all integers p^m, for ail positive integers r, and for all the values of z.
Hence the series converges unconditionally and uniformly within the specified
region of variation of z ; let it be denoted by S (z), so that
We have
1-
r
--z 2
60.] TRANSCENDENTAL INTEGRAL FUNCTIONS 111
Each step of this process is reversible in all cases in which the original product
f (g\
converges. If, therefore, it can be shewn of a function f(z) that J ,, , / takes
/W
this form, the function is thereby proved to be of class n.
7*
If there be no zero at the origin, the term - is absent.
2!
If the exponential factor G (z) be a constant so that G' (z) is zero, the
function f(z) is said to be a simple function of class n.
61. There are several criteria, used to determine the class of a function :
the simplest of them is contained in the following proposition, due to
Laguerre*.
If, as z tends to the value oo , a very great value of z can be found for
f ( z \
which the limit of z ~ n ~f7^\ > where f(z) is a transcendental integral function,
tends uniformly to the value zero, thenf(z) is of class n.
Take a circle, centre the origin and of radius R equal to this value of | z j ;
then, by 24, II., the integral
1 f If'(t) dt
2m J t n f(t) t-z'
taken round the circle, is zero when R becomes indefinitely great. But the
value of the integral is, by the Corollary in 20,
1 f'> I/HO dt J_ p lf'(t) dt J^ I p> lf'(t) dt
2m J t n f(t) t-z 2-rri <, J t n
2m t n /(O *-* 2m J t n f(t) t-z 2-rri <, t n f(t) t-z'
taken round small circles enclosing the origin, the point z, and the points
a{, which are the infinities of the subject of integration; the origin being
supposed a zero of f(t) of multiplicity r. Now
w **
2m J t n f(t) t-z z n f(z) '
1 [Wlf'(t) dt
2-jrij t n f(t) t-z
_
t n f(t)t-z
where <f> (z) denotes the polynomial
(/' (Q _ r] d_ \f'(t) _r\ z n ~* d n ~ l (f (t) _ r
\f(t) t\^ Z dt\f(t) I]* + (n-I)}dt"->\f(t) t
when t is made zero. Hence
Comptes Eendus, t. xciv, (1882), p. 636 ; (Euvres Completes, t. i, p. 172.
112 CLASS-PROPERTIES OF [61.
and therefore
J -^ = d>(z)+--z n S(z)
which, by 60, shews that/(f) is of class n.
COROLLARY. The product of any finite number of functions of the same
class n is a function of class not higher than n ; and the class of the product
of any finite number of functions of different classes is not greater than the
highest class of the component functions.
NOTE 1. In connection with Weierstrass's theorem in 52, one remark
may be made as to its influence upon the class of a function ; it will be
sufficiently illustrated by taking e* sin z as an example. Laguerre's test
shews that the class is two, whereas by the test of 60 the class apparently
is unity. The explanation of the difference is that, in 60, the zeros of the
generalising factor e^ (Z} of 52 are not taken into account. It is true that all
these zeros are at infinity ; but their existence may affect the integer, which
is the least that secures the convergence of the series 2 | a t j" 71 " 1 . Thus the
zeros of the function e 2 * sin z are WTT, where m = 0, 1, ..., 00, arising
from sin 2: and
7, ^n2 _ 7,71'^
each occurring p times, where p is an infinite positive integer : the latter
arising from e 2 *, by regarding it as the limit of
.
when p is an infinite positive integer. In order that the critical series may
converge, it is necessary that, as these new zeros are at infinity, the integer n
should be chosen so as to make
p \ (ip)-"- 1 \+p\(-
vanish. The lowest value of n is two ; and therefore the function really is of
class two, agreeing with the result of Laguerre's test.
More generally, consider a function
where f(z) is of class n, and G (z} is itself an integral function. On the
application of Laguerre's test, the limit of
F(Z) >
when \z\ increases indefinitely, is the limit of z~ n G' (z). Thus F(z} is not
of class 7i, unless G (z) is a polynomial in z of degree ^ n. If G (z) is a
polynomial of degree m>n, then F(z) is of class m. If G(z) is a transcen-
dental integral function, F(z) is of infinite class.
61.] TRANSCENDENTAL INTEGRAL FUNCTIONS 113
Of course, this is not the only manner in which functions of infinite class
can arise. Thus consider an integral function having log 2, log 3, log 4, ... for
its infinite succession of zeros. It has been noted (p. 95, foot-note) that no finite
00
integer s exists such that the series S (log n}~ 8 converges ; consequently the
class of the series is infinite*.
NOTE 2. Borel f introduces the notion of the order of an integral function
as distinct from the class of the function. In the preceding investigation ( 59),
the class of the equation is taken to be the lowest integer s (if any) for which
the series
2 a*
(where a 1} a. 2 , ... are the zeros arranged in non-descending magnitude of
moduli) converges absolutely. Borel takes the order of the function to be the
lowest real quantity for which the same series converges absolutely ; so that,
if fj, be the class and // the order of a function,
fJb' ^ fji< fjf + 1.
Thus the class of the product
is unity, because 2 is the lowest integer which makes the series 2 n~ s converge;
n-\
its order is 1 + k, where k is any quantity greater than zero but as small as
00
we please, because the series 2 n~ l ~ k converges.
The following are the chief references to memoirs discussing the class of functions :
Laguerre, Comptes Rendus, t. xciv, (1882), pp. 160163, pp. 635638, ib. t. xcv, (1882),
pp. 828831, ib. t. xcviii, (1884), pp. 79 81 J ; Poincare, Bull, des Sciences Math., t. xi,
(1883), pp. 136144; Cesaro, Comptes Rendus, t. xcix, (1884), pp. 2627 (followed
(p. 27) by a note by Hermite), Oiornale di Battaglini, t. xxii, (1884), pp. 191 200 ;
Vivanti, Oiornale di Battaglini, t. xxii, (1884), pp. 243261, pp. 378380, ib. t. xxiii,
(1885), pp. 96122, ib. t. xxvi, (1888), pp. 303314; Hermite, Cours a la faculte
des Sciences (4 me ed., 1891), pp. 9193; Hadamard, Liouville, 4 me Se"r., t. ix, (1893),
pp. 171214 ; Borel, Acta Math., t. xx, (1897), pp. 357396, Lecons sur les fonctions
entieres, (1900), ch. ii.
Ex. 1. Prove that the class of the functions sin z, 1 + z sin z is unity. ,
Ex. 2. The function
n
2
where the quantities c are constants, n is a finite integer, and the functions f t (z) are
polynomials, is of class unity.
* For functions of infinite class, reference may he made to Blumenthal's monograph
Principes de la theorie des fonctions entieres d'ordre infini (1910).
t Lecons sur les fonctions entieres, p. 26.
J All these are included in the first volume of the (Euvres de Laguerre, (1898, Gauthier-
Villars).
F. F. 8
114 EXAMPLES [61.
Ex. 3. If a simple function be of class , its derivative is also of class n.
Ex. 4. Discuss the conditions under which the sum of two functions, each of class n,
is also of class n.
Ex. 5. Examine the following test for the class of a function, due to Poincare".
Let a be any number, no matter how small provided its argument be such that 6***
vanishes when z tends towards infinity. Then f(z) is of class n, if the limit of
~ n + '/(*)
vanish with indefinite increase of z.
00
A possible value of a is 2 c i a i ~ n ~ l t where c t - is a constant of modulus unity.
<=i
E:c. 6. Verify the following test for the class of a function, due to de Sparre*.
Let X be any positive non-infinitesimal quantity ; then the function f(z) is of class n,
if the limit, for m= oo , of
be not less than X. Thus sin z is of class unity.
Ex. 7. Let the roots of d n t l =l be 1, o, a 2 , ...... , a"; and let f(z) be a function
of class n. Then forming the product
n
8=0
we evidently have an integral function of z" + 1 ; let it be denoted by F(z n + 1 ). The roots
of F(z n + l ) = Q are a ( a s , for i=l, 2, ...... , and s = 0, 1, ...... , n; and therefore, replacing z* + 1
by z, the roots of F(z)=0 are af + l , for i= 1, 2, .......
Since f(z) is of class n, the series
converges unconditionally. This series is the sum of the first powers of the reciprocals of
the roots of F(z}=0; hence, according to the definition (p. 109), F(z) is of class zero.
It therefore follows that from a function of any class, a function of class zero with a
modified variable can be deduced. Conversely, by appropriately modifying the variable of
a given function of class zero, it is possible to deduce functions of any required class.
Ex. 8. If all the zeros of the function
n=i
be real, then all the zeros of its derivative are also real. (Witting.)
* Comptes Rendus, t. cii, (1886), p. 741.
CHAPTER VI.
FUNCTIONS WITH A LIMITED NUMBER OF ESSENTIAL SINGULARITIES.
62. SOME indications regarding the character of a function at an
essential singularity have already been given. Thus, though the function
is regular in the vicinity of such a point a, it may, like sn (1/z) at the origin,
have a zero of unlimited multiplicity or an infinity of unlimited multiplicity
at the point ; and in either case the point is such that there is no factor of
the form (z a) A , which can be associated with the function so as to make the
point an ordinary point for the modified function. Moreover, even when the
path of approach to the essential singularity is specified, the value acquired
may not be definite : thus, as z approaches the origin along the axis of x,
so that its value may be taken to be 1 -r (4<mK + x), the value of sn (l/#) is not
definite in the limit when ra is made infinite. One characteristic of the
point is the indefmiteness of value of the function at the essential singu-
larity, though in the vicinity the function is uniform.
A brief statement and a proof of this characteristic were given in 32 ;
the theorem there proved that a uniform analytical function can assume
any value at an essential singularity may also be proved as follows. The
essential singularity will be taken at infinity a supposition that does not
detract from generality.
Let f(z) be a function having any number of zeros and any number
of accidental singularities and z = oo for its sole essential singularity ; then
it can be expressed in the form
G 2 (z)
where G l (z} is polynomial or transcendental according as the number of zeros
is finite or infinite, and G 2 (z} is polynomial or transcendental according as
the number of accidental singularities is finite or infinite.
If G 2 (z) be transcendental, we can omit the generalising factor e fflz -.
Then/(z) has an infinite number of accidental singularities; each of them
89
- 1
116 FORM OF A FUNCTION NEAR [62.
in the finite part of the plane is of only finite multiplicity and therefore some
of them must be at infinity. At each such point, the function G 2 (z) vanishes
and G l (z) does not vanish ; and so/ (2) has infinite values for z = oo .
If G 2 (z) be polynomial and G^ (z) be also polynomial, then the factor e g (e}
may not be omitted, for its omission would make f(z) a rational function.
Now z = oo is either an ordinary point or an accidental singularity of
hence as g (z) is integral, there are infinite values of z which make
<?,(*)
infinite.
If 6r 2 (z) be polynomial and G^ (z} be transcendental, the factor e' J te) may
be omitted. Let a 3 , a 2 , . . . , a n be the roots of G 2 (z) : then taking
/(*)= /' -- +(*),
1* = 1 * Ct'j-
(TJ (tt r )
we have J ,. = 7=^ ,
Cr 2 (Or)
a non-vanishing constant ; and so
where r n (z) is a transcendental integral function. When z= oo , the value
of G s (z)/G 2 (z) is zero, but G n (z) is infinite ; hence /(V) has infinite values for
z = oo .
Similarly it may be shewn, as follows, ih&tf(z) has zero values for z= oo .
In the first of the preceding cases, if G 1 (z) be transcendental, so that/ (z)
has an infinite number of zeros, then some of them must be at an infinite
distance ; f(z) has a zero value for each such point. And if G^ (z) be
polynomial, then there are infinite values of z which, not being zeros of
G 2 (z), make f(z) vanish.
In the second case, when z is made infinite with such an argument as to
make the highest term in g(z) a real negative quantity, then f(z) vanishes
for that infinite value of z.
In the third case, f(z) vanishes for a zero of G l (z) that is at infinity.
Hence the value of f(z) for z = oo is not definite. If, moreover, there
be any value neither zero nor infinity, say C, which f(z) cannot acquire
for z = oo , then
f(z}-C
is a function which cannot be zero at infinity, and therefore all its zeros are
in the finite part of the plane : no one of them is an essential singularity, for
62.] AN ESSENTIAL SINGULARITY 117
f(z) has only a single value at any point in the finite part of the plane;
hence they are finite in number and are isolated points. Let H l (z) be
the polynomial having them for its zeros. The accidental singularities of
f(z) C are the accidental singularities of f(z) ; hence
where, if G*(z} be polynomial, the exponential h (z) must occur, since f(z),
and therefore f(z) C, is transcendental. The function
F (z\ - - ^ 2 ^ c~ h w
- ~
evidently has z = oo for an essential singularity, so that, by the second or
the third case above, it certainly has an infinite value for z = <x> , that is,
f(z) certainly acquires the value G for z = oo .
Hence the function can acquire any value at an essential singularity.
63. We now proceed to obtain the character of the expression of a
function at a point z which, lying in the region of continuity, is in the
vicinity of an essential singularity b in the finite part of the plane.
With 6 as centre describe two circles, so that their circumferences and
the whole area between them lie entirely within the region of continuity.
The radius of the inner circle is to be as small as possible consistent with
this condition ; and therefore, as it will be assumed that b is the only
singularity in its own immediate vicinity, this radius may be made very
small.
The ordinary point z of the function may be taken as lying within the
circular ring-formed part of the region of continuity. At all such points in
this band, the function is holomorphic ; and therefore, by Laurent's Theorem
( 28), it can be expanded in a converging series of positive and negative
integral powers of z b, in the form
u + % (z b) + u z (z - 6) 2 + . . .
+ V! (z - b)- 1 + v 2 (z- b)~ 2 + . . . ;
the coefficients u n are determined by the equation
u n
the integrals being taken positively round the outer circle, and the coefficients
v n are determined by the equation
the integrals being taken positively round the inner circle.
118 FORM OF A FUNCTION NEAR [63.
The series of positive powers converges everywhere within the outer circle
of centre b, and so ( 26) it may be denoted by P (z b) ; and the function P
may be either polynomial or transcendental.
The series of negative powers converges everywhere without the inner
circle of centre b ; and, since 6 is not an accidental but an essential singularity
of the function, the series of negative powers contains an infinite number of
terms. It may be denoted by $( r) a series converging for all points
in the plane except z = b, and vanishing when z b = oo .
Thus f(z) =
is the analytical representation of the function in the vicinity of its essential
singularity b ; the function is transcendental and converges everywhere in
the plane outside an infinitesimal circle round b, and the function P, if
transcendental, converges for sufficiently small values of \ z b \.
Had the singularity at b been accidental, the function G would have been
polynomial.
COROLLARY I. If the function have any essential singularity other than
b, it is an essential singularity of P (z b) continued outside the outer circle ;
but it is not an essential singularity of G[ -- j], for the latter function
\2 ~~ O/
converges everywhere in the plane outside the inner circle.
COROLLARY II. Suppose the function has no singularity in the plane
except at the point b ; then the outer circle can have its radius made infinite.
In that case, all positive powers except the constant term U Q disappear :
and even this term survives only in case the function have a finite value at
infinity. The expression for the function is
and the transcendental series converges everywhere outside the infinitesimal
circle round 6, that is, at every point in the plane for which . j-. remains
Z ~~ 4
less than any assigned quantity, however large. Hence the function can be
represented by
This special result is deduced by Weierstrass from the earlier in-
vestigations*, as follows. If f(z) be such a function with an essential
* Weierstrass, Ges. Werkt, t. ii, p. 102.
63.] AN ESSENTIAL SINGULARITY 119
singularity at b, and if we change the independent variable by the
relation
1
z =
z-b'
then f(z) changes into a function of z', the only essential singularity of which
is at z = oo . It has no other singularity in the plane ; and the form of the
function is therefore G(z'), that is, a function having an essential singularity
at 6, but no other singularity in the plane, is
G
COROLLARY III. The most general expression of a function having its
sole essential singularity at b, a point in the finite part of the plane, and any
number of accidental singularities, is
-b
where the zeros of the function are the zeros of G lt the accidental singularities
of the function are the zeros of G 2 , and the function g in the exponential is a
function which is finite for all finite values of 7 .
Z ^
This can be derived in the same way as before ; or it can be deduced
from the corresponding theorem relating to transcendental integral functions,
as above. It would be necessary to construct an integral function G 2 (z'),
having as its zeros
1 1
and then to replace z' by ; and G 2 is polynomial or transcendental,
z o
according as the number of zeros is finite or infinite.
Similarly we obtain the following result :
COROLLARY IV. A uniform function of z, which has its sole essential
singularity at b, a point in the finite part of the plane, and no accidental
singularities, can be represented in the form of an infinite product of primary
factors of the form
,
z b
which converges uniformly and unconditionally everywhere in the plane outside
an infinitesimal circle drawn round the point b.
120 FUNCTIONS WITH A LIMITED NUMBER [63.
The function g { -- j\ is an integral function of ^-, vanishing when
r vanishes ; and k and I are constants. In particular factors, q ( - r )
z b \z b.l
may vanish ; and either k or I (but not both k and I) may vanish, with or
without a vanishing exponent g ( , j .
If a t - be any zero, the corresponding primary factor may evidently be
expressed in the form
Similarly, for a uniform function of z with its sole essential singularity at b
and any number of accidental singularities, the product-form is at once
derivable by applying the result of the present Corollary to the result given
in Corollary III.
These results, combined with the results of Chapter V., give the general
theory of uniform functions with only one essential singularity.
64. We now proceed to the consideration of functions, which have a
limited number of assigned essential singularities.
The theorem of 63 gives an expression for the function at any point in
the band between the two circles there drawn.
Let c be such a point, which is thus an ordinary point for the function ;
then in the domain of c, the function is expansible in a form Pj (z c).
This domain may extend as far as an infinitesimal circle round an essential
singularity 6, or it may be limited by a pole d which is nearer to c than b is,
or it may be limited by an essential singularity / which is nearer to c than b
is. In the first case, we form a continuation of the function in a direction
away from b ; in the second case, we continue the function by associating
with the function a factor (z d) n which takes account of the accidental
singularity; in the third case, we form a continuation of the function
towards f. Taking the continuations for successive domains of points in the
vicinity of f, we can obtain the value of the function for points on two circles
that have f for their common centre. Using these values, as in 63, to
obtain coefficients, we ultimately construct a series of positive and negative
powers converging outside an infinitesimal circle round/. Different express-
ions in different parts of the plane will thus be obtained, each being valid
only in a particular portion : the aggregate of all of them is the analytical
expression of the function for the whole of the region of the plane where the
function exists.
We thus have one mode of representation of the function ; its chief
advantage is that it indicates the form in the vicinity of any point, though it
64.] OF ESSENTIAL SINGULARITIES 121
gives no suggestion of the possible modification of character elsewhere. This
deficiency renders the representation insufficiently precise and complete ; and
it is therefore necessary to have another mode of representation.
65. Suppose that the function has n essential singularities a 1} a 2 , ..., a n ,
and that it has no other singularity. Let a circle, or any simple closed
curve, be drawn enclosing them all, every point of the boundary as well
as the included area (with the exception of the n singularities*) lying in
the region of continuity of the function.
Let z be any ordinary point in the interior of the circle or curve ; and
consider the integral
t z
taken round the curve. If we surround z and each of the n singularities by
small circles with the respective points for centres, then the integral round
the outer curve is equal to the sum of the values of the integral taken round
the n + 1 circles. Thus
and therefore
\JJV _. .1 UVU ~ . *-l I .
. t z 2-Trt J s t z Im J ar * ~" z
The left-hand side of the equation is f(z).
Evaluating the integrals, we have
2?n .',.
where G> is, as before, a transcendental function of - - vanishing when
^~
1 .
- is zero.
z a r
Now, of these functions, 0*1 -I converges everywhere in the plane
\Z CL r /
outside the infinitesimal circle round a,., (say except at a r ) : and therefore, as
n is finite,
r =i
z-a
is a function which converges everywhere in the plane except at the n points
. , Ct n .
Because z = x> is not an essential singularity of/ (2), the radius of the
circle in the integral -^ - I ^-^ dt may be indefinitely increased. The value
2?n J s t z
* This phrase will frequently be used as an abbreviation for " the infinitesimal regions
enclosed by infinitesimal circles round the singularities."
122 FUNCTIONS WITH A LIMITED NUMBER [65.
of f(t) tends, with unlimited increase of t, to some determinate value C which
is not infinite ; hence, as in 24, II., Corollary, the value of the integral is
C, We therefore have the result that f(z) can be expressed in the form
1
or, absorbing the constant C into the functions G' and replacing the limitation
that the function G r ( ) shall vanish for - = 0, by the limitation
\z - aj z- or
that, for the same value - = 0, it shall be finite, we have the theorem * :
z a r
If a given function f(z) have n singularities a^, ..., a n , all of which are in
the finite part of the plane and are essential singularities, it can be expressed
in the form
/
/ 1 \
9 rs. [_L_i
~ "t \ >
r =i \za r !
where G r is a transcendental function, converging everywhere in the plane
outside an infinitesimal circle round a r , and having a determinate finite
1 n
value g r for - = 0, such that 2 g r is the finite value of the given func-
Z> ~*~ Cty J* = l
tion at infinity.
COROLLARY. If the given function have a singularity at oo , and n singu-
larities in the finite part of the plane, then the function can be expressed in
the form
1
G r
r=l \Z ~
where G r is a transcendental or a polynomial function, according as a r is an
essential or an accidental singularity : and so also for G (z), according to the
character of the singularity at infinity.
66. Any uniform function, which has an essential singularity at z = a,
can ( 63) be expressed in the form
for points z in the vicinity of a. Suppose that, for points in this vicinity,
the function f(z) has no zero, and that it has no accidental singularity.
Therefore, among such points z, the function
1 df(z)
/<*) dz
* The method of proof, by an integration, is used for brevity : the theorem can be established
by purely algebraical reasoning.
66.] OF ESSENTIAL SINGULARITIES 123
has no pole, and therefore no singularity except that at a which is essential.
Hence it can be expanded in the form
where G converges everywhere in the plane except at a, and vanishes for
- = 0. Let
z a
1
' z-a^ dz I 'U-
x& Cv tvx I \x
where Gil - j converges everywhere in the plane except at a, and vanishes
for = 0.
z a
Then c, evidently not an infinite quantity, is an integer. To prove this,
describe a small circle of radius p round a : then taking z a = pe 6 *, so that
- = idd, we have
z a
1 df(z) j , d ( n ( 1 \) ,
J \ J fjy p / ~ n \ J ~ I Mflfj I J fj. \ ] I fly
"TTT r j \JU& " -1. \& \M ) \AJ& ~t t/t'Cvt' i -j i Ui i I f W/x&.
/ (^) a^ a^ ( \z a/J
and therefore
Now jP(z a)dz is a uniform function : and so is f(z). But a change
of into 9 + 2?r does not alter z or any of the functions : thus
eon* _ j .
and therefore c is an integer.
67. If the function /(*) have essential singularities a l ,..., a n and no
others, then it can be expressed in the form
n / 1 \
C+ 2 g r (- .
r =i \z-aj
If there be no zeros for this function f(z) anywhere (except of course such
as may enter through the indeterminateness at the essential singularities),
then
J_ <*/(*)
f(z} dz
has n essential singularities a l , ..., a n and no other singularities of any kind.
Hence it can be expressed in the form
r=l
124 EXPRESSION OF [67
where the function G r vanishes with - - . Let
z - a r
f 1 \ C r d (= / 1 \)
Gr\ I ~ ~ + j~ \ > I f >
\z a r / z a r dz { \z a r / }
where G r ( ) is a function of the same kind as G r {
\z a r l
z a,
Then all the coefficients c r , evidently not infinite quantities, are integers.
For v let a small circle of radius p be drawn round a r : then, if z a r = pe 9 *, w
have
c r dz
z a.
= c r idd,
anfl = dP s (z a r ).
z a s
We proceed as before : the expression for the function in the forme
case is changed so that now the sum ^P 8 (z a r ) for s = l, ..., r 1,
r + 1, ..., n is a uniform function ; there is no other change. In exactly the
same way as before, we shew that every one of the coefficients c r is an
integer.
Hence it appears that if a given function f(z) have, in the finite part of
the plane, n essential singularities a l} ...,a n and no other singularities, and if
it have no zeros anywhere in the plane, then
-
/(*) dz i= i z - di i=l
where all the coefficients c t - are integers, the functions G converge everywher
in the plane except at the essential singularities, and Gi vanishes for
Z - a t
Now, since f(z) has no singularity at oo , we have for very large values of
2i ( 2
and therefore, for very large values of z,
df(z) = _Vi 1,^4.
+ ^
dz
Thus there is no constant term in - 7 -r , , and there is no term in - . But
/(*) dz z
the above expression for it gives C as the constant term, which must therefore
67.] A FUNCTION 125
vanish ; and it gives 2c, : as the coefficient of - , for ^- \G,- (-- H will begin
dz { \z OiJ)
with at least ; thus 2c t - must therefore also vanish.
z
Hence for a function f(z), which has no singularity at z = <x> and no
zeros anywhere in the plane, and of which the only singularities are the n
essential singularities at a 1} a 2 , .... a n , we have
f(z) dz i=l z-ai <=1 dz
where the coefficients Cf are integers subject to the condition
n
2 Cf=0.
1=1
If a n = oo , so that z = oo is an essential singularity in addition to Oi, a 2r
. . . , _!, there is a term (r (2) instead of 6r n ( -- ) ; there is no term, that
\z a n /
corresponds to - , but there may be a constant C. Writing
a
with the condition that G(z) vanishes when z = 0, we then have
/ \ i * ' T^ 7 M-* v^ / ' ** 7
/ (z) dz 1=1 z a t dz i=1 dz
where the coefficients C{ are integers, but are no longer subject to the
condition that their sum vanishes.
Let R* (z) denote the function
i=l
the product extending over the factors associated with the essential
singularities of f(z) that lie in the finite part of the plane; thus R* (z}
is a rational meromorphic function. Since
1 d*0) = v G J
R* (z) dz i=\ z Ui '
we have
d f(*l _ L_ dR*(z) 5 d_ (^
\ ^-* r
_
f(z) dz R*(z} dz id* l \z-a
where G n ( - - ] is to be replaced by G (z) if a n = oo , that is, if z = oo be an
\z - aj
essential singularity of f(z). Hence, except as to an undetermined constant
factor, we have
which is therefore an analytical representation of a function with n essential
126 PRODUCT-EXPRESSION OF [67.
singularities, no accidental singularities, and no zeros : and the rations
function R* (z) becomes zero or oo only at the singularities of f(z\
If z = oo be not an essential singularity, then R* (z) for z = oo is equal t<
M
unity because S c,- = 0.
COROLLARY. It is easy to see, from 43, that, if the point Oi be only an
accidental singularity, then c is a negative integer and Gi [ ) is zero: si
\Z ' fti/
that the polar property at o- is determined by the occurrence of a factor
(z a f ) c solely in the denominator of the rational meromorphic function R* (z
And, in general, each of the integral coefficients c t - is determined from th
expansion of the function /' (z} -f(z) in the vicinity of the singularit
with which it is associated.
68. Another form of expression for the function can be obtained from
the preceding; and it is valid even when the function possesses zeros
not absorbed into the essential singularities^.
Consider a function with one essential singularity, and let a be the
point. Suppose that, within a finite circle of centre a (or within a finite
simple curve which encloses a), there are m simple zeros o, $, . .., A, of the
function f(z)\ assume m to be finite, and also assume that there are n<
accidental singularities within or on the circle, or at a merely infinitesimal
distance from its circumference. Then, if
the function F (z) has a for an essential singularity and has no zeros within
the circle. Hence, for points z within the circle,
z-a
where G^ ( ) converges uniformly everywhere in the plane outside a
small circle round a and vanishes with - , and P (z a) is an integral
2 ~~ CL
function converging uniformly within the circle ; moreover, c is an integer.
Thus
G> -
- a e
t See Guichard. Theorie des points singuliers essentiels, (These, Gauthier-Villars, Paris, 1888),
especially the first part.
68.] A FUNCTION 127
then
\ u//
-} SP(z-a)dz
/ 1 \ G.f^-}
= A(z a) m+c g l ( -}e \*-*J e
\z - aj
Now of this product-expression forf(z) it should be noted :
(i) That m -f c is an integer, finite because m and c are finite :
o t (-L,}
(ii) The function e ^ z ~ a ' can be expressed in the form of a series con-
verging uniformly everywhere outside a small circle round a, and proceeding
in powers of --- in the form
z-a
6,
1+ _ _+
....
z a (z ay
It has no zero within the circle considered, for F(z} has no zero. Also g [ - )
\z a/
is a polynomial in - , beginning with unity and containing only a finite
2> "~~ a
number of terms : hence, multiplying the two series together, we have as the
product a series proceeding in powers of --- in the form
z ~~ a
...,
z a (z af
which converges uniformly everywhere outside any small circle round a. Let
this series be denoted by H ( - ] ; it has an essential singularity at a and
\z-aj'
its only zeros are the points a, ft, ..., X, because the series multiplied by
0i (- ] has no zeros :
y \z-aj
(iii) The function / P {z a) dz is a series of positive powers of z a,
converging uniformly in the vicinity of a; and therefore $/?(*-)*' can be
expanded in a series of positive integral powers of z - a, which converges
in the vicinity of a. Let it be denoted by Q (z a) which, since it is a
factor of F(z), has no zeros within the circle.
Hence we have
where /JL is an integer ; H ( - ) is a series that converges everywhere
outside an infinitesimal circle round a, is equal to unity when - - vanishes,
Z ~~ CL
and has as its zeros the (finite) number of zeros assigned to f(z) within a
128 GENERAL FORM OF A FUNCTION [68.
finite circle of centre a ; and Q (z a) is a series of positive powers of z a '
beginning with unity which converges (but has no zero) within the circle.
The foregoing function f(z) is supposed to have no essential singularity
except at a. If, however, a given function have singularities at points
other than a, then the circle would be taken of radius less than the distance
of a from the nearest essential singularity.
Introducing a new function /j (z) defined by the equation
the value of /i (z) is Q (z a) within the circle, but it is not determined by
the foregoing analysis for points without the circle. Moreover, as (z of-
I 1 x
and also H ( - ) are finite everywhere except in the immediate vicinity of
\z - a'
the isolated singularity at a, it follows that essential singularities of f(z)
other than a must be essential singularities of _/i (z). Also since /, (z) is
Q (z - a) in the immediate vicinity of a, this point is not an essential
singularity of f^ (z).
Thus /j (z) is a function of the same kind as f(z) ; it has all the essential
singularities of f(z) except a, but it has fewer zeros, on account of the m'
zeros of f(z) possessed by H ( - - ) . The foregoing expression for f(z) is
\z dj
the one referred to at the beginning of the section.
Gi ( J_\
If we choose to absorb into fi(z) the factors e ' **-' and e ^ p(z ~ a ^ dz ,
which occur in
&l k- J
A(z- a) m+e
\z-a
an expression that is valid within the circle considered, then we obtain a
result that is otherwise obvious, by taking
where now q^ { - ) is polynomial in - . and -has for its zeros all the
\z-aj z -a
zeros within the circle ; /u, is an integer ; and /i (z) is a function of the same
kind as f(z), which now possesses all the essential singularities of f(z), but
/ I \
its zeros are fewer by the m zeros that are possessed by g l ( - J .
69. Next, consider a function f(z) with n essential singularities Ojj
a 2 ,..., ci n but without accidental singularities; and let it have any number
of zeros.
.69.] WITH ESSENTIAL SINGULARITIES 129
When the zeros are limited in number, they may be taken to be isolated
points, distinct in position from the essential singularities.
When the zeros are unlimited in number, then at least one of the
singularities must be such that the zeros in infinite number lie within
a circle of finite radius, described round it as centre and containing no other
singularity. For if there be not an infinite number in such a vicinity of
some one point (which must be an essential singularity : the only alternative
is that the zeros should form a continuous aggregate, and then the function
would be zero everywhere), the points are isolated and there must be an
infinite number outside a circle \z = JR, where R is a finite quantity that
can be made as large as we please, say an infinite number at z = <x> . If
2 = 00 be an essential singularity, the above alternative is satisfied: if not,
the function, as in the preceding alternative, must be zero at all other parts
of the plane. Hence it follows that, if a uniform function have a finite number
of essential singularities and an infinite number of zeros, all but a finite
number of the zeros lie within circles of finite radii described round the
essential singularities as centres ; at least one of the circles contains an
infinite number of the zeros, und some of the circles may contain only a finite
number of them.
We divide the whole plane into regions, each containing one but only one
singularity and containing also the circle round the singularity ; let the
region containing a t - be denoted by Ci, and let the region G n be the part of
the plane other than Cj, C 2 , ..., C n -i-
If the region C l contain only a limited number of the zeros, then, by 68,
we can choose a new function /i (z) such that, if
the function /i (2) has i for an ordinary point, has no zeros within the region
(7u and has a 2 , a s , ..., a n for its essential singularities.
If the region C l contain an unlimited number of the zeros, then, as in
Corollaries II. and III. of 63, we construct any transcendental function
(TJ [ - ] , having a^ for its sole essential singularity and the zeros in C l for
* 2 Of j /
all its zeros. When we introduce a function g 1 (2), defined by the equation
the function g- i (z) has no zeros in C^ and certainly has a^, a s , ..., a n for
essential singularities; in the absence of the generalising factor of G 1} it can
have Oj for an essential singularity. By 67, the function g l (z), defined by
F. F.
130
GENERAL FORM OF A FUNCTION
[69.
has no zero and no accidental singularity, and it has a l as its sole essential
singularity : hence, properly choosing d and Aj , we may take
9i 0) = 9i (*)/i (z\
so that yj (2) does not have a^ as an essential singularity, but it has all the
remaining singularities of g^ (z), and it has no zeros within CV
In either case, we have a new function /j (z} given by
where /^ is an integer. The zeros of/0) that lie in (7 a are the zeros of G^, the
function /j 0) has 0%, a 3 , ..., a n (but not a,) for its essential singularities,
and it has the zeros off(z) in the remaining regions for its zeros.
Similarly, considering (7 2 , we obtain a function f^(z), such that
, (z) = (z-
G 2
Z
00,
where /^ is an integer, G 2 is a transcendental function finite everywhere except
at 2 and has for its zeros all the zeros of/i (z) and therefore all the zeros of
f(z) that lie in (7 2 . Then/ 2 (z) possesses all the zeros of f(z) in the regions
other than G l and (7 2 , and has a 3 , 4 , ..., a n for its essential singularities.
Proceeding in this manner, we ultimately obtain a function f n (z) which
has none of the zeros of f(z) in any of the n regions C 1} C 2 , ..., C n , that is,
has no zeros in the plane, and it has no essential singularities; it has no
accidental singularities, and therefore f n (z) is a constant. Hence, when we
n
substitute, and denote by $* (z) the product II (z-ai)*', we have
which is the most general form of a function with n essential singularities, no
accidental singularities, and any number of zeros. The function S* (z) is a
rational function of z, usually meromorphic in form, and it has the essential
singularities of f(z) as its zeros and poles; and the zeros of f(z) are dis-
tributed among the functions &,-.
As however the distribution of the zeros by the regions C and therefore
the functions G ( j are somewhat arbitrary, the above form though general
\z aj
is not unique.
If any one of the singularities, say a m , had been accidental and not
essential, then in the corresponding form the function G m ( - - j would be
\ z ~ a m/
polynomial and not transcendental.
70.] WITH ESSENTIAL SINGULARITIES 131
70. A function f(z), which has any finite number of accidental singu-
larities in addition to n assigned essential singularities and any number of
anxi <jned zeros, can be constructed as follows.
Let A (z) be the polynomial which has, for its zeros, the accidental
singularities off(z), each in its proper multiplicity. Then the product
f(z}A(z)
is a function which has no accidental singularities ; its zeros and its essential
singularities are the assigned zeros and the assigned essential singularities of
f(z), and therefore it is included in the form
n ( / 1
s* (z) n \Gi f
\z
where 8* (z) is a rational meromorphic function having the points o 1} a 2 , . .. , a n
for zeros and poles. The form of the function /(V) is therefore
A (z) i=l } l \z-
71. A function f(z), which has an unlimited number of accidental singu-
larities in addition to n assigned essential singularities and any number of
assigned zeros, can be constructed as follows.
Let the accidental singularities be a', /3', Construct a function f (z),
having the n essential singularities assigned to f(z), no accidental singu-
larities, and the series a', /3', ... of zeros. It will, by 69, be of the form of a
product of n transcendental functions G n +i, G m , which are such that a
function G has for its zeros the zeros of /j (z) lying within a region of the plane,
divided as in 69 ; and the function G n+ t is associated with the point a.
Thus
i-i
where T* (z) is a rational meromorphic function having its zeros and its
poles, each of finite multiplicity, at the essential singularities off(z).
Because the accidental singularities of/ (z} are the same points and have
the same multiplicity as the zeros of ^(z), the function f(z)f 1 (z) has no
accidental singularities. This new function has all the zeros of f(z), and
Oj, ..., a n are its essential singularities; moreover, it has no accidental singu-
larities. Hence the product f(z)fi (z) can be represented in the form
z -
and therefore we have
as an expression of the function.
92
132 GENERAL FORM OF A FUNCTION [71.
But, as by their distribution through the n selected regions of the plane
in 69, the zeros can to some extent be arbitrarily associated with the
functions (?,, G 3 , ..., G n and likewise the accidental singularities can to some
extent be arbitrarily associated with the functions 6r n +i, G n+ ^, ..., G m , the
product-expression just obtained, though definite in character and general,
is not unique in the detailed form of the functions which occur.
The fraction 7pW7\
1 (z)
is rational, neither S* nor T* being transcendental ; it vanishes or becomes
infinite only at the essential singularities a u a^, ..., a n , being the product
of factors of the form (z a^i, for i = 1 , 2, . . . , n. Let the power (z a^) m i
be absorbed into the function Gi/G n+ i for each of the n values of i; no
substantial change in the transcendental character of Gi and of G n+i is
thereby caused, and we may therefore use the same symbol to denote the
modified function after the absorption. Hence f the most general product-
expression of a uniform function of z, which has n essential singularities
Oj, Oa, ..., a n , any unlimited number of assigned zeros, and any unlimited
number of assigned accidental singularities, is
r (
"* &i
-r-I
The resolution of a transcendental function with one essential singularity
into its primary factors, each of which gives only a single zero of the function,
has been obtained in 63, Corollary IV.
We therefore resolve each of the functions G lt ..., G m into its primary
factors. Each factor of the first n functions will contain one and only one zero
of the original functions f(z) ; and each factor of the second n functions will
contain one and only one of the poles of f(z\ The sole essential singularity
of each primary factor is one of the essential singularities off(z). Hence we
have a method of constructing a uniform function with any finite number of
essential singularities as a product of any number of primary factors, each
of which has one of the essential singularities as its sole essential singularity
and either (i) has as its sole zero either one of the zeros or one of the
accidental singularities of f(z), so that it is of the form
z
or (ii) it has no zero and thBn it is of the form
t Weierstrass, Ges. Werke, t. if, p. 121.
71.] WITH ESSENTIAL SINGULARITIES 133
When all the primary factors of the latter form are combined, they constitute
a generalising factor in exactly the same way as in 52 and in 63,
Cor. III., except that now the number of essential singularities is not
limited to unity. The product converges uniformly for all finite values of z
that lie outside small circles round the singularities ; and similarly for infinite
values, if the function is regular for z = oo .
Two forms of expression of a function with a limited number of essential
singularities have been obtained : one ( 65) as a sum, the other. ( 69) as a
product, of functions each of which has only one essential singularity. Inter-
mediate expressions, partly product and partly sum, can be derived, e.g.
expressions of the form
1
-c
But the pure product-expression is the most general, in that it brings into
evidence not merely the n essential singularities but also the zeros and the
accidental singularities, whereas the expression as a sum tacitly requires that
the function shall have no singularities other than the n which are essential.
Note. The formation of the various elements, the aggregate of which is the complete
representation of the function with a limited number of essential singularities, can be
carried out in the same manner as in 34 ; each element is associated with a particular
domain, the range of the domaiu is limited by the nearest singularities, and the aggregate
of the singularities determines the boundary of the region of continuity.
To avoid the practical difficulty of the gradual formation of the region of continuity
by the construction of the successive domains when there is a limited number of
singularities (and also, if desirable to be considered, of branch-points), Fuchs devised
a method which simplifies the process. The basis of the method is an appropriate change
of the independent variable. The result of that change is to divide the plane of the
modified variable f into two portions, one of which, G%, is finite in area and the other of
which, GI, occupies the rest of the plane; and the boundary, common to G 1 and G%, is
a circle of finite radius, called the discriminating circle* of the function. In G 2 the
modified function is holomorphic; in G l the function is holomorphic except at f=oo ;
and all the singularities (and the branch-points, if any) lie on the discriminating circle.
The theory is given in Fuchs's memoir " Ueber die Darstellung der Functionen com-
plexer Variabeln, ," Crelle, t. Ixxv, (1872), pp. 176 223. It is corrected in details
and is amplified in Crelle, t. cvi, (1890), pp. 1 4, and in Crelle, t. cviii, (1891),
pp. 181 192; see also Nekrassoff, Math. Ann., t. xxxviii, (1891), pp. 82 90, and"
Anissimoff, Math. Ann., t. xl, (1892), pp. 145148.
* Fuchs calls it Grenzkreis.
CHAPTER VII.
FUNCTIONS WITH UNLIMITED ESSENTIAL SINGULARITIES, AND EXPANSION
IN SERIES OF FUNCTIONS.
IN addition to the memoirs mentioned below, as being the basis of the present chapter,
there are several others (alluded to at the end of 35) of the greatest importance, dealing
with the general theory of uniform analytic functions and particularly with their analytical
representation by an infinite series of polynomials in the variable. Among these, specially
worthy of note, are :
Runge, Acta Math., t. vi, (1885), pp. 229248 ;
Hilbert, Gott. Nachr., (1897), pp. 6370;
Painleve, Comptes Rendus, t. cxxvi, (1898), pp. 200202, 318321, 385388, 459-461,
ib. t. cxxviii, (1899), pp. 1277 1280, ib. t. cxxix, (1899), pp. 2731 ; see also his
thesis, quoted in 86 ;
Phragme"n, Comptes Rendus, t. cxxviii, (1899), pp. 1434 1437;
Mittag-Leffler, Acta Math., t. xxiii, (1900), pp. 43 62, where references are given
to earlier records of the investigations; also Camb. Phil. Trans., (Stokes Jubilee
volume), t. xviii, (1900), pp. 111; and Act* Math., t. xxiv, (1901), pp. 183244.
See also Borel, Lecons sur la theorie des fonctions, (Gauthier-Villars, Paris, 1898),
ch. vi.
A comprehensive reference may here be given to the Collection de monographies sur la
theorie des fonctions, publiee sous la direction de M. fimile Borel. The earliest of
them is the monograph by Borel just quoted; and some of them deal solely with
functions of real variables.
72. It now remains to consider functions which have an infinite number
of essential singularities*. It will, in the first place, be assumed that the
essential singularities are isolated points, that is, that they do not form a
continuous line, however short, and that they do not constitute a continuous
* The results in the present chapter are founded, except where other particular references are
given, upon the researches of Mittag-Leffler and Weierstrass. The most important investigations
of Mittag-Leffler are contained in a series of short notes, constituting the memoir "Sur la theorie
des fonctions uniformes d'une variable," Comptes Rendus, t. xciv, (1882), pp. 414, 511, 713, 781,
938, 1040, 1105, 1163, t. xcv, (1882), p. 335 ; and in a memoir " Sur la representation analytique
des fonctions monogenes uniformes," Acta Math., t. iv, (1884), pp. 1 79. The investigations of
Weierstrass referred to are contained in his two memoirs " Ueber einen functionentheoretischen
Satz des Herrn G. Mittag-Leffler," (1880), and " Zur Functionenlehre," (1880), both included in
the volume Abhandlungen aus der Functionenlehre, pp. 53 66, 67 101, 102 104, Ges. Werke,
t. ii, pp. 189 199, 201 233. A memoir by Hermite, " Sur quelques points de la theorie des
fonctions," Acta Soc. Fenn., t. xii, pp. 67 94, Crelle, t. xci, (1881), pp. 54 78, maybe consulted
with great advantage.
72.] MITTAG-LEFFLER'S THEOREM 135
area, however small, in the plane. Since their number is unlimited and
their distance from one another is finite, there must be at least one point in
the plane (it may be at z = oo ) where there is an. infinite aggregate of such
points. But no special note need be taken of this fact, for the character of an
essential singularity does not enter into the question at this stage ; the
essential singularity at such a point would merely be of a nature different
from the essential singularity at some other point.
We take, therefore, an infinite series of quantities a 1} a 2 , a 3 , ... arranged in
order of increasing moduli, and such that no two are the same : and so we
have infinity as the limit of j | when v = oo .
Let there be an associated series of uniform functions of z such that
for all values of i, the function Gt(- -} , vanishing with '- , has a* as
\z ail z cii
its sole singularity; the singularity is essential or accidental according as
G{ is transcendental or polynomial. These functions can be constructed
by theorems already provod. Then we have the theorem, due to Mittag-
Leffler: It is always possible to construct a uniform analytic function F(z),
having no singularities other than a 1} a 2 , a s , ... and such that for each
determinate value of v, the difference F(z) G v f ) is finite for z = a v
\z-aj
and therefore, in the vicinity of a v , is expressible in the form P(z ).
73. To prove Mittag-Leffler's theorem, we first form subsidiary functions
F v (z}, derived from the functions G as follows. The function G
converges everywhere in the plane except within an infinitesimal circle round
the point a v ; hence within a circle | z \ = p, where p is less than a v , it is a
monogenic analytic function of z, and can therefore be expanded in a series
of positive powers of z which converges uniformly within the circle z\= p,
say
G v
z-a ,1=0
for values of z such that z ^ p < j a v . If a v be zero, there is evidently no
expansion.
Let e be a positive quantity less than 1, and let e 1} e 2 , e 3 , ... be arbitrarily
chosen positive decreasing quantities, subject to the single condition that Se
is a converging series, say of sum A : and let e be a positive quantity inter-
mediate between 1 and e. Let a be the greatest value of G v ( )
y & \z a...l
for
points on or within the circumference z \ = e 1 a v ; then, because the series
00
2 v^z* is a converging series, we have, by 29,
136
or
MITTAG-LEFFLER S
[^ ic 9
n O>v
Hence, with values of z satisfying the condition
z ^ e
[73.
, we have, for
2 v f
i*. = m
I--
n
since e < e . Take the smallest integral value of m such that
9
- <
it will be finite and may be denoted by m v . Thus we have
7f- e
Z < t v ,
for values of z satisfying the condition z \ ^ e a v \.
We now construct a subsidiary function F v (z} such that, for all values of z,
m v 1
then, for values of z \ which are 3: e\a v ,
-!
Moreover, the function X v^zf- is finite for all finite values of z ; so that, if we
^=o
take
then <f> v (z) is zero at infinity, because, when z= oo , G v ( - ) is finite by
\0 Cty/
hypothesis. Evidently <$> v (z) is infinite only at z = a v , and its singularity is
of the same kind as that of G v ( ) .
U - aj
74. Now let c be any point in the plane, which is not one of the points
tt 1} Oz, a 3 , ... ; it is possible to choose a positive quantity p such that all the
points-a lie without the circle | z c \ = p. Let be the singularity, which
is. the point nearest to the origin satisfying the condition a, >|c +p', then,
for points within or on the circle, we have
z_
a. '
74.]
THEOREM
137
when s has the values v, v + 1, v + 2, Introducing the subsidiary functions
F v (z\ we have, for such values of z,
and therefore
a finite quantity. Also let B denote any assigned finite positive quantity,
H+r
however small ; an integer // can be chosen so that 2 s < 8, for all integers
s=p.
ft ^ /i', and for all positive integers r. For these same integers, we have
r /n+r fi+r
F s (z) < 2, F g (z} < 2 e g ^8.
>i = /* S = fJ.
00
It therefore follows that the series S ^ (z) converges uniformly for all
S = v
values of z which satisfy the condition z - c ^ p. Moreover, all the functions
F-i(z), F 2 (z),..., F r _i(z) are finite for such values of z, because their singularities
lie without the circle j z c = p ; and therefore the series
I F r (z\
r = l
converges uniformly for all points z within or on the circle | z c \ = p, where
p is chosen so that all the points a lie without the circle.
The function, represented by the series, can therefore be expanded in the
form P (z c), in the domain of the point c.
If a m denote any one of the points a 1} a 2 , ..., and we take p' so small that
all the points, other than a m , lie without the circle
z- a,
= P
then, since F m (z) is the only one of the functions F which has a singularity
at a m , the series
where 2 implies that F m (z) is omitted, converges uniformly in the vicinity
of a, and therefore it can be expressed in the form P (z a m ). Hence
r=l
138 MITTAG-LEFFLER'S [74.
the difference of F m and G m being absorbed into the series P to make P l . It
00
thus appears that the series S F r (z) is a function which has infinities only
r=l
at the points Oj , a a , . . . , and is such that
can be expressed in the vicinity of a m in the form P(z a m ). Hence 2 F r (z)
is a function of the required kind.
r=l
75. It may be remarked that the function is not unique. As the
positive quantities e were subjected to merely the single condition that they
form a converging series, there is the possibility of wide variation in their
choice : and a difference of choice might easily lead to a difference in the
ultimate expression of the function.
This latitude of ultimate expression is not, however, entirely unlimited.
For, suppose there are two functions F(z} and P '(z), enjoying all the assigned
properties. Then as any point c, other than aj, Og, . . . , is an ordinary point for
both F (z) and F (z), it is an ordinary point for their difference : and so
F(z)-F(z) = P(z-c}
for points in the immediate vicinity of c. The points a are, however,
singularities for each of the functions: in the vicinity of such a point a t -,
we have
since the functions are of the required form : hence
or the point c^ is an ordinary point for the difference of the functions. Hence
every finite point in the plane, whether an ordinary point or a singularity
for each of the functions, is an ordinary point for the difference of the
functions : and therefore that difference is a uniform integral function of z.
It thus appears that, if F (z) be a function with the required properties, then
every other function with those properties is of the form
F(z)+G(z\
where G (z} is a uniform integral function of z either transcendental or
polynomial.
The converse of this theorem is also true.
75.] THEOREM 139
oo
Moreover, the function G (z) can always be expressed in a form S g v (z), if
v=l
it be desirable to do so : and therefore it follows that any function with the
assigned characteristics can be expressed in the form
*=i
Note. In the preceding investigation, the integers m v have not been limited to be the
same for each of the functions G. The simplest sets of functions evidently arise when a
common value can be assigned to the integers; they then correspond to Weierstrass's
converging infinite products ( 50, 59 61), arranged according to their class. But as
with the converging infinite products ( 51), it may happen that no common value can
be assigned : and then the preceding investigation, in its most general form, establishes
the existence of the functions.
It does not, however, indicate that the expression is unique. If, for instance, the
series of functions G be
for 7i = l, 2,..., the function formed by the preceding method is
* 1 /I I7\m ~ 1 ~ | '
and there is no finite integer which, when assigned as the common value of the
integers m n , will make the series converge.
But we may use the function
_
n =inlog e n(x-\og e ri)'
which satisfies all the conditions and is a converging series*.
76. The following applications, due to Weierstrass, can be made so as
to give a new expression for functions, already considered in Chapter VI.,
having z = oo as their sole essential singularity and an unlimited number
of poles at points c^, a 2 ,
If the pole at a t - be of multiplicity ra f , then (z ai) m if(z) is regular at
the point a t and can therefore be expressed in the form
mi -I
Hence, if we take / (z) = 2 cv (z - ftf
/*=o
we have f(z) =fi (z) + P(z - at).
Now deduce from fi(z) a function F t (z) as in 73, and let this deduction be
effected for each of the functions f { (z). Then we know that
This remark was made to me by Prof. A. C. Dixon.
140 FUNCTIONS POSSESSING [76.
is a uniform function of z having the points a^, a*,, ... for poles in the proper
multiplicity and no essential singularity except z = oc . The most general
form of the function therefore is
r=l
Hence any uniform analytical function which has no essential singularity
except at infinity can be expressed as a sum of functions each of which has only
one singularity in the finite part of the plane. The form of F r (z) is
f r (z)-G r (z\
where f r (z) is infinite at z = a,., and Q r (z) is a properly chosen integral
function.
We pass to the case of a function, having a single essential singularit} r at
c and at no other point, and any number of accidental singularities, by taking
/ = - as in 63, Cor. II. : and so we obtain the theorem :
z c
Any uniform function which has only one essential singularity, say at c,
can be expressed as a sum of uniform functions each of which has only one
singularity different from c.
Evidently the typical summative function F r (z) for the present case is of
the form
77. The results, which have been obtained for functions possessed of
an infinitude of singularities, are valid on the supposition, stated in 72,
that the limit of a v with indefinite increase of v is infinite ; the terms
in the sequence a l} a^, ... tend to one definite limiting point which is
=oo and, by the substitution z'(z c) = \, can be made any point c in
the finite part of the plane.
Such a sequence-, however, does not necessarily tend to one definite limiting
point : it may, for instance, tend to condensation on a curve, though the
condensation does not imply that all points of the continuous arc of the
curve must be included in the sequence. We shall not enter into the dis-
cussion of the most general case, but shall consider that case in which the
sequence of moduli a^ \, \a 2 \, ... tends to one definite limiting value so that,
with indefinite increase of v, the limit of j a v \ is finite and equal to R ;
the points a^, a?, ... tend to condense on the circle \z\=R.
Such a sequence is given by
77.] UNLIMITED SINGULARITIES 141
for /!- = 0, 1,..., n, and w = l, 2, ... ad inf. ; and another* by
where e is a positive proper fraction.
With each point a m we associate the point on the circumference of the
circle, say b m , to which a^ is nearest : let ._
so that p w approaches the limit zero with indefinite increase of m. There
cannot be an infinitude of points a p , such that -p p ^ , any assigned positive
quantity ; for then either there would be an infinitude of points a within or
on the circle z \ = R 6, or there would be an infinitude of points a within
or on the circle z \ = R + , both of which are contrary to the hypothesis
that, with indefinite increase of i>, the limit of a, \ is R. Hence it follows
that a finite integer n exists for every assigned positive quantity , such that
when 7H ^ 71.
Then the theorem, which corresponds to Mittag-Leffler's as stated in 72
and which also is due to him, is as follows :
It is always possible to construct a uniform analytical function of z which is
definite over the whole plane, except within infinitesimal circles round the points
a <nid b, and which, in the immediate vicinity of each one of the singularities a,
can be expressed in the form
Gt( }+P(z-a i ),
\Z diJ
where the functions Gi are assigned functions, vanishing with - - , and finite
z a{
everywhere in the plane except at the single points a t with which they are
respectively associated.
In establishing this theorem, we shall need a positive quantity e less than
unity and a converging series e 1} e 2 , e 3 , ... of positive quantities, all less than
unity.
Let the expression of the function G n be
n ( \ Cn ' 1 4- Cn ' 8 4- Cn ' 3 4-
\jTvt 1 " ~~ I / \o "l / \o I
n \z-aj z-a n (z-a n f (z - a n ) 3
/ i \ 1 1 n n I
Then, since z - a n = (z - b n ) U --- r- h ,
( Z O n )
the function G n can be expressed! in the form
n _ 7)
n "
z-a
* The first of these examples is given by Mittag-Leffler, Acta Math., t. iv, p. 11 ; the second
was stated to me by Prof. Buruside.
t The justification of this statement is to be found in the proposition in 82.
142 FUNCTIONS POSSESSING
for values of z such that
[77.
a n - b n
z-b n
< e;
and the coefficients A are given by the equations
% C n ,r (/*-]
-" ~
Now, because (r n is finite everywhere in the plane except at a n , the series
has a finite value, say g, for any non-zero value of the positive quantity n ;
then
tc
Hence
>n,r "* </n
Kr| U-l)!
r=i
n
< 9% n 1 1
^ 7 I S - 1 - T^
61
n[
Introducing a positive quantity a such that
we choose so that n < a j a n 6 n | ;
and then | A n< M < go. (1 + a)"" 1 .
Because (1 + a) e is less than unity, a quantity 6 exists such that
Then for values of z determined by the condition
a n -b n
\ z-b n
< e, we have
a n b n
z-b n
go.
a 1-0'
Let the integer m n be chosen so that
go.
-i , -i /j < c n 3
1 + a 1 a
it will be a finite integer, because < 1. Then
I,M
We now construct, as in 73, a subsidiary function F n (z), defining it by
the equation
z-a
n' M =
"' M V z - b n J '
77.]
UNLIMITED SINGULARITIES
143
a n b n
< e, we have
so that for points z determined by the condition
\F n (z)\<e n .
A function with the required properties is
F(z) = 2 F m (z).
To prove it, let c be any point in the plane distinct from any of the points
a and b ; we can always find a value of p such that the circle
I v c I n
C I P
contains none of the points a and 6. Let I be the shortest distance between
this circle and the circle of radius R, on which all the points b lie ; then for
all points z within or on the circle | z c = p, we have
\z-b m \>l.
Now we have seen that, for any assigned positive quantity @, there is a
finite integer n such that
a m -b m \< S,
when m^n. Taking @ = el, we have
a m -b ri
z-b r ,
when m^n,n being the finite integer associated with the positive quantity el.
It therefore follows that, for points z within or on the circle | z c =p,
when m is not less than the finite integer n ; hence
00
2 \F m (z) <e n + e n+1 + e n +2+ ....
m=n
Now the series of positive quantities e 1} e 2 , ... converges; and therefore
1 F m (z)
m=n
is a series, which converges uniformly and unconditionally. Each of the
functions F 1 (z), F 2 (z), ... , F n ^ (z) is finite when z c \ ^. p ; and therefore
5 F m (z)
OT = 1
is a series which converges uniformly and unconditionally for all values of z
within the circle
| z - c \ = p.
144 TRANSCENDENTAL FUNCTION AS [77.
Hence the function represented by the series can be expressed in the form
P (z c) for all such values of z. The function therefore exists over the
whole plane except at the points a. and 6.
It may be proved, exactly as in 74, that, for points z in the immediate
vicinity of a singularity a m ,
+ p <* ~ a -)-
The theorem is thus completely established.
The function thus obtained is not unique, for' a wide variation of choice of
the converging series e l + e 2 4- . .. is possible. But, in the same way as in the
corresponding case in 75, it is proved that, if F(z) be a function with the
required properties, every other function with those properties is of the form
F(z} + G(z\
where G (z) behaves regularly in the immediate vicinity of every point in the
plane except the points b.
Ex. If the points a in Mittag-Leffler's theorem are given by
2^i
n > (*=0, 1, ...,rc-l; n=l,2, ...,00),
and if G m ( -- ) = , shew that
m \z-aj z-a m '
is a function of the character specified in the theorem.
Discuss the nature of the function defined by
(Math. Trip., Part II., 1899.)
78. The theorem just given regards the function in the light of an
infinite converging series of functions of the variable : it is natural to suppose
that a corresponding theorem holds when the function is expressed as an
infinite converging product. With the same series of singularities as in
77, when the limit of a v \ with indefinite increase of v is finite and
equal to R, the theorem* is:
It is always possible to construct a uniform analytic function, which
behaves regularly everywhere in the plane except within infinitesimal circles
* Mittag-Leffler, Ada Math., t. iv, p. 32 ; it may Tbe compared with Weierstrass's theorem
in 67.
78.]
AN INFINITE PRODUCT OF FUNCTIONS
145
round the points a and b, and which in the vicinity of any one of the points a v
can be expressed in the form
where the numbers n^n Z) ... are any assigned integers.
The proof is similar in details to proofs of other propositions and it will
therefore be given only in outline. We have
a v - b v \*
provided
a v
z o v
such values of z,
If we denote
i T r~ ^> I ~~
z a v z b v z b v M=1 \z
< e, the notation being the same as in 77. Hence, for
fZ d v \ n v - n v S
(^b) =e "-'
\ z~b v )
by E v (z), we have E v (z) = e p-mrtt A*
Hence, if F(z) denote the infinite product
we have
n E.(Z\
v=l
- 1 <n v %
i = "=i '
and F(z) is a determinate function provided the double series in the index of
the exponential converges.
Because n v is a finite integer, and because
is a converging series, it is possible to choose an integer m v so that
n v S -
Z
where rj v is any assigned positive quantity. We take a converging series of
positive quantities r) v ; and then the moduli of the terms in the double series
form a uniformly converging series. The double series itself therefore
converges uniformly; and then the infinite product F (z) converges uniformly
for points z such that
a u
< e.
F. F.
10
146 TRANSCENDENTAL FUNCTION HAVING [78.
As in 77, let c be any point in the plane, distinct from any of the
points a and b. We take a finite value of p such that all the points a and b
lie outside the circle j z c \ = p ; and then, for all points within or on this
circle,
a m -b r ,
when m ^ n, n being the finite integer associated with the positive quantity
fl. The product
00
nV / ~ \
/ \Z)
is therefore finite, for its modulus is less than
2 !
* =n
n-l
the product II E, (z)
K=l
is finite, because the circle z c \ = p contains none of the points a and b ;
and therefore the function F(z) is finite for all points within or on the circle.
Hence in the vicinity of c, the function can be expanded in the form P (z c) ;
and therefore the function is definite everywhere in the plane except within
infinitesimal circles round the points a and b.
The infinite product converges uniformly and unconditionally. As in 51,
it can be zero only at points which make one of the factors zero and, from the
form of the factors, this can take place only at the points a v with positive
integers ?. In the vicinity of a,, all the factors of F (z) except E v (z) are
regular; hence F(z)/E l ..(z) can be expressed as a function of z a v in the
vicinity. But the function has no zeros there, and therefore the form of the
function is
e P t (z-a v )
Hence, in the vicinity of a v , we have
on combining the exponential index in E v (z) with P x (z a v ). This is the
required property.
Other general theorems will be found in Mittag-Leffler's memoir just
quoted.
79. The investigations in 72 75 have led to the construction of a
function with assigned properties. It is important to be able to change, into
the chosen form, the expression of a given function, having an infinite series
of singularities tending to a definite limiting point, say to z = oo . It is
79.] AN INFINITE SET OF SINGULARITIES 147
necessary for this purpose to determine (i) the functions F r (z) so that the
or
series 2 F r (z) may converge uniformly and unconditionally, and (ii) the
r = l
function G(z).
Let 4> (z) be the given function, and let S be a simple contour embracing
the origin and /z of the singularities, viz., a n ...,a M : then, if t be any
point, we have
r**w (if dt = r m ft- * + r *w
J t-z\tJ J t-z\t) J t-z
/<)
where I implies an integral taken round a very small circle centre a.
If the origin be one of the points a lt a z , ..., then the first term will be
included in the summation.
Assuming that z is neither the origin nor any one of the points a 1} ..., a^,
\vt- have
so.that
AT
Now
. 1 . f '*
2?rt t
z\t
1 f w 4(O
- . ^ / -
2?riJ t z\t
(m - 1) i !
rm-)
"'
o
~|
' " J
[
w+i*
unless ^ = be a singularity and then there will be no term G (z). Similarly,
it can be shewn" that
z\ m j
/ 1 \ m-l / z \ A.
is equal to G v ( ) - 2 v K { } = F V (z\
\z-aj A=0 \aj
i i \ i r ^ a vt ^) (t}
where G ( I = -~ I dt
102
148 TRANSCENDENTAL FUNCTION AS [79.
and the subtractive sum of m terms is the sum of the first m terms in the
development of G v in ascending powers of z. Hence
M
*(.)-G(,) + S.W
If, for an infinitely large contour, m can be chosen so that the integral
J_ r*<
27ri!t-
z(t
diminishes indefinitely with increasing contours enclosing successive singu-
larities, then
The integer m may be called the critical integer.
If the origin be a singularity, we take
and there is then no term G (z) : hence, including the origin in the summa-
tion, we have
so that if, for this case also, there be some finite value of m which makes
the integral vanish, then
*(*)= S F v (z).
v =
Other expressions can be obtained by choosing for m a value greater than
the critical integer ; but it is usually most advantageous to take m equal to
its lowest effective value.
Ex. 1. The singularities of the function ircotirz are given by 2=X, for all integer
values of X from oo to +x including zero, so that the origin is a singularity.
The integral to be considered is
-f 1 t ' ' it \s\J \J H ts I ~ \ i
/= ~ . | 1 - I at.
2iri J t-Z \tj
AVe take the contour to be a circle of very large radius R chosen so that the circumference
does not pass infinitesimally near any one of the singularities of ir cot irt at infinity ; this
is, of course, possible because there is a finite distance between any two of them. Then,
round the circumference so taken, IT cot irt is never infinite : hence its modulus is never
greater than some finite quantity M.
M dt
Let t=Re , so that = idd; then
r = J_ f 2 '
2*r J
r 9.]
AN INFINITE SERIES OF FUNCTIONS
149
and therefore
W jr.
for some point on the circle. Now, as the circle is very large, we have 1 1 z \ infinite :
hence \J\ can be made zero merely by taking m. unity.
Thus, for the function IT cot irz, the critical integer is unity.
Hence, by the general theorem, we have the equation
1 fir cot irt z ,
TT cot irz= - . 2 I - at.
2irl J t-z t
the summation extending to all the points X for integer values of X= oo to + oo , and
each integral being taken round a small circle centre X.
IT cot irt z ,
if, m
we take t=\ + , we have
where
when f=0; and therefore the value of the integral is
,.
X-Z
In the limit when | f is infinitesimal, this integral
2
__
~X^ X'
and therefore
if X be not zero.
And for the zero of X, the value of the integral is
(z) = - + ^- ,
z A A
so that F (z) is . In fact, in the notation of 72, we have
and the expansion of (? A needs to be carried only to one term.
150 REGION OF CONTINUITY [79.
1 A=oo /I 1\
We thus have IT cot nz = - + 2' r + r ) >
z A=-oo \ z ~ '*- V
the summation not including the zero value of X.
Ex. 2. Obtain, ab initio, the relation
1 _ A=oo 1
' 9~ ^ /_ N~
r. 3. Shew that, if
Trcotirz I , * 1
then =-+2g 2
(Gylden, Mittag-Leffler.
r. 4. Obtain an expression, in the form of a sum, for
TT cot irz
/ 2\ 2 / 2\ 3 / 2\ n
where #(*) denotes (!_*)( i_f] (1 ) ...... (1--J .
\ 2 / \ */ \ nJ
Ex. 5. Construct a uniform analytical function F(x), which is ^finite at all finite
points except at the points 0, 1, 2, 3, ..., at which it is infinite in such a way that
F(x)-e x cotirx
is finite at each' of the points.
(Math. Trip., Part II., 1897.)
80. The results obtained in the present chapter relating to functions
which have an unlimited number of singularities, whether distributed over
the whole plane or distributed over only a finite portion of it, shew that
analytical functions can be represented, not merely as infinite converging
series of powers of the variable, but also as infinite converging series of
functions of the variable. The properties of functions when represented by
series of powers of the variable depended in their proof on the condition that
the series proceeded in powers ; and it is therefore necessary at least to
revise those properties in the case of functions when represented as series
of functions of the variable.
Let there be a series of uniform functions /j (z), / 2 (z), . . . ; then the
aggregate of values of z, for which the series
has a finite value, is the region of continuity of the series. If a positive
quantity p can be determined such that, for all points z within the circle
,z - a I = p,
00
the series fi (z) converges uniformly*, the series is said to converge
t=i
* In connection with most of the investigations in the remainder of this chapter, Weierstrass's
memoir " Zur Functionenlehre ;> already quoted (p. 134, note) should be consulted.
80.] OF A SERIES OF POWERS 151
uniformly in the vicinity of a. If R be the greatest value of p for which this
holds, then the area within the circle
j z a | = R
is called the domain of a; and the series converges uniformly in the vicinity
of any point in the domain of a.
It will be proved in 82 that the function, represented by the series of
functions, can be represented by power-series, each such series being equiva-
lent to the function within the domain of some one point. In order to be
able to obtain all the power-series, it is necessary to distribute the region of
continuity of the function into domains of points where it has a uniform
finite value. We therefore form the domain of a point b in the domain of a
from a knowledge of the singularities of the function, then the domain of
a point c in the domain of b, and so on ; the aggregate of these domains is a
continuous part of the plane which has isolated points and which has one or
several lines for its boundaries. Let this part be denoted by A^.
For most of the functions, which have already been considered, the region
A-,, thus obtained, is the complete region of continuity. But examples will
be adduced almost immediately to shew that A l does not necessarily include
all the region of continuity of the series under consideration. Let a be a
point not in A lt within whose vicinity the function has a uniform finite
value ; then a second portion A 2 can be separated from the whole plane, by
proceeding from a as before from a. The limits of A 1 and A 2 may be wholly
or partially the same, or may be independent of one another : but no point
within either can belong to the other. If there be points in the region of
continuity which belong to neither A l nor A 2 , then there must be at least
another part of the plane A 3 with properties similar to A l and A 2 . And so
00
on. The series 2 f t (z) converges uniformly in the vicinity of every point
i=l
within each of the separate portions of its region of continuity.
It was proved that a function represented by a series of powers has a
definite finite derivative at every point lying actually within the circle
of convergence of the series, but that this result cannot be affirmed for a
point on the boundary of the circle of convergence even though the value of
the series itself should be finite at the point, an illustration being provided
by the hypergeometric series at a point on the circumference of its circle of
convergence. It will appear that a function represented by a series of
functions has a definite finite derivative at every point lying actually within
its region of continuity, but that the result cannot be affirmed for a point
on the boundary ; and an example will be given ( 83) in which the derivative
is indefinite.
Again, it has been seen that a function, initially defined by a given power-
series, is, in most cases, represented by different analytical expressions in
152 REGION OF CONTINUITY OF [80.
different parts of the plane, each of the elements being a valid expression of
the function within a certain region. The questions arise whether a given
analytical expression, either a series of powers or a series of functions :
(i) can represent different functions in the same continuous part of its region
of continuity, (ii) can represent different functions in distinct, that is, non-
continuous, parts of its region of continuity.
81. Consider first a function defined by a given series of powers.
Let there be a region A' in the plane and let the region of continuity of
the function, say g (z}, have parts common with A'. Then if a be any point
in one of these common parts, we can express g (z) in the form P (z a ) in
the domain of .
As already explained, the function can be continued from the domain of
a by' a series of elements, so that the whole region of continuity is gradually
covered by domains of successive points ; to find the value in the domain of
any point a, it is sufficient to know any one element, say, the element in the
domain of a . The function is the same through its region of continuity.
Two distinct cases may occur in the continuations.
First, it may happen that the region of continuity of the function g (z)
extends beyond A'. Then we can obtain elements for points outside A',
their aggregate being a uniform analytical function. The aggregate of
elements then represents within A ' a single analytical function : but as that
function has elements for points without A', the aggregate within A' does not
completely represent the function. Hence :
If a function be defined within a continuous region of a plane by an
aggregate of elements in the form of power-series, which are continuations of
one another, the aggregate represents in that part of the plane one (and only
one) analytical function : but if the power-series can be continued beyond the
boundary of the region, the aggregate of elements within the region is not the
complete representation of the analytical function.
This is the more common case, so that examples need not be given.
Secondly, it may happen that the region of continuity of the function does
not extend beyond A' in any direction. There are then no elements of the
function for points outside A' and the function cannot be continued beyond
the boundary of A'. The aggregate of elements is then the complete repre-
sentation of the function and therefore :
If a function be defined within a continuous region of a plane by an
aggregate of elements in the form of power-series, which are continuations of
one another, and if the power-series cannot be continued across the boundary
of that region, the aggregate of elements in the region is the complete repre-
sentation of a single uniform monogenic function which exists only for values
of the variable within the region.
81.] A SERIES OF FUNCTIONS 153
The boundary of the region of continuity of the function is, in the latter
case, called the natural limit of the function*, as it is a line beyond which
the function cannot be continued. Such a line arises for the series
in the circle \z = 1, a remark due to Kronecker; other illustrations occur
in connection with the modular functions, the axis of real variables being
the natural limit, and in connection with the automorphic functions (see
Chapter XXII.) when the fundamental circle is the natural limit. A few
examples will be given at the end of the present chapter.
It appears that Weierstrass was the first to announce the existence of natural limits
for analytic functions, Berlin. Monatsber. (1866), p. 617; see also Schwarz, Ges. Werke,
t. ii, pp. 240 242, who adduces other illustrations and gives some references ; Klein and
Fricke, Vorl. iiber die Theorie der elliptischen Modulfunctionen, t. i, (1890), p. 110. Some
interesting examples and discussions of functions, which have the axis of real variables
for a natural limit, are given by Hankel, " Untersuchungen iiber die unendlich oft
oscillirenden und unstetigen Functionen," Math. Ann., t. xx, (1870), pp. 63 112.
82. Consider next a series of functions f L (z), / 2 (z), fa (z), of the
variable z.
In the first place, let each of them occur in the form of power-series in z,
with (it may be) positive and negative indices, say in the form
Assume that the power-series for the separate functions, as well as the series
i/.<*)
s=l
of functions, have a common region of continuity in the vicinity of the origin
such that, for values of z given by
R< z = r<R',
the function-series and each of the power-series converge uniformly. Then
the sum
00
2 a^
8=1
has a definite finite value, say A^; for the values of z considered, the series
converges ; and we have
!/.(*) -24^.
* Die natilrliche Grenze, according to German mathematicians.
154
REGION OF CONTINUITY OF
[82.
Let k denote any arbitrary positive quantity, taken as small as we please.
In consequence of the uniform convergence of the function-series, it is possible
to choose an integer m, such that
% /.(*) < P,
i *=M
for all integers n ^ m, and for all values of z such that R < r^ < r < r < R',
where ^ R and R' r 2 are non-vanishing quantities, no matter how small
they may be assigned ; and therefore for the same range of variables, it is
possible to choose an integer m so that, for all integers n^m and for all
finite positive integers p, we have
n+p I oo <
2 /.(*) = ! 2 /.(*)- 2 /.(*)
s=n s=n n+p+l
2 / f (*)
=
< p + p ^ fr.
Owing to the finiteness of the integer p, we have
2 /.(*)
w+p+l
n+p /n+p
2/,(*)-2( 2 t
s=n M \s=n
2 2
^ \s=
so that
for all integers n^m, and for all positive integers p. Hence (Corollary, 29)
< kr-*,
where j z = r ; because &, being greater than the upper limit of the modulus
of the above series for all the values of z considered, is greater than the upper
limit of its modulus for values of z such that \z\=r. It therefore follows
that, because kr~* is an arbitrary quantity assigned as small as we please,
and because an integer m can be chosen such that the above inequality holds
00
for all integers n^m and for all positive integers p, the series 2 a M converges
to a unique finite limit. Denote this by A^.
s=l
Let
- A ' 5 1 n A "
. *! u > U'xn ./I n }
then regarding kr{~* and &r 2 ~** as two assigned quantities, as small as we
please (because k can be assigned as small as we please, and r lt r. 2 are finite
non-vanishing magnitudes), the convergence of the series whose sum is A^
enables us to choose an integer n such that A^" is smaller than each of the
quantities kr-ri* and kr 2 -*; thus
J' < krr*,
<
82.]
Now consider the series
We have
A SERIES OF FUNCTIONS 155
for a value of z such that r l < z r< r 2 .
<k
-i
2
1= -00
_, r
v 1 z. (L\* i~ TI
<2* A/ I " I ^ nt
\ >* / /> ^^. /*
i= ao \'V ' 'i
and therefore
2
r z r
Hence the series S-4/V converges. Moreover, each of the power-series
fi(z), ,/n-i(^) converges uniformly; therefore
= 1
and the latter series converges uniformly. The two series
can therefore be combined into the series
which accordingly is a converging series.
Finally, we have
'Z f (?\ _ T A rt* 5 1 f f>'\
* /s W * -"-M- 2 ^ ~ ^ Js \ z )
*=1 M s=l
and therefore
As the assigned quantity A; is at our disposal, we can choose it so that the
quantity on the right-hand side is smaller than any assignable magnitude :
consequently, for the values of z under consideration, we have
83. In the second place, consider the series of functions fi(z), fz(z),
f 3 (z), ... more generally. The region of continuity may be supposed to
consist of one part or of more than one part : let such a part be denoted by
156 REGION OF CONTINUITY OF [83.
A, and let F(z} denote the function represented by the series within A, so
that
and assume that within A (though not necessarily at points on its boundary)
the function-series converges uniformly. Let a denote any arbitrarily
assumed position within A ; each of the functions f g (z) is regular in the
vicinity of a and is expressible in the form of a power-series P x (z a)
containing only positive powers of z a. By the preceding investigation, the
function-series can be represented as a power-series, and we have
F(z) = P(z- a).
00
In P(z a), the coefficient of (z a,y is A^, which is 2 a M , where a SHL is the
*=i
coefficient of (z ay in f g (z) ; accordingly
dz*
for all values of /z. Since a is any arbitrarily chosen point in A, it follows
that, for all points within A, we have
dz* , =1 dz*
00
As the function-series f g (z) converges uniformly, and as/ (z) is regular in
*=i
the vicinity of a, it is easy to see that the series
s =i z
also converges uniformly ; and therefore the derivatives of the function-series
within the region of continuity are the derivatives of the function the series
represents.
The expression P (z a) is an Element of the function F (z) : and within
the domain of a, contained in the region A, it represents the function. It can
be used for the continuation of F (z) so long as the domains of successive
points lie within A ; but this restriction is necessary, and the full continuation
of P (z a) as an element of a power-series is not necessarily limited by
the region A. It is solely in that part of its region of continuity which is
included within A that it represents the function F (z} ; the boundary of the
region A must not be crossed in forming the continuations of P (z a).
It therefore appears that a converging series of functions of a variable
can be expressed in the form of series of powers of the variable, which
converge within the parts of the plane where the series of functions
converges uniformly ; but the equivalence of the two expressions is limitec
83.] A SERIES OF FUNCTIONS 157
to such parts of the plane, and cannot be extended beyond the boundary of
the region of continuity of the series of functions.
If the region of continuity of a series of functions consist of several parts
of the plane, then the series of functions can in each part be expressed in
the form of a set of converging series of powers : but the sets of series of
powers are not necessarily the same for the different parts, and they are not
necessarily continuations of one another, regarded as power-series.
Suppose, then, that the region of continuity of a series of functions
J= 2 /.(*)
5=1
consists of several parts A lt A 2 , .... Within the part A l let F ' (z) be
represented, as above, by a set of power-series. At every point within A 1}
the values of F (z) and of its derivatives are each definite and unique ; so
that, at every point which lies in the regions of convergence of two of the
power-series, the values which the two power-series, as the equivalents of F(z}
in their respective regions, furnish for F (z) and for its derivatives must be
the same. Hence the various power-series, which are the equivalents of F(z)
in the region A ly are continuations of one another: and they are sufficient to
determine a uniform monogenic analytic function, say F 1 (z). The functions
F(z) and F 1 (z} are equivalent in the region A l ; and therefore, by 81, the
series of functions represents one and the same function for all points within
one continuous part of its region of continuity. It may (and frequently does)
happen that the region of continuity of the analytical function F^ {z} extends
beyond A^; and then F l (z) can be continued beyond the boundary of A^ by
a succession of elements. Or it may happen that the region of continuity
of F l (z) is completely bounded by the boundary of A l ; and then that function
cannot be continued across that boundary. In either case, the equivalence
00
of F l (z] and 2 f s (z) does not extend beyond the boundary of A l} one
' *=i
00
complete and distinct part of the region of continuity of 2 f 8 (z) ; and
*=1
therefore, by using the theorem proved in 81, it follows that:
A series of functions of a variable, which converges within a continuous
part of the plane of the variable z, is either a partial or a complete
representation of a single uniform analytic function of the variable in that
part of the plane.
Further, it has just been proved that the converging series of functions
can, in any of the regions A, be changed into an equivalent uniform analytic
function, the equivalence being valid for all points in that region, say
158 A CONVERGING SERIES OF FUNCTIONS [83.
We have seen that every derivative of F^ (z) at any point within A is the
sum of the corresponding derivatives of f s (z), this sum converging uniformly
within A. The equivalence of the analytic function and the series of
functions has not been proved for points on the boundary ; even if they are
equivalent there, the function F l {z} cannot be proved to have a uniform
finite derivative at every point on the boundary of A, and therefore it cannot
00
be affirmed that 2 f s (2) has, of necessity, a uniform finite derivative at points
=i
oo
on. the boundary of A, even though the value of 2 f s (z) be uniform and finite
at every point on the boundary*.
8=1
Ex. In illustration of the last inference, regarding the derivative of a function at
a point on the boundary of its region of continuity, consider the series
g(z)= 2 to B ,
71=0
where b is a positive quantity less than unity, and a is a positive quantity which will be
taken to be an odd integer.
For points within and on the .circumference of the circle | z \ = 1, the series converges
uniformly and unconditionally ; and for all points without the circle the series diverges.
It thus defines a function for points within the circle and on the circumference, but not
for points without the circle.
Moreover, for points actually within the circle, the function has a first derivative and
consequently has any number of derivatives. But it cannot be declared to have a
derivative for points on the circle : and it will in fact now be proved that, if a certain
condition be satisfied, the derivative for variations at any point on the circle is not merely
infinite but that the sign of the infinite value depends upon the direction of the variation,
so that the function is not monogenic for the circumference t.
Let z=e ei : then, as the function converges unconditionally for all points along the
circle, we take
f(6)= 2 &V* ne *,
71=0
where 6 is a real variable. Hence
n=0
+ 26"
=0
* It should be remarked here, as at the end of 21, that the result in itself does not
contravene Riemann's definition of a function, according to which (8) must have the same
value whatever be the direction of the vanishing quantity dz ; at a point on the boundary of
the region there are outward directions for which dw is not defined.
t The following investigation is due to Weierstrass, who communicated it to Du Bois-
Reymond : see Crelle, t. Ixxix, (1875), pp. 29 31 ; Weierstrass, Ges. Werke, t. ii, pp. 7174.
83.]
NOT POSSESSING A DERIVATIVE
159
assuming //<, in the first place, to be any positive integer. To transform the first sum on
the right-hand side, we take
and therefore
2
n=0
m-1
2
tt=0
m-l
< 2 (&)<
=o
if ab > 1. Hence, on this hypothesis, we have
where y is a complex quantity with modulus < 1.
d
To transform the second sum on the right-hand side, let the integer nearest to a m -
TT
be a,,,, so that
i>"--o*>-i
7T
for any value of m : then taking
a: = a m 5 7ra m ,
we have , JTT ^ .r > ATT ,
and cos .r is not negative. We choose the quantity </> so that
n a
and therefore
which, by taking m sufficiently large (a is >1), can be made as small as we please. We
now have
if a be an odd integer, and
Hence
o m
e a m+ *0i = e a"i(x+ 7ra m ) _ / _ j \ o^ e "^
= - ( - 1 )
(f)
and therefore 2 b m +
n=0
-
9 J
= - ( - 1)"- - - 2 6(1 + e ).
7T ^ M=0
The real part of the series on the right-hand side is
every term of this is positive and therefore, as the first term is 1 +cos x, the real part
> 1+cos.r
160 A CONVERGING SERIES OF FUNCTIONS [8
for cos x is not negative. Also it is finite, for it is
< 2 2 b n
n=0
2
l-b'
Moreover kir < IT -x < iiTr,
Q
so that - is positive and > = . Hence
TT - u; 3
where r; is a finite complex quantity, the real part of which is positive and greater than;
unity. We thus have
where j y' \ < 1, and the real part of 17 is positive and > 1.
Proceeding in the same way and taking
so that
Y=
A
where j y t ' \ < 1 and the real part of T^, a finite complex quantity, is positive and greater
than unity.
If now we take ab 1 > f IT,
the real parts of ~+y' ^~~r ^ ? Sa 7 f f'
3 7T 050 1
and of |li +yi '_L_, sa yofCi,
are both positive and different from zero. Then, since
and "
A.
?n being at present any positive integer, we have the right-hand sides essentially different
quantities, because the real part of the first is of sign opposite to the real part of the
second.
Now let m be indefinitely increased ; then and x are infinitesimal quantities which
ultimately vanish; and the limit of [/(# + <) -/($)] for $ = is a complex infinite
quantity with its real part opposite in sign to the real part of the complex infinite
quantity which is the limit of ~[f(0- x )-f(0}] for x = 0. If/(<9) had a differential
A
coefficient, these two limits would be equal : hence /(#) has not, for any value of i
a determinate differential coefficient.
83.] NOT POSSESSING A DERIVATIVE 161
From this result, a remarkable inference relating to real functions may be at once
derived. The real part of f(6) is
2 6"cos(a n 0),
=o
which is a series converging uniformly and unconditionally. The real parts of
and of + (-l) a "(a&) m fi
are the corresponding magnitudes for the series of real quantities : and they are of opposite
signs. Hence for no value of 6 has the series
2 b n cos(a n 6)
M =
a determinate differential coefficient, that is, we can choose an increase $ and a decrease x
of d, both being made as small as we please and ultimately zero, such that the limits of
the expressions
<> ~X
are different from one another, provided a be an odd integer and
The chief interest of the above investigation lies in its application to functions of real
variables, continuity in the value of which is thus shewn not necessarily to imply the
existence of a determinate differential coefficient defined in the ordinary way. The
application is due to Weierstrass, as has already been stated. Further discussions will
be found in a paper by Wiener, Crelle, t. xc, (1881), pp. 221 252, in a remark by
Weierstrass, Ges. Werke, t. ii, p. 229, and in a paper by Lerch, Crelle, t. ciii, (1888),
pp. 126 138, who constructs other examples of continuous functions of real variables;
and an example of a continuous function without a derivative is given by Schwarz,
Ges. Werke, t. ii, pp. 269274.
The simplest classes of ordinary functions are characterised by the properties: ,
(i) Within some region of the plane of the variable they are uniform, finite, and
continuous :
(ii) At all points within that region (but not necessarily on its boundary) they have
a differential coefficient :
(iii) When the variable is real, the number of maximum values and the number of
minimum values within any given range is finite.
00
The function 2 b n cos (a"0), suggested by Weierstrass, possesses the first but not the
re=0
second of these properties. Kopcke (Math. Ann., t. xxix, pp. 123 140) gives an example
of a function which possesses the first and the second but not the third of these
properties.
84. In each of the distinct portions A l} A 2 , ... of the complete region
of continuity of a series of functions, the series can be represented by a
monogenic analytic function, the elements of which are converging power-
series. But the equivalence of the function-series and the monogenic
analytic function for any portion A l is limited to that region. When the
monogenic analytic function can be continued from A l into A 2 , the continua-
tion is not necessarily the same as the monogenic analytic function which is
F. F. 11
162 ANALYTICAL EXPRESSION [84.
00
the equivalent of the series 2 f g (z) in A 2 . Hence, if the monogenic analytic
=i
functions for the two portions A^ and A 2 be different, the function-series
represents different functions in the distinct parts of its region of continuity.
A simple example will be an effective indication of the actual existence
of such variety of representation in particular cases ; that, which follows, is
due to Tannery*.
Let a, b, c be any three constants ; then the fraction
a + bcz m
1 + bz m '
when m is infinite, is equal to a if | z \ < 1, and is equal to c if j z j > 1.
Let*m , m l , ra 2 , ... be any set of positive integers arranged in ascending
order and be such that the limit of m n , when n = oo , is infinite. Then,
since
a + bcz* a + bcz m 3 (a -f bcz m i a +
+
+ bz m " I + bz m
_ a + bcz , f (z^-i-i - 1)
'
, f (z^-i-i - 1) z m i-i \
*> ^i ((1 + bzi) (T+bt"^))
the function <j> (z), defined by the equation
_ a + bcz m
V ( =H
converges uniformly to a value a if \z < p < 1, and converges uniformly to a
value c if \ z \ ^ p' > 1. But if j z \ = 1, the value to which the series tends
depends upon the argument of z : the series cannot be said to converge for
values of z such that z \ = 1.
The simplest case occurs when b = 1 and ra f = 2 1 ' ; then, denoting the
function by <f>(z), we have
a-cz , z*
,1+1 T
;=o2 2 1
that is, the function <f> (z) is equal to a if j z \ < 1, and it is equal to c if
* It is contained in a letter of Tannery's to Weierstrass, who communicated it to the
Berlin Academy in 1881, Ges. Werke, t. ii, pp. 231 233. A similar series, which indeed is
equivalent to the special form of <f>(z), was given by Schroder, Schlom. Zeitschrift, t. xxii, (1876),
p. 184; and Pringsheim, Math. Ann., t. xxii, (1883), p. 110, remarks that it can be deduced,
without material modifications, from an expression given by Seidel, Crelle, t. Ixxiii, (1871),
pp. 297299.
S4.) REPRESENTING DIFFERENT FUNCTIONS 163
When j z \ = 1, the function can have any value whatever. Hence a circle
of radius unity is a line of singularities, that is, it is a line of discontinuity
for the series. The circle evidently has the property of dividing the plane
into two parts such that the analytical expression represents different
functions in the two parts.
If we introduce a new variable connected with z by the relation*
then, if = + irj and z x + iy, we have
l-a'
2
t hat % is positive when z < 1, and is negative when z > 1. If then
the function %() is equal to -a or to c according as the real part of is
positive or negative.
And, generally, if we take a rational function of z and denote the
modified form of <f>(), which will be a sum of rational functions of z, by
$1(2), then (f>i(z) will be equal to a in some parts of the plane and to c
in other parts of the plane. The boundaries between these parts are lines
of singular points : and they are constituted by the ^-curves which correspond
to | = 1.
85. Now let F(z) and G(z) be two functions of z with any number of
singularities in the plane : it is possible to construct a function which shall
be equal to F(z) within a circle centre the origin and to G(z) without the
circle, the circumference being a line of singularities. For, when we make
a = l and c = in <f> (z) of 84, the function
1 z z~ z*
0(*)=r- ~ +^~ T + _i---i +J T+
1 z z* l z* l z s l
is unity for all points within the circle and is zero for all points without it :
and therefore
G(z) + {F(z)-G(z)}8(z)
is a function which has the required property.
Similarly F 3 (z) + {F, (z} - F 2 (z}} 6 (z} + [F 2 (z) - F 3 (z)} 6
is a function which has the value F l (z) within a circle of radius unity, the value F 2 (z)
between a circle of radius unity and a concentric circle of radius r greater than unity, and
the value F 3 (z) without the latter circle. All the singularities of the functions ^i, F 2 , F 3
are singularities of the function thus represented : and it has, in addition to these, the
two lines of singularities given by the circles.
* The significance of a relation of this form will be discussed in Chapter XIX.
112
164
MONOGENIC FUNCTIONALITY
[85.
Again,
G(z)+{F(z}-G(z}}6<
*-^
is a function of z, which is equal to F(z) on the positive side of the axis of y, and is equal
to G (z) on the negative side of that axis.
1+2
f~ Ia > ~Pi T^~ >
Also, if we take
where aj and p^ are real constants, as an equation denning a new variable +ir), we have
so that the two regions of the 2-plane determined by | z \ < 1 and | z \ > 1 correspond to the
two regions of the -plane into which the line cos c^+j; sin a l p i =0 divides it. Let
M _J /ft-'"- Pi-l\
''((*-*> -
fi
so that on the positive side of the line
and on the negative side of that line it is zero.
a i> Pi) a 2ip2', 3> PS respectively; then
(2)
^=0 the function 6 l is unity
Take any three lines defined by
is a function which has the value F within
the triangle, the value F in three of the
spaces without it, and the value zero in the
remaining three spaces without it, as indi-
cated in the figure (fig. 13).
And for every division of the plane by
lines, into which a circle can be transformed
by rational equations, as will be explained
when conformal representation is discussed
hereafter, there is a possibility of represent-
ing discontinuous functions, by expressions similar to those just given.
These examples are sufficient to lead to the following result*, which is
complementary to the theorem of 82 :
When the region of continuity of an infinite series of functions consists
of several distinct parts, the series represents a single function in each part
but it does not necessarily represent the same function in different parts.
It thus appears that an analytical expression of given form, which con-
verges uniformly and unconditionally in different parts of the plane separated
from one another, can represent different functions of the variable in those
different parts ; and hence the idea of monogenic functionality of a complex
variable is not coextensive with the idea of functional dependence expressible
through arithmetical operations, a distinction first established by Weierstrass.
86. We have seen that an analytic function has not a definite value at
an essential singularity and that, therefore, every essential singularity is
excluded from the region of definition of the function.
* Weierstrass, Ges. Werke, t. ii, p. 221.
86.] LINE OF SINGULARITIES 165
Again, it has appeared that not merely must single points be on occasion
excluded from the region of definition but also that functions exist with
continuous lines of essential singularities which must therefore be excluded.
( >M<> method for the construction of such functions has just been indicated:
but it is possible to obtain other analytical expressions for functions which
possess what may be called a singular line. Thus let a function have a
circle of radius c as a line of essential singularity*; let it have no other
singularities in the plane and let its zeros be a 1 , a 2 , a 3 , ..., supposed arranged
in such order that, if p n e ie = a n , then
| Pn c ^ I Pn+i C \ ,
so that the limit of p n , when n is infinite, is c.
Let c n = ce ie , a point on the singular circle, corresponding to a n which is
assumed not to lie on it. Then, proceeding as in Weierstrass's theory in 51,
if
= 00 (y _ n
G(z) = n J- ^*
n = l \Z-C n
. . + 1 /^r-'
m n -l\z-c n j
G (z) is a uniform function, continuous everywhere in the plane except along
the circumference of the circle which may be a line of essential singularities.
Special simpler forms can be derived according to the character of the
series of quantities constituted by j On c n \. If there be a finite integer m,
oc
.such that 2 \a n c n \ m is a converging series, then in g n (z) only the first
n=l
m 1 terms need be retained.
Ex. Construct the function when
2mirt '
m being a given positive integer and r a positive quantity. ,
Again, the point c n was associated with a n so that they have the same
argument : but this distribution of points on the circle is not necessary, and
it can be made in any manner which satisfies the condition that in the limited
00
case just quoted the series S j a n c n j m is a converging series.
=i
Singular lines of other classes, for example, sections^ in connection with functions
defined by integrals, arise in connection with analytical functions. They are discussed
by Painleve, Sur les lignes singulieres des fonctions analytiques, (These, Gauthier-Villars,
Paris, 1887).
Ex. 1. Shew that, if the zeros of a function be the points
. _b + c (a d)i
~ a + d(b c)i'
* This investigation is due to Picard, Comptes Rendus, t. xci, (1881), pp. 690692.
t Called coupures by Hermite ; see 103.
166 LACUNARY [86.
where a, 6, c, d are integers satisfying the condition ad bc=l, so that the function
has a circle of radius unity for an essential singular line, then if
D _
~
n -T -j. **"'[
the function
where the product extends to all positive integers subject to the foregoing condition
ad bc=l, is a uniform function finite for all points in the plane not lying on the
circle of radius unity. (Picard.)
Ex. 2. Examine the character of the distribution of points z n in the plane of :
which are given by
Z = I 1 4- - I <A/2ttiri (t) = 1 9 3 ^
11 -t- IB , (u i, ^j, .j, ...;.
Consider especially the neighbourhood of the circle whose centre is the origin and wlio.-t
radius is 1.
Shew that
1
W
where the constants A n and b n are subject to the conditions
Xl
(i) The series 2 A n converges unconditionally:
;i =
(ii) Each of the points b n is either within or upon the curve C :
(in) When any arc whatever of C is taken, as small as we please, that
arc contains an unlimited number of the points b n .
* Acta Soc. Fenn., t. xii, (1883), pp. 341350; Amer. Journ. Math., t. xiv, (1892),
pp. 201221.
represents a monogenic function of z at all points within the circle ; and investigate the
possibility of an analytical continuation of this function beyond the circle. I
(Math. Trip., Part II., 1896.)
87. In the earlier examples, instances were given of functions which
have only isolated points for their essential singularities: and, in the latter
examples, instances have been given of functions which have lines of
essential singularities, that is, there are continuous lines for which the
functions do not exist. We now proceed to shew how functions can be
constructed which do not exist in assigned continuous spaces in the plane.
Weierstrass was the first to draw attention to lacunary functions, as they
may be called; the following investigation in illustration of Weierstrass's
theorem is due to Poincare*.
Take any convex curve in the plane, say C : and consider a function-
series of the form
87.]
FUNCTIONS
167
It will be seen that, for values of z outside C, <f>(z) is represented by a
power-series, which cannot be continued across the curve C into the interior,
and which therefore has the area of C for a lacunary space.
00
Let S denote the sum of the converging series 2 | A n \ : then denoting by
=o
K any assigned quantity, as small as we please, an integer p can always be
determined so that
00
S p = 2 \A n \<K.
n=p
Consider the function-series in the vicinity of any point c outside (7. Let
R denote the distance of c from the nearest point of the boundary* of C, so
that R is a finite non-vanishing quantity ; and draw a circle of radius R and
centre c, which thus touches C externally. Thus for all the points b except
at the point of contact, we have
b n c | > R.
Let z be any point within the circle, so that
z c < R
say, where is a positive quantity less than 1. Then
| b n c \ \ z c \
and therefore
Consequently
v n
An ^ \A n
Z-bn "R(l~0)'
A iSf
^ -sf --n ^, *3
w=o zb n
n=Q -R (1 0) .R (1 0]
so that the function-series converges unconditionally. Also
A
-"-n
A n \
O n
^ R(l-0)'
and therefore the function-series converges uniformly : that is,
oo A
2 n
w =0 Z b n
* This will be either the shortest normal from c to the boundary, or the distance of c from
some point of abrupt change of direction, as for instance at the angular point of a polygon ; for
brevity of description we shall assume the former to bs the case.
168 LACUNARY [87.
converges uniformly and unconditionally within any circle concentric with
the circle of radius R and lying within it. Accordingly, by Weierstrass's
investigation ( 82, 88), this is expressible in the form of a converging series
P (z c) ; manifestly
We have
-_ " , (z - c) r '
(b n
\b n - c I
and therefore
ac oo
2
m=0 n=0
r\ m + 1
CO 00
2
7=0n=0
s
that is, the series P (z c) converges unconditionally. Let G m denote
b n -c)--*\ then
n=0
m=0
The point c is any arbitrarily chosen point outside the curve C ; and therefore
the function represented by <f> (z} for points z outside the curve C is a uniform
analytic function.
Any power-series representing this function can be used as an element
for continuation outside C and away from C: we proceed to prove that it
cannot be continued across the boundary of C. If this were possible, it would
arise through the construction of the domain of some point z , where z is a
point outside C (say within such a circle as the above, centre c), and where
the circle bounding the domain of z would cut off some arc from the boundary
of C. The preceding analysis shews that, in the domain of z , the function is
represented by a power-series
Q (z - z,} = - 2 B m (z - z ) m ,
m=0
00
where B m = 2 A n (b n - z )- m ~ l :
n=0
it must be shewn that the series diverges for points z within C.
In the first place, consider the series P (z c); in order that it may
converge, only such values of z are admissible as. make the limit of
Cm (^ c) m zero, when m is infinite. Let a point be taken on the circum-
ference of the circle C of radius R ; then the above limit can only be zero if
Lt C m R m =Q,
87.J
FUNCTIONS
169
a condition that is not satisfied, as will now be proved. This circle touches
C externally ; let the point of contact be a point b k (such a circle can always
be constructed, by drawing the outward normal at a point b and choosing
some point c upon it). Let any arbitrary quantity e be assigned, as small
as we please ; and let an integer p be chosen large enough to secure that
this being possible because $p,the remainder of the converging series 2|-4 n |
can (by choice of p) be made less than any assigned quantity. Either the
chosen number p is greater than k: or if it is less than k, then some other
number (> p) can be chosen so that it is greater than k: we may therefore
assume p > k,
Draw a circle, centre c and radius R' greater than R, so as to include the
point b k , and exclude the points b , , b p with the exception of &&. This can
be done : for if
where A, is some positive quantity as small as we please (but not absolutely
zero), we can take
and then
b n c <R', for n = 0, 1, ...,/; 1, k + 1, ..., p.
Let q denote a number sufficiently large to secure that
&
R'\R'
Then as
we have
and therefore
C q = 2
"2 1
+ 2
= p (b n -
k-l
\R{C q -A k (b k -c)-*~i}\< 2
n=0
p-l
+
M =0
_i 4.
-i '
-"n |
R
k-l
1'i 1
s
170 LACUNAEY FUNCTIONS [87.
a quantity arbitrarily assigned as small as we please. Accordingly we have
Limit \&{C q -A t (b k - c)-<?-'} = 0,
=
that is,
Limit i C q Ri \ = Limit | A k R* (b k - c)-?- 1 = L~*J ,
q=<B ' q=cc ' ">
so that CqRv does not tend to zero when q is infinitely large, as it should if
P (z e) converges. Thus P (z c) does not converge for points given by
z c j = R.
Consider now the domain of z , assumed to include points within G and
therefore some arc of C; the function is represented throughout that domain
by Q( Z z o)- On the included arc of C take any one (say 6 t ) of the un-
limited number of points b ; at b k draw an outward normal to C and choose a
point 2-1 on it such that the circle
\Z-ZT. = I b k - z l |
lies wholly within the domain of Z Q . The function is represented by a power-
series in z Z-L throughout this circle ; and as the circle lies wholly within
the domain of z , the representation is included in Q(z z ). But, by the
preceding investigation, the power-series does not converge on the circum-
ference of the circle j z . z l \ = b k z l \ : contradicting the supposition that
Q (z z ) converges in a domain of z enclosing this circle. Hence the power-
series P(z c) cannot be continued across the boundary of C ; in other words,
the . function represented by P (z c) and its continuations has the area of C
for a lacunary space.
The discussion of the significance (if any) of <j> (z) for points z within
G depends on the distribution of the points b n within G, as to which no
hypothesis has been made.
As an example, take a convex polygon having e^, ...... , a p for its angular points;
then any point
p
2 m r a r
r=l _
p "
2 m r
r=l
by proper choice of m t , ...... , m p , a choice which can be made in an infinite number
of ways.
Let HI, ...... , u p be given quantities, the modulus of each of which is less than unity :
then the series
where m l , , in p are positive integers or zero (simultaneous zeros being excluded), is
either within the polygon or on its boundary : and any rational point within the polygon
or on its boundary can be represented by
87.] LACUNARY SPACES 171
converges unconditionally. Then all tbe assigned conditions are satisfied for the function
\
L '
and therefore it is a function which converges uniformly and unconditionally everywhere
outside the polygon and which has the polygonal space (including the boundary) for
a lacunary space.
If, in particular, p = 2, we obtain a function which has the straight line
joining a, and a 2 as a line of essential singularity. When we take a 1 = 0,
=!, and slightly modify the summation, we obtain the function
- 1 &
n=l ?n=0
which, when j % < 1 and \u 2 < 1, converges uniformly and unconditionally
everywhere in the plane except at points between and 1 on the axis of real
quantities, this part of the axis being a line of essential singularity.
For the general case, the following remarks may be made :
(i) The quantities u 1} u 2 , ... need not be the same for every term; a
numerator, quite different in form, might be chosen, such as
(m^ + . . . +.m p 2 )~' t where 2/i > p ; all that is requisite is that the
series, made up of the numerators, should converge uncondition-
ally.
(ii) The preceding is only a particular illustration, and is not necessarily
the most general form of function having the assigned lacunary
space.
It is evident that one mode of constructing a function, which shall have
any assigned lacunary space, would begin by the formation of some expression
which, by the variation of the constants it contains, can be made to represent
indefinitely nearly any point within or on the contour of the space. Thus
for the space between two concentric circles, of radii a and c and centre the
origin, we could take
m^a + (n m^) b f 2
n
which, by giving m l all values from to n, m 2 all values from to n 1, and
n all values from 1 to infinity, will represent all rational points in the space :
and a function, having the space between the circles as lacunary, would be
given by
oo n w-1
2 2 2
m.a + ^-m^bJ^'.
& ~~~ &
n
provided | u < 1, u^ < 1, | u 2 < 1.
172 EXAMPLES . [87.
In particular, if a =6, then the common circumference is a line of essential singularity
for the corresponding function. It is easy to see that the function
m n
-1 U V
^ _ m, n m,n
^^^^^m "^"'
n=0 m=0 iri
Z Ct6 **
oo 2n 1 m n
provided the series 2 2 u v
n=l m=0 m,n m,n
converges unconditionally, is a function having the circle \z =a as a line of essential
singularity. It can be expressed as an analytic function within the circle, and as another
analytic function without the circle.
Other examples will be found in memoirs by Goursat*, Poincaret, and HomenJ.
Ex. 1. Shew that the function
2 2 (w+7i?)- 2 ~ r ,
where r is a real positive quantity and the summation is for all integers m and n between
the positive and the negative infinities, is a uniform function in all parts of the plane
except the axis of real quantities which is a line of essential singularity.
Ex. 2. Discuss the region in which the function
2 2 2 -'
n n
is definite. (Homen.)
Ex. 3. Prove that the function
2
=o
exists only within a circle of radius unity and centre the origin. (Poincare.)
Ex. 4. Prove that the series
1 -!_
,,=i z - a n
represents a uniform meromorphic function, if the quantities | a n | increase without limit
as n increases and if the series | A n la n \ converges.
Ex. 5. An infinite number of points a lt a 2 , a 3 , ...... are taken on the circumference of
a given circle, centre the origin, so that they form the aggregate of rational points on the
circumference. Shew that the series
can be expanded in a series of ascending powers of z which converges for points within the
circle, but that the function cannot be continued across the circumference of the circle.
(Stieltjes.)
* Comptes Rendus, t. xciv, (1882), pp. 715718; Bulletin de Darboux, 2 me Ser., t. xi, (1887),
pp. 109114.
t In the memoirs, quoted p. 166, and Comptes Rendus, t. xcvi, (1883), pp. 1134 1136.
J Ada Soc. Fenn., t. xii, (1883), pp. 445464.
87.] EXAMPLES 173
Ex. 6. Prove that the infinite continued fraction
1111
a^\-
converges for all values of 2, provided the series
2 a n
=i
diverges, the quantities a being real. Discuss, in particular, the cases, (i) when z has real
positive values, (ii) when z has real negative values.
(Stieltjes.)
Ex. 7. Denoting by t n a positive quantity less than 1, prove that the infinite product
ao ( / 2
H \( 1 --
*=ll\
converges ; and that the series
I
converges.
Shew that, if a new series be constructed by separating the two fractions in the single
term so as to provide two terms, this new series does not converge when t n = n~ 3 . Does
the same consequence follow when f n =n~ 2 1
(Borel.)
Ex. 8. Prove that the series
9 9 oo oo ( 1
/.J..-1NJ." v 5 J Z _
TT ^ TT _t _ \(1 - 2m - 2n2i) (2m + 2/m) 2 J
- 2m - 22 - J i) (2m + 20 - *
where the summation extends over all positive and negative integral values of m and of n
except simultaneous zeros, converges uniformly and unconditionally for all points in
the finite part of the plane which do not lie on the axis of y ; and that it has the
value +1 or 1, according as the real part of z is positive or negative.
(Weierstrass.)
Ex. 9. Prove that the region of continuity of the series
consists of two parts, separated by the circle z \ 1 which is a line of infinities for
the series : and that, in these two parts of the plane, it represents two different
functions.
ia'ir
If two complex quantities u> and &>' be taken, such that z=e "* and the real part of
. is positive, and if they be associated with the elliptic function & (u) as its half-periods,
wt
then for values of z, which lie within the circle | z | = 1,
s __. 3 . i
'*<)*" + *-" 27T o- ()"*"
in the usual notation of Weierstrass's theory of elliptic functions.
Find the function which the series represents for values of z without the circle | z \ = 1.
(Weierstrass.)
174 EXAMPLES [87.
Ex* 10. Discuss the descriptive properties of the functions represented by the
expressions :
/<*>
.
for all values of the complex argument z. (Math. Trip., Part II., 1893.)
Ex. 11. Four circles are drawn each of radius -r= having their centres at the points
v 2
1, i, 1, i respectively ; the two parts of the plane, excluded by the four circumferences,
are denoted the interior and the exterior parts. Shew that the function
1 1 1 1
(1-*)* < l +)"
is equal to IT in the interior part and is zero in the exterior part. (Appell.)
Ex. 12. Obtain the values of the function
v.
1
in the two parts of the area within a circle centre the origin and radius 2 which lie
without two circles of radius unity, having their centres at the points 1 and - 1
respectively. (Appell.)
Ex.13. If f(z)=U 1+ U 2 + ...... + U n ,
and U m = F m (,) --L- + (z - am -
where the regions of continuity of the functions F extend over the whole plane, then f(z)
is a function existing everywhere except within the circles of radius unity described round
the points i , 2 > ...... > a n- (Teixeira. )
Ex. 14. Let there be n circles having the origin for a common centre, and let
C 15 C 2 , ...... , (7 n , C n+ i be 7i4-l arbitrary constants ; also let a ls a 2 , ...... , be any n points
lying respectively on the circumferences of the first, the second, ...... , the wth circles.
Shew that the expression
has the value C m for points z lying between the (m l)th and the with circles, and the
value C n + i for points lying without the th circle.
Construct a function which shall have any assigned values in the various bands into
which the plane is divided by the circles. (Pincherle.)
Ex. 15. Examine the nature of the functions denned by the series
=i 2 (z - a) 2 " - 5 (a 2 - a 2 )" + 2 (s + a) 2 '
() 2 ,-
where a is a real positive constant. (Math. Trip., Part II., 1897.)
88.] CLASSIFICATION OF SINGULARITIES 175
88. In 32 it was remarked that the discrimination of the various
species of essential singularities could be effected by means of the properties
of the function in the immediate vicinity of the point.
Now it was proved, in 63, that in the vicinity of an isolated essential
singularity b the function could be represented by an expression of the form
for all points in the space without a circle centre b of small radius and within
a concentric circle of radius not large enough to include singularities at
a finite distance from b. Because the essential singularity at b is isolated,
the radius of the inner circle can be diminished to be all but infinitesimal :
the series P(zb) is then unimportant compared with G ( -- 7), which
can be regarded as characteristic for the singularity of the function.
Another method of obtaining a function, which is characteristic of the
singularity, is provided by 68. It was there proved that, in the vicinity of
an essential singularity a, the function could be represented by an expression
of the form
(,-) JET JLQ(,- a),
where, within a circle of centre a and radius not sufficiently large to include
the nearest singularity at a finite distance from a, the function Q (z a) is
finite and has no zeros : all the zeros of the given function within this circle
(except such as are absorbed into the essential singularity at a) are zeros of
the factor H( ), and the integer-index n is affected by the number of
these zeros. When the circle is made small, the function
.can be regarded as characteristic of the immediate vicinity of a or, more
briefly, as characteristic of a.
It is easily seen that the two characteristic functions are distinct. For
if F and F-^ be two functions, which have essential singularities at a of the
same kind as determined by the first characteristic, then
F(z) - F, (z} = P(z-a)-P 1 (z-a)
= P,(*-a),
while if their singularities at a be of the same kind as determined by the
second characteristic, then
176 CLASSIFICATION [88.
in the immediate vicinity of a, since Qi has no zeros. Two such equations
cannot subsist simultaneously, except in one instance.
Without entering into detailed discussion, the results obtained in thej
preceding chapters are sufficient to lead to an indication of the classification
of singularities*.
Singularities are said to be of the first class when they are accidental ;
and a function is said to be of the first class when all its singularities are oi
the first class. It can, by 48, have only a finite number of such singularities,
each singularity being isolated.
It' is for this case alone that the two characteristic functions are in
accord.
When a function, otherwise of the first class, fails to satisfy the last
condition, solely owing to failure of finiteness of multiplicity at some point,
say at z = oo , then that point ceases to be an accidental singularity. It has
been called ( 32) an essential singularity ; it belongs to the simplest kind of
essential singularity ; and it is called a singularity of the second class.
A function is said to be of the second class when it has some singularities
of the second class ; it may possess singularities of the first class. By an
argument similar to that adopted in 48, a function of the second class
can have only a limited number of singularities of the second class, each
singularity being isolated.
When a function, otherwise of the second class, fails to satisfy the last
condition solely owing to unlimited condensation at some point, say at z = oo ,
of singularities of the second class, that point ceases to be a singularity
of the second class: it is called a singularity (necessarily essential) of the
third class.
A function is said to be of the third class when it has some singularities
of the third class ; it may possess singularities of the first and the second
classes. But it can have only a limited number of singularities of the third
class, each singularity being isolated.
Proceeding in this gradual sequence, we obtain an unlimited number of
classes of singularities : and functions of the various classes can be constructed
by means of the theorems which have been proved. A function of class n
has a limited number of singularities of class n, each singularity being
isolated, and any number of singularities of lower classes which, except in so
far as they are absorbed in the singularities of class n, are isolated points.
* For a detailed discussion, reference should be made to Guichard, Theorie des pointt
singuliers essentiels (These, Gauthier-Villars, Paris, 1883), who gives adequate references to the
investigations of Mittag-Leffler in the introduction of the classification and to the researches of
Cantor. See also Mittag-Leffler, Acta Math., t. iv, (1884), pp. 179 ; Cantor, Crelle, t. Ixxxiv,
(1878), pp. 242258, Acta Math., t. ii, (1883), pp. 311328.
88.] OF SINGULARITIES 177
The effective limit of this sequence of classes is attained when the
number of the class increases beyond any integer, however large. When
once such a limit is attained, we have functions with essential singularities of
unlimited class, each singularity being isolated ; when we pass to functions
which have their essential singularities no longer isolated but, as in previous
class-developments, of infinite condensation, it is necessary to add to the
arrangement in classes an arrangement in a wider group, say, in species*.
Galling, then, all the preceding classes of functions functions of the first
species, we may, after Guichard (I.e.), construct, by the theorems already
proved, a function which has at the points a 1} a 2 , ... singularities of classes
1, 2, .. , both series being continued to infinity. Such a function is called
a function of the second species.
By a combination of classes in species, this arrangement can be continued
indefinitely ; each species will contain an infinitely increasing number of
classes ; and when an unlimited number of species is ultimately obtained,
another wider group must be introduced.
This gradual construction, relative to essential singularities, can be carried
out without limit ; the singularities are the characteristics of the functions.
* Guichard (I.e.) uses the term genre.
P. F. 12
CHAPTER VIII. .
MULTIFORM FUNCTIONS.
89. HAVING now discussed some of the more important general properties
tt' uniform functions, we proceed to discuss some of the properties of multiform
functions.
Deviations from uniformity in character may arise through various causes :
the most common is the existence of those points in the 2-plane, which have
already ( 12) been defined as branch -points.
As an example, consider the two power-series
u=\-\z ' -\z'-- ..., v = -(\-\z -\z- -...),
which, for points in the plane such that | z is less than unity, are the two
values of (1 2')^: they may be regarded as representing the two branches
of the function w, say w-^ and w 2 , defined by the equation
w 2 = 1 z = z.
Let z describe a small curve (say a circle of radius ?*) round the point
z' = 1, beginning on the axis of x ; the point 1 is the origin for z. Then z
is r initially, and at the end of the first description of the circle z is re zni .
The branch of the function, which initially is equal to u, changes continuously
during the description of the circle. The series for u, and the continuations
of that series, give rise to the complete variation of the branch of the function
which originally is u. Its initial value is r^ t and its final value is r^e, that
is, r * ; so that the final value of the branch is v. Similarly for the branch
of the function, which initially is equal to v, it is continuously changed
during the description of the circle ; the series for v, and the continuations
of that series, give rise to the complete variation of the branch of the function
which originally is v ; and the branch acquires u as its final value. Thus the
effect of the single circuit is, to change w 1 into w 2 and w 2 into w, , that is, the
effect of a circuit round the point, at which w l and w 2 coincide in value, is to
interchange the values of the two branches.
If, however, z describe a circuit which does not include the branch-point,
Wi and w-t return each to its initial value.
89.] CONTINUATION OF MULTIFORM FUNCTION 179
Instances have already occurred, e.g. integrals of uniform functions, in
which a variation in the path of the variable has made a difference in the
result; but this interchange of value is distinct from any of the effects
produced by points belonging to the families of critical points which have
been considered. The critical point is of a new nature ; it is, in fact, a
characteristic of multiform functions at certain associated points.
We now proceed to indicate more generally the character of the relation
of such points to functions affected by them.
The method of constructing a monogenic analytic function, described in
34, by forming all the continuations of a power-series, regarded as a given
initial element of the function, leads to the aggregate of the elements of the
function and determines its region of continuity. When the process of con-
tinuation has been completely carried out, two distinct cases may occur.
In the first case, the function is such that any and every path, leading
from one point a to another point z by the construction of a series of
successive domains of points along the path, gives a single value at z as the
continuation of one initial value at a. When, therefore, there is only a
single value of the function at a, the process of continuation leads to only a
single value of the function at any other point in the plane. The function is
uniform throughout its region of continuity. The detailed properties of such
functions have been considered in the preceding chapters.
In the second case, the function is such that different paths, leading from
a to z, do not give a single value at z as the continuation of one and the
same initial value at a. There are different sets of elements of the function,
associated with different sets of consecutive domains of points on paths from
a to z, which lead to different values of the function at z ; but any change
in a path from a to z does not necessarily cause a change in the value of the
function at z. The function is multiform in its region of continuity. The
detailed properties of such functions will now be considered.
90. In order that the process of continuation may be completely carried
out, continuations must be effected, beginning at the domain of any point a
and proceeding to the domain of any other point b by all possible paths in
the region of continuity, and they must be effected for all points a and 6.
Continuations must be effected, beginning in the domain of every point a
and returning to that domain by all possible closed paths in the region of
continuity. When they are effected from the domain of one point a to that
of another point b, all the values at any point z in the domain of a (and not
merely a single value at such points) must be continued : and similarly when
they are effected, beginning in the domain of a and returning to that domain.
The complete region of the plane will then be obtained in which the function
can be represented by a series of positive integral powers : and the boundary
of that region will be indicated.
122
180 BRANCHES OF [90.
In the first instance, let the boundary of the region be constituted by a
number, either finite or infinite, of
isolated points, say L lt L z , L 3 ,
Take any point A in the region, so
that its distance from any of the
points L is not infinitesimal ; and
in the region draw a closed path
ABC...EFA so as to enclose one
point, say L l} but only one point, of
the boundary and to have no point
of the curve at a merely infinitesimal distance from 7^. Let such curves be
drawn, beginning and ending at A, so that each of them encloses one and
only one of the points of the boundary : and let K r be the curve which
encloses the point L r .
Let w^ be one of the power-series defining the function in a domain with
its centre at A : let this series be continued along each of the curves K g by
successive domains of points along the curve returning to A. The result
of the description of all the curves will be that the series w l cannot be
reproduced at A for all the curves, though it may be reproduced for some
of them ; otherwise, w l would be a uniform function. Suppose that w 2 , w 3 , ...,
each in the form of a power-series, are the aggregate of new distinct values
thus obtained at A ; let the same process be effected on w 2 , w 3 , ... as has
been effected on w t . and let it further be effected on any new distinct values
obtained at A through w z , w s , ..., and so on/ When the process has
been carried out so far that all values obtained at A, by continuing any
series round any of the curves K back to A , are included in values already
obtained, the aggregate of the values of the function at A is complete : they
are the values at A of the branches of the function.
We shall now assume that the number of values thus obtained is finite,
J
say n, so that the function has n branches at A : if their values be denoted
by w lt w 2 , ..., w n , these n quantities are all the values of the function at A.
Moreover, n is the same for all points in the plane, as may be seen by con-
tinuing the series at A to any other point and taking account of the corollaries
at the end of the present section.
The boundary-points L may be of two kinds. It may (and not infre-
quently does) happen that a point L g is such that, whatever branch is taken
at A as the initial value for the description of the circuit K g , that branch is
reproduced at the end of the circuit. Let the aggregate of such points be
/i, / 2 , .... Then each of the remaining points L is such that a description
of the circuit round it effects a change on at least one of the branches, taken
as an initial value for the description ; let the aggregate of these points be
B l} B 2 , .... They are the branc h -points ; their association with the definition
in 12 will be made later.
90.]
MULTIFORM FUNCTIONS
181
Fig. 15.
When account is taken of the continuations of the function from a point
A to another point B, we have n values at B as the continuations of n values
at A. The selection of the individual branch at B, which is the continuation
of ;i particular branch at A, depends upon the path of z between A and B;
it is governed by the following fundamental proposition :
The final value of a branch of a function for two paths of variation of the
Independent variable from one point to another will be the same, if one path
ciin, be deformed into the other without passing over a branch-point.
Let the initial and the final points be a and b, and let one path of
variation be acb. Let another path of variation be aeb, A
both paths lying in the region in which the function can
be expressed by series of positive integral powers : the two
paths are assumed to have no point within an infinitesimal
distance of any of the boundary-points L and to be taken
so close together, that the circles of convergence of pairs of
points (such as c x and e 1} c 2 and e 2 , and so on) along the two
paths have common areas. When we begin at a with a
branch of the function, values at c, and at e l are obtained,
depending upon the values of the branch and its derivatives at a and upon
the positions of c 1 and e l ; hence, at any point in the area common to the
circles of convergence of these two points, only a single value arises as
derived through the initial value at a. Proceeding in this way, only a single
value is obtained at any point in an area common to the circles of con-
vergence of points in the two paths. Hence ultimately one and the same
value will be obtained at- b as the continuation of the value of the one branch
at a by' the two different paths of variation which have been taken so that
no boundary-point L lies between them or infmitesimally near to them.
Now consider any two paths from a to b, say acb and adb, such that
neither of them is near a boundary-point and that the
contour they constitute does not enclose a boundary-point.
Then by a series of successive infinitesimal deformations we
can change the path acb to adb ; and as at b the same value
of w is obtained for variations of z from a to b along the
successive deformations, it follows that the same value of w
is obtained at b for variations of z along acb as for varia-
tions along adb.
Next, let there be two paths acb, adb constituting a closed contour,
enclosing one (but not more than one) of the points / and none of the points
B. When the original curve K which contains the point / is described, the
initial value is restored : and hence the branches of the function obtained at
any point of K by the two paths from any point, taken as initial point, are
the same. By what precedes, the parts of this curve K can be deformed
Fig. 16.
182 EFFECT OF DEFORMATION OF [90.
into the parts of acbda without affecting the branches of the function : hence
the value obtained at b, by continuation along acb, is the same as the value
there obtained by continuation along adb. It therefore follows that a path
between two points a and b can be deformed over any point / without
affecting the value of the function at b ; so that, when the preceding
results are combined, the proposition enunciated is proved.
By the continued application of the theorem, we are led to the following
results :
COROLLARY I. Whatever be the effect of the description of a circuit on the
initial value of a function, a reversal of the circuit restores the original value
of the function.
For the circuit, when described positively and negatively, may be re-
garded as the contour of an area of infinitesimal breadth, which encloses no
branch-point within itself and the description of the contour of which
therefore restores the initial value of the function.
COROLLARY II. A circuit can be deformed inty any other circuit without
affecting the final value of the function, provided that no branch-point be crossed
in the process of deformation.
It is thus justifiable, and it is often convenient, to deform a path con-
taining a single branch -point into a loop round the
point. A loop* consists of a line nearly to the point, o
nearly the whole of a very small circle round the point, F - 17
and a line back to the initial point ; see figure 17.
COROLLARY III. The value of a function is unchanged when the variable
describes a closed circuit containing no branch-point ; it is likewise unchanged
luhen the variable describes a closed circuit containing all the branch-points.
The first part is at once proved by remarking that, without altering the
value of the function, the circuit can be deformed into a point.
For the second part, the simplest plan is to represent the variable on
Neumann's sphere. The circuit is then a curve on the sphere enclosing all
the branch-points : the effect on the value of the function is unaltered by
any deformation of this curve which does not make it cross a branch-point.
The curve can, without crossing a branch-point, be deformed into a point
in that other part of the area of the sphere which contains none of the
branch-points ; and the point, which is the limit of the curve, is not a
branch-point. At such a point, the value of the function is unaltered ; and
therefore the description of a circuit, which encloses all the branch-points,
restores the initial value of the function.
COROLLARY IV. If the values of w at b for variations along two path*
* French writers use the word lacet, German writers the vfor&Schleife.
90.]
PATH OF THE VARIABLE
183
acb, adb be not the same, then a description of acbda will not restore the initial
ralue of w at a.
In particular, let the path be the loop OeceO (fig. 17), and let it change w
at into w'. Since the values of w at are different and because there is
no branch-point in Oe (or in the evanescent circuit OeO\ the values of w at
e cannot be the same : that is, the value with which the infinitesimal circle
round a begins to be described is changed by the description of that circle.
Hence the part of the laop that is effective for the change in the value of w is
the small circle round the point; and it is because the description of a small
circle changes the value of w that the value of w is changed at after the
description of a loop.
If/(/) be the value of w which is changed into/i (z) by the description of
the loop, so that / (z) and /j (z} are the values at 0, then the foregoing
explanation shews that /(e) and /i (e) are the values at e, the branch /(e)
being changed by the description of the circle into the branch yi (e).
From this result the inference can be derived that the points B l} B. 2 , ...
are branch-points as defined in 12. Let a be any one of the points, and
let/ (2) be the value of w which is changed into /i (2) by the description of
a very small circle round a. Then as the branch of w is monogenic, the
difference between f (z) and /i (z) is an infinitesimal quantity of the same
order as the length of the circumference of the circle : so that, as the circle
is infinitesimal and ultimately evanescent, \f(z) fi (z) \ can be made as small
as we please with decrease of z a\ or, in the limit, the values of f (a) and
/i (a) at the branch-point are equal. Hence each of the points B is such
that two or more branches of the function have the same value at the point,
and there is interchange among these branches when the variable describes a
small circuit round the point : which affords a definition of a branch-point,
more complete than that given in 12.
COROLLARY V. If a closed circuit contain several branch-points, the effect
which it produces can be obtained by a combination of the effects produced in
succession by a set of loops each going round only one of the branch-points.
If the circuit contain several branch-points, say three as at a, 6, c, then
a path such as AEFD, in fig. 18, can without
crossing any branch-point, be deformed into the
loops A aB, BbC, CcD ; and therefore the complete
circuit AEFDA can be deformed validly into
AaBbGcDA, and the same effect will be produced
by the two forms of circuit. When D is made
practically to coincide with A, the whole of the
second circuit is composed of the three loops. Hence the corollary.
This corollary is of especial importance in the consideration of integrals
of multiform functions.
184 BRANCHES OF [90.
COROLLARY VI. In a continuous part of the plane where there are no
branch-points, each branch of a multiform function is uniform.
Each branch is monogenic and, except at isolated points, continuous ;
hence, in such regions of the plane, all the propositions which have been
proved for monogenic analytic functions can be applied to each of the
branches of a multiform function.
91. If there be a branch-point within the circuit, then the value of the
function at 6 consequent on variations along acb may, but will not necessarily,
differ from its value at the same point consequent on variations along adb.
Should the values be different, then the description of the whole curve acbda
will lead at a not to the initial value of w, but to a different value.
The test as to whether such a change is effected by the description is
immediately derivable from the foregoing proposition; and as in Corollary
IV., 90, it is proved that the value is or is not changed by the loop,
according as the value of w for a point near the circle of the loop is or
is not changed by the description of that circle. Hence it follows that, if
there be a branch-point which affects the branch of the function, a path of
variation of the independent variable cannot be deformed across the branch-
point without a change in the value of w at the extremity of the path.
And it is evident that a point can be regarded as a branch-point for a
function only if a circuit rounfl the point interchange some (or all) of the
branches of the function which are equal at the point. It is not necessary that
all the branches of the function should be thus affected by the point : it is
sufficient that some should be interchanged*.
Further, the change in the value of w for a single description of a circuit
enclosing a branch-point is unique.
For, if a circuit could change w into w' or w", then, beginning with w"
and describing it in the negative sense we should return to w and afterwards
describing it in the positive sense with w as the initial value we should
obtain w'. Hence the circuit, described and then re versed,, does not restore
the original value w" but gives a different branch w' ; and no point on
the circuit is a branch-point. This result is in opposition to Corollary L,
of 90; and therefore the hypothesis of alternative values at the end of
the circuit is not valid, that is, the change for a single description is
unique.
But repetitions of the circuit may, of course, give different values at the
end of successive descriptions.
* In what precedes, certain points were considered which were regular singularities (see
p. 192, note) and certain which were branch-points. Frequently points will occur which are
at once branch-points and infinities ; proper account must of course be taken of them.
92.]
MULTIFORM FUNCTIONS
185
Fig. 19.
92. Let be any ordinary point of the function ; join it to all the
branch-points (generally assumed finite in
number) in succession by lines which do not
meet each other: then each branch is uniform
f >r each path of variation of the variable which
meets none of these lines. The effects pro-
duced by the various branch-points and their
relations on the various branches can be indi-
cated by describing curves, each of which
begins at a point indefinitely near and
returns to another point indefinitely near it
after passing round one of the branch-points,
and by noting the value of each branch of the function after each of these
curves has been described.
The law of interchange of branches of a function after description of a
circuit round a branch-point is as follows :
All the branches of a function, which are affected by a branch-point as such,
can either be arranged so that the order of interchange (for description of a
path round the point} is cyclical, or be divided into sets in each of which the
order of interchange is cyclical.
Let Wi, w 2 , w s ,... be the branches of a function for values of z near a
branch-point a which are affected by the description of a small closed curve
C round a : they are not necessarily all the branches of the function, but only
those affected by the branch-point.
The branch Wj is changed after a description of C ; let w 2 be the branch
into which it is changed. Then w. 2 cannot be unchanged by C ; for a reversed
description of C, which ought to restore w 1} would otherwise leave w 2 un-
changed. Hence w. 2 is changed after a description of C\ it may be changed
either into w 1 or into a new branch, say w s . If into w lt then w 1 and w 2 form
a cyclical set.
If the change be into w 3 , then w 3 cannot remain unchanged after a
description of C, for reasons similar to those that before applied to the
change of w 2 ; and it cannot be changed into w 2 , for then a reversed de-
scription of C would change w 2 into w s , and it ought to change w z into w x .
Hence, after a description of C, w 3 is changed either into w 1 or into a new
branch, say w. If into w 1} then w l , w. 2 , w 3 form a cyclical set.
If the change be into w 4 , then w 4 cannot remain unchanged after a
description of C ; and it cannot be. changed into w z or w 3 , for by a reversal
of the circuit that earlier branch would be changed into w 4 whereas it ought
to be changed into the branch, which gave rise to it by the forward descrip-
tion a branch which is not w 4 . Hence, after a description of C, w 4 is
changed either into w 1 or into a new branch. If into w lt then w 1} w 2 , w s> w 4
form a cyclical set.
186 INTERCHANGE OF BRANCHES [92.
If w 4 be changed into a new branch, we proceed as before with that new
branch and either complete a cyclical set or add one more to the set. By
repetition of the process, we complete a cyclical set sooner or later.
If all the branches be included, then evidently their complete system
taken in the order in which they come in the foregoing investigation is a
system in which the interchange is cyclical.
If only some of the branches be included, the remark applies to the set
constituted by them. We then begin with one of the branches not included
in that set and evidently not inclusible in it, and proceed as at first, until
we complete another set which may include all the remaining branches or
only some of them. In the latter case, we begin again with a new branch
and repeat the process; and so on, until ultimately all the branches are
included. The whole system is then arranged in sets, in each of which the
order of interchange is cyclical.
93. The analytical test of a branch-point is easily obtained by con-
structing the general expression for the branches of a function which are
interchanged there.
Let z = a be a branch-point where n branches w lt w 2 , ..., w n are cyclically
interchanged. Since by a first description of a small curve round a, the
branch w^ changes into w 2 , the branch w 2 into w 3 , and so on, it follows that
by r descriptions w l is changed into w r+1 and by n descriptions w l reverts to
its initial value. Similarly for each of the branches. Hence each branch
returns to its initial value after n descriptions of a circuit round a branch-
point where n branches of the function are interchangeable.
Now let z a = Z n ;
then, when z describes circles round a, Z moves in a circular arc round its
origin. For each circumference described by z, the variable Z describes
- th part of its circumference ; and the complete circle is described bv Z
round its origin when n complete circles are described by z round a. Xow
the substitution changes w r as a function of z into a function of Z, sav into
W r \ and, after n complete descriptions of the ^-circle round a, w r returns
to its initial value. Hence, after the description of a ^-circle round its
origin, W r returns to its initial value, that is, Z=0 ceases to be a branch-
point for W r . Similarly for all the branches W.
But no other condition has been associated with a as a point for the
function w ; and therefore Z = may be any point for the function W, that
is, it may be an ordinary point, or a singularity. In every case, we have W
a uniform function of Z in the immediate vicinity of the origin ; and therefore
in that vicinity it can be expressed in the form
93.] ANALYTICAL TEST .187
with the significations of P and G already adopted. When Z=Q is an
ordinary point, G is a constant or zero ; when it is an accidental singularity,
G is a polynomial function ; and, when it is an essential singularity, G is
a transcendental function.
The simpler cases are, of course, those in which the form of G is poly-
nomial or constant or zero ; and then W can be put into the form
Z m P (Z),
where P is an infinite series of positive powers and m is an integer. As this
is the form of W in the vicinity of Z = 0, it follows that the form of w in the
vicinity of z = a is
W 1
(z-a) n P{(z-a} n ] ;
and the various n branches of the function are easily seen to be given by
substituting in the above for (z a) n the values
2JTM _!
e n (z - a) n ,
where .9 = 0, 1, . . . , n 1. We therefore infer that the general expression for
the n branches of a function, ivhich are interchanged by circuits round a
branch-point z = a, assumed not to be an essential singularity, is
where in is an integer, and where to {z a) n its n values are in turn assigned
to obtain the different branches of the function.
There may be, however, more than one cyclical set of branches. If there
be another set of r branches, then it may similarly be proved that their
general expression is
where ?w x is an integer, and Q is an integral function ; the various branches
i
are obtained by assigning to (z a) r its r values in turn.
And so on, for each of the sets, the members of which are cyclically
interchangeable at the branch-point.
When the branch-point is at infinity, a different form is obtained. Thus
in the case of a set of n cyclically interchangeable branches we take
z = u~ n ,
so that n negative descriptions of a closed ^-curve, excluding infinity and no
other branch-point, require a single positive description of a closed curve
round the w-origin. These n descriptions restore the value of w as a function
of z to its initial value ; and therefore the single description of the w-curve
round the origin restores the value of U the equivalent of w after the
188 BRANCHES OF [93.
change of the independent variable as a function of u. Thus u = Q ceases
to be a branch -point for the function U', and therefore the form of U is
where the symbols have the same general signification as before.
If, in particular, z = oo be a branch-point but not an essential singularity,
then G is either a constant or a polynomial function ; and then U can be
expressed in the form
u~ m P (u),
where ra is an integer. When the variable is changed from u to z, then the
general expression for the n branches of a function which are interchangeable
at z oo , assumed not to be an essential singularity, is
m _1
z n P(z ),
1
where m is an integer and where to z n its n values are assigned to obtain the
different branches of the function.
If, however, the branch-point z = a in the former case or z = oo in the
latter be an essential singularity, the forms of the expressions in the vicinity
of the point are
__
and m G(z n } + P(z n \
respectively.
Note. When a multiform function is defined, either explicitly or im-
plicitly, it is practically always necessary to consider the relations of the
branches of the function for z = oo as well as their relations for points that
are infinities of the function. The former can be determined by either
of the processes suggested in 4 for dealing with z = GO ; the latter can be
determined as in the present section.
Moreover, the total number of branches of the function has been assumed
to be finite. The cases, in which the number of branches is unlimited, need
not be discussed in general : it will be sufficient to consider them when they
arise, as they do arise, e.g., when the function is of the form of an algebraical
irrational with an irrational index such as z^ hardly a function in the
ordinary sense , or when the function is the logarithm of a function of z,
or is the inverse of a periodic function. In the nature of their multiplicity
of branching and of their sequence of interchange, they are for the most part
distinct from the multiform functions with only a finite number of branches.
Ex. The simplest illustrations of multiform functions are furnished by functions
denned by algebraical equations, in particular, by algebraic irrationals.
93.]
MULTIFORM FUNCTIONS
189
The general type of the algebraical irrational is the product of a number of functions
j,
of the form w = {A (* a 1 )(z a 2 ) ...... (2-a n )}>, m and n being integers.
This particular function has m branches; the points Oj, a 2 , ...... , a n are branch-points.
To find the law of interchange, we take z a r pe et ; then when a small circle of radius p
is described round a r , so that 2 returns to its initial position, the value of 6 increases by
2n and the new value of w is aw, where a is the mth root of unity defined by em "*. Taking
then the various branches as given by w, aw, a'%, ...... , a w ~%, we have the law of inter-
change for description of a small curve round any one branch-point as given by this
succession in cyclical order. The law of succession for a circuit enclosing more than
one of the branch-points is derivable by means of Corollary V., 90.
To find the relation of 2=00 to w, we take zz' \ and consider the new function W in
the vicinity of the z'-origin. We have
1 _*
W={A (l-a,2')(l-a 2 2')......(l-X)} OT s' '"
If the variable z' describe a very small circle round the origin in the negative sense, then
*
z' is multiplied by e~* and so W acquires a factor 8 **, that is, W is changed unless
this acquired factor is unity. It can be unity only when n/m is an integer; arid therefore,
except when n/m is an integer, 2 = 00 is a branch-point of the function. The law of
succession is the same as that for negative description of the /-circle, viz., w, a n w, a? n w, ......
the m values form a single cycle only if n be prime to m, and a set of cycles if n be not
prime to m.
Thus 2=00 is a branch-point for w=(4z s g^;-g 3 )~^; it is not a branch-point for
%={(! -2 2 )(1 2 2 2 )} 2 ; an d z b i s a branch-point for the function defined by
(z 6) w 2 = 2 a,
but 2 = 6 is not a branch-point for the function defined by (2 ft) 2 w> 2 = 2 a.
Again, if p denote a particular value of *, when 2 has a given value, and q similarly
denote a particular value of ( -- =) , then w=p + q is a six- valued function, the values
V'T I/
being
where a is a primitive cube root of unity. The branch-points are 1,0, 1, oo ; and the
orders of change for small circuits round one (and only one) of these points are as
follows :
For a small circuit round
-1
1
QC
I/PI changes to
w b
w z
W 3
w 2
w 2
W
Wi
W 4
Wi
MS
Wi
iv t
W 5
w t
"V
g
W 3
WQ
W 3
Wo
,
W 6
W 1
IV 6
W'6
tt' 4
MS
U'2
w.
190 ALGEBRAIC [93.
Combinations can at once be effected ; thus, for a positive circuit enclosing both 1 and QC
but* not -- 1 or 0, the succession is
Wl, M-4, 7 5 , ID}, W 3 , W G
in cyclical order.
94. It has already been remarked that algebraic irrationals are a special
class of functions denned by algebraical equations. Functions thus generally
defined by equations, which are polynomial so far as concerns the dependent
variable but need not be so in reference to the independent variable, are
often called algebraical. The term, in one sense, cannot be strictly applied
to the roots of an equation of every degree, seeing that the solution
of equations of the fifth and higher degrees can be effected only by
transcendental functions ; but what is implied is that a finite number of
determinations of the dependent variable is given by the equation f.
The equation is polynomial in relation to the dependent variable w, that
is, it will be taken to be of finite degree n in iv. The coefficients of the
different powers will be supposed to be uniform functions of z : were they
multiform (with a limited number of values for each value of 2) in any given
equation, the equation could be transformed into another, the coefficients of
which are uniform functions. And the equation is supposed to be irreducible,
that is, if the equation be taken in the form
f(w,z) = 0,
the left-hand member f(w, z) cannot be resolved into factors of a form and
character as regards' w and z similar to / itself.
The existence of equal roots of the equation for general values of z
requires that
,, , ()f(w, z)
f(w, z} and :L V-
ow
shall have a common factor, which will be uniform owing to the form of
f(w, z}. This form of factor is excluded by the irreducibility of the equation ;
so that /= 0, as an equation in w, has not equal roots for general values
of z. But though the two equations are not both satisfied in virtue of a
simpler equation, they are two equations determining values of w and z :
and their form is such that they will give equal values of w for special
values of z.
Since the equation is of degree n, it may be taken to be
w n + w n-i Fi ^ + W n-2f 2 (^) + . . . + lu p n _ i (>) + f n (^) = ,
where the functions F l} F 2t ... are uniform. If all their singularities be
accidental, they are rational meromorphic functions of z (unless z = oo is the
* Such a circuit, if drawn on the Neumann's sphere, maybe regarded as excluding - 1 and 0,
or taking account of the other portion of the surface of the sphere, it may be regarded as a
negative circuit including - 1 and 0, the cyclical interchange for which is easily proved to be
'!, ic 4 , w 5 , w%, w 3 , w e as in the text.
t Such a function is called bien defini by Liouville.
94.1 FUNCTIONS 191
only singularity, in which case they are holomorphic) ; and the equation can
then be replaced by one which is equivalent and has all its coefficients
holomorphic, the coefficient of w n being the least common multiple of all the
denominators of the meromorphic functions in the first form. This form
cannot however be deduced, if any of the singularities be essential.
The equation, as an equation in w, has n roots, all functions of z\ let
these be denoted by w 1} w,,, . . ., w n , which are the n branches of the function w.
When the geometrical interpretation is associated with the analytical relation,
there* are n points in the w-plane, say a l ,...,a n , which correspond with a point
in the ^-plane, say with j ; and in general these n points are distinct.
Further, as will appear from the investigations in 97 (p. 207), the n roots w
are continuous functions of z ; that is to say, any small change in the value
of z entails corresponding small changes in the value of each of the n roots w.
Hence, when z varies so as to move in its own plane, each of the w-points
moves in their common plane ; and thus there are n w-paths corresponding
to a given ^-path. These n curves may or may not meet one another.
If they do not, there are n distinct w-paths, leading from a l , ...,a n to
/3 T , ..., /3 n , respectively corresponding to the single z-path leading from a to 6.
If two or more of the w-paths do meet one another, and if the describing
w-points coincide at their point of intersection, then at such a point of
intersection in the w-plane, the associated branches w are equal; and
therefore the point in the ^-plane is a point that gives equal values for w.
It is one of the roots of the equation obtained by the elimination of w
between
the analytical test as to whether the point is a branch-point will be
considered later. The march of the concurrent w-branches from such a
point of intersection of two w-paths depends upon their relations in its
immediate vicinity.
When no such point lies on a z-paih from a to 6, no two of the w-points
coincide during the description of their paths. By 90, the -path can be
deformed (provided that, in the deformation, it does not cross a branch-point)
without causing any two of the w-points to coincide. Further, if z describe
a closed curve which includes none of the branch-points, then each of the
w-branches describes a closed curve and no two of the tracing points ever
coincide.
Note. The limitation for a branch-point, that the tracing w-points
coincide at the point of intersection of the w-curves, is of essential im-
portance.
W T hat is required to establish a point in the ^-plane as a branch-point,
is not a mere geometrical intersection of a couple of completed zo-paths
192 ALGEBRAIC FUNCTIONS [94.
but the coincidence of the w-points as those paths are traced, together with
interchange of the branches for a small circuit round the point. Thus let there
be such a geometrical intersection of two w-curves, without coincidence of the
tracing points. There are two points in the ^-plane corresponding to the
geometrical intersection; one belongs to the intersection as a point of the
w-path which first passed through it, and the other to the intersection as a
point of the w-path which was the second to pass through it. The two
branches of w for the respective values of z are undoubtedly equal ; but the
equality would not be for the same value of z. And unless the equality of
branches subsists for the same value of z, the point is not a branch-point.
A simple example will serve to illustrate these remarks. Let w be defined by the
equation
/=<J*(W*-2M?) -* = (),
so that the branches w l and w 2 are given by
c, cw 2 = cz - z
it is easy to prove that the equation resulting from the elimination of w between /=0 and
and that only the two points z= +ic are branch-points.
The values of z which make w v equal to the value of w. 2 for z - a (supposed not equal to
either 0, d or - ct) are given by
cz + z (z 2 +c 2 )^ = ca - a (a 2 + c 2 )* ,
which evidently has not z= a for a root. Rationalising the equation so far as concerns z
and removing the factor z - a, as it has just been seen not to furnish a root, we find that z
is determined by
z s + Z 2 a + to* + a s + aac 2 - 2ac (a 2 + c 2 )* = 0,
the three roots of which are distinct from a, the assumed point, and from ci, the branch-
point. Each of these three values of z will make Wj equal to the value of w 2 for z = a: we
have geometrical intersection without coincidence of the tracing points.
95. When the characteristics of a function are required, the most im-
portant class are its infinities : these must therefore now be investigated.
It is preferable to obtain the infinities of the function rather than the
singularities alone, in the vicinity of which each branch of the function
is uniform* : for the former will include these singularities as well as those
branch-points which, giving infinite values, lead to regular singularities when
the variables are transformed as in 93. The theorem which determines
them is :
The infinities of a function determined by an algebraical equation (ire the
singularities of the coefficients of the equation.
Let the equation be
w n + w n - l F^ (z} + w n ~ 2 F 2 (*) + ...+ wF n _, (z) + F n (z) = 0,
* Tbese singularities will, for the sake of brevity, be called regular.
95.]
and let w' be
determines the
w n ~ l +
we have
THEIR INFINITIES
any branch of the function; then, if the equation
remaining branches be
u n ~* G! (z) + w n ~ 3 (r 2 (z) + ... + wG n -2 (z) + G n -i (z) = 0,
F n (z} = -w'G n _,(z),
193
which
Now suppose that a is an infinity of w'', then, unless it be a zero of order
at least equal to that of G n -\ (z), a, is an infinity of F n (z). If, however, it be
a zero of G n -i (z) of sufficient order, then from the second equation it is an
infinity of F n ^(z) unless it- is a zero of order at least equal to that of
G n -v(z)\ and so on. The infinity must be an infinity of some coefficient not
earlier than Fi (z) in the equation, or it must be a zero of all the functions
G which are later than G^ (z). If it be a zero of all the functions G r , so
that we may not, without knowing the order, assert that it is of rank at
least equal to its order as an infinity of w, still from the last equation it
follows that a must be an infinity of F! (z). Hence any infinity of w is an
infinity of at least one of the coefficients of the equation.
Conversely, from the same equations it follows that a singularity of one
of the coefficients is an infinity either of w' or of at least one of the co-
efficients G. Similarly the latter alternative leads to an inference that the
infinity is either an infinity of another branch w" or of the coefficients of the
(theoretical) equation which survives when the two branches have been
removed. Proceeding in this way, we ultimately find that the infinity either
is an infinity of one of the branches or is an infinity of the coefficient in the
last equation, that is, of the last of the branches. Hence any singularity
of a coefficient is an infinity of at least one of the branches of the function.
It thus appears that all the infinities of the function are included among,
and include, all the singularities of the coefficients ; but the order of the
infinity for a branch does not necessarily make that point a regular
singularity nor, if it be a regular singularity, is the order necessarily the
same as for the coefficient.
The following method is effective for the determination of the order of
the infinity of the branch.
Let a be an accidental singularity of one or more of the F functions,
say of order nii for the function Fi ; and assume that, in the vicinity of a,
we have
F t (z} = (z- a)~ m i [a + di(z - a) + ei(z- of +...'].
F. F. 13
194
INFINITIES OF
[95.
Then the equation which determines the first term of the expansion of w in
a series in the vicinity of a is
w n + d (z - a)~ m iw n ~ l + c 2 (z - #)- w *w M - 2 + ...
+ c n _! (z - a)-"-* w + c n (z - a)- m = 0.
Mark in a plane, referred to two rectangular axes, points n, 0; nl,
m l ' ) n 2, m 2 ; . . . , 0, m n ; let these __
be A , AI, ..., A n respectively. Any line
through At has its equation of the form
y + mi = \ [x (n i)}.
that is,
y \oc = X (w i)
A
An-,-
y ,
Fig- 20.
If then w = (z a)~*f(z), where f(z) is
finite when z = a, the intercept of the
foregoing line on the negative side of the axis of y is equal to the order of
the infinity in the term
This being so, we take a line through A n coinciding in direction with the
negative part of the axis of y, and we turn it about A n in a trigonometrically
positive direction until it first meets one of the other points, say A n _ r ] then
we turn it about An^ r until it meets one of the other points, say A n _ s ; and
so on until it passes through A Q . There will thus be a line from A n to
A , generally consisting of a number of parts; and none of the points A
will be outside the figure bounded by this line and the axes.
The perpendicular from the origin on the line through A n _ r and A n _ g is
evidently greater than the perpendicular on any parallel line through a
point A, that is, on any line through a point A with the same value
of \; and, as this perpendicular is
it follows that the order of the infinite terms in the equation, when the par-
ticular substitution is made for w, is greater for terms corresponding to points
lying on the line than it is for any other terms.
If f(z) = when z = a, then the terms of lowest order after the substitu-
tion of (z a}~ K f(z} for w are
as many terms occurring in the bracket as there are points A on the line
joining A n _ r to A n _ s . Since the equation determining w must be satisfied,
terms of all orders must disappear, and therefore
c n _ 8 0*-'-+...+c n _ r = 0,
an equation determining s r values of 0, that is, the first terms in the;
expansions of s r branches w.
95.] ALGEBRAIC FUNCTIONS 195
Similarly for each part of the line : for the first part, there are r branches
with an associated value of X; for the second, s r. branches with another
associated value ; for the third, t s branches with a third associated value ;
and so on.
The order of the infinity for the branches is measured by the tangent
of the angle which the corresponding part of the broken line makes with the
axis of x\ thus for the line joining A n ^. to A n _ g the order of the infinity
for the s r branches is ^
s r
where m n _ r and mn_ g are the orders of the accidental singularities of F n _ r (z)
and F n _ s (z).
If any part of the broken line should have its inclination to the axis of
x greater than \TT so that the tangent is negative and equal to p, then the
form of the corresponding set of branches w is (z a)* 4 g (z) for all of them,
that is, the point is not an infinity for those branches. But when the
inclination of a part of the line to the axis is < ^TT, so that the tangent is
positive and equal to \, then the form of the corresponding set of branches
w is (z a)~ >i f(z) for all of them, that is, the point is an infinity of order A,
for those branches.
In passing from A n to A , there may be parts of the broken line which
have the tangential coordinate negative, implying therefore that a is not an
infinity of the corresponding set or sets of branches w. But as the revolving
line has to change its direction from A n y' to some direction through A ,
there must evidently be some part or parts of the broken line which have
their tangential coordinate positive, implying therefore that a is an infinity
of the corresponding set or sets of branches.
Moreover, the point a is, by hypothesis, an accidental singularity of at
least one of the coefficients, and it has been supposed to be an essential
singularity of none of them; hence the points A , A l} ..., A n are all in the
finite part of the plane. And as no two of their abscissae are equal, no line
joining two of them can be parallel to the axis of y, that is, the inclination
of the broken line is never -|TT and therefore the tangential coordinate is
finite, that is, the order of the infinity for the branches is finite for any
accidental singularity of the coefficients.
If the singularity at a be essential for some of the coefficients, the
corresponding result can be inferred by passing to the limit which is
obtained by making the corresponding value or values of m infinite. In
that case the corresponding points A move to infinity and then parts of the
broken line pass through A (which is always on the axis of x) parallel to
the axis of y, that is, the tangential coordinate is infinite and the order of
132
196 INFINITIES [95.
the infinity at a for the corresponding branches is also infinite. The point is
then an essential singularity (and it may be also a branch-point).
It has been assumed implicitly that the singularity is at a finite point in
the .z-plane ; if, however, it be at oo , we can, by using the transformation
zz' = 1 and discussing as above the function in the vicinity of the origin,
obtain the relation of the singularity to the various branches. We thus
have the further proposition :
The order of the infinity of a branch of an algebraical function at a
singularity of a coefficient of the equation, which determines the function, is
finite or infinite according as the singularity is accidental or essential.
If the coefficients F { of the equation be holomorphic functions, then
z = oo is their only singularity and it is consequently the only infinity for
branches of the function. If some of or all the coefficients F{ be mero-
morphic functions, the singularities of the coefficients are the zeros of
the denominators and, possibly, z=cc; and, if the functions be rational,
all such singularities are accidental. In that case, the equation can be
modified to
h (z) w n + h 1 (z) w n ~ l + h z (z) w n ~*+ ... = 0,
where h Q (z) is the least common multiple of all the denominators of the
functions F t . The preceding results therefore lead to the more limited
theorem :
When a function w is determined by an algebraical equation the coefficients
of which are holomorphic functions of z, then each of the zeros of the coefficient
of the highest power of w is an infinity of some of (and it may be of all) the
branches of the function w, each such infinity being of finite order. The point
z = oo may also be an infinity of the function w ; the order of that infinity is
finite or infinite according as z = oo is an accidental or an essential singularity
of any of the coefficients.
It will be noticed that no precise determination of the forms of the
branches w at an infinity has been made. The determination has, however,
only been deferred: the infinities of the branches for a singularity of the
coefficients are usually associated with a branch-point of the function, and
therefore the relations of the, branches at such a point will be of a general
character independent of the fact that the point is an infinity.
If, however, in any case a singularity of a coefficient should prove to be.
not a branch-point of w but only a regular singularity, then in the vicinity of
that point the branch of w is a uniform function. A necessary (but not suffi-
cient) condition for uniformity is that (w n _ r m n _g) -(s r) be an integer.
Note. The preceding method can be applied to determine the leading
terms of the branches in the vicinity of a point a which is an ordinary point
for each of the coefficients F.
96.] BRANCH-POINTS 197
96. There remains therefore the consideration of the branch-points of a
function determined by an algebraical equation.
The characteristic property of a branch-point is the equality of branches
of the function for the associated value of the variable, coupled with the
interchange of some of (or all) the equal branches after description by the
variable of a small contour enclosing the point.
So far as concerns the first part, the general indication of the form of the
value has already ( 93) been given. The points, for which values of w
determined as a function of z by the equation
f(w,z) =
are equal, are determined by the solution of this equation treated simul-
taneously with
- o >
OW
and when a point z is thus determined, the corresponding values of w, which
are equal there, are obtained by substituting that value of z and taking M,
?>f
the greatest common measure of f and =^- . The factors of M then lead to
the value or the values of w at the point ; the index m of a linear factor
gives at the point the multiplicity of the value which it determines, and
shews that m + 1 values of w have a common value there, though they are
distinct at infinitesimal distances from the point. Values of w, determined
by f but not occurring in a factor of M, are isolated values ; each of them
determines a branch that is uniform at the point.
Let z = a, W OL be a value of z and a value of w thus obtained ; and
suppose that m is the number of values of w that are equal to one another.
The point z = a is not a branch-point unless some interchange among the
m values of w is effected by a small circuit round a; and it is therefore
necessary to investigate the values of the branches* in the vicinity of z = a.
Let w = a + w', z = a + z' ; then we have
that is, on the supposition that f(w, z) has been freed from fractions,
/(a, a) + 22A rs z' r w' 8 = 0,
r,s ,
so that, since a is a value of w corresponding to the value a of z, we have
w' and z' connected by the relation
Az' r w' s = 0.
* The following investigations are founded on the researches of Puiseux on algebraic
functions; they are contained in two memoirs, Liouville, 1 S^r., t. xv, (1850), pp. 365 480,
ib., t. xvi, (1851), pp. 228240. See also the chapters on algebraic functions, pp. 1976,
in the second edition of Briot and Bouquet's Tbeorie des /auctions elliptiques.
198 BRANCH-POINTS [96.
When / is 0, the zero value of w' must occur m times, since a is a root
m times repeated; hence there are terms in the foregoing equation inde-
pendent of /, and the term of lowest index among them is w' m . Also when
w' = 0, / = is a possible root ; hence there must be a term or terms
independent of w' in the equation.
First, suppose that the lowest power of / among the terms independent
of w' is the first. The equation has the form
Az + higher powers of /
+ Bw' m + higher powers of w'
+ terms involving z and w = 0,
where A is the value of ^ ' for w = a, z = a. Let z 1 = % m , w' = v; the
oz
last form changes to
(A + Bv m ) m + terms with m+1 as a factor = ;
and therefore A + Bv m + terms involving = 0.
Hence in the immediate vicinity of z = a, that is, of = 0, we have
A + Bv m = 0.
Neither A nor B is zero, so that all the m values of v are finite. Let them
be #!,...* v m , so arranged that their arguments increase by 2?r/m through
the succession. The corresponding values of w' are
w r ' = v r
= v r z' m ,
for r=l, ..., m. Now a ^-circuit round a, that is, a ^'-circuit round its
origin, increases the argument of z' by 2?r; hence after such a circuit, we
1. ? _i
have the new value of w r ' as v r z' m e m , that is, it is v r+1 z' m which is the value
of w'r+i. Hence the set of values w\, w' 2 ,..., w' m form a complete set of
interchangeable values in their cyclical succession; all the m values, which
are equal at a, form a single cycle and the point is a branch-point.
Next, suppose that the lowest power of z among the terms independent :
of w' is z /l , where I > 1. The equation now has the form
= Az' 1 4- higher powers of /
+ Bw' m 4- higher powers of w'
1-1 m-l
where in the last summation r and s are not zero and in every term either
(i), r is equal to or greater than I or (ii), s is equal to or greater than m
or (iii), both (i) and (ii) are satisfied. As only terms of the lowest orders
96.]
BRANCH-POINTS
199
need be retained for the present purpose, which is the derivation of the first
term of w' in its expansion in powers of z, we may use the foregoing equation
in the form
l-l m-l
Az' l + 2 2 A rg z' r w' s + Bw' m = 0.
r=l s=l
To obtain this first term we proceed in a manner similar to that in 95 *.
Points A ,...,A m are taken in a plane
referred to rectangular axes having as co-
ordinates 0, 1 ; . . . ; s, r ; . . . ; ra, respectively.
A line is taken through A m and is made to
turn round A m from the position A m O until
it first meets one of the other points ; then
round the last point which lies in this
direction, say round Aj, until it first meets
another; and so on.
Any line through A t (the point s i} r t -) is
of the form
Fig. 21.
The intercept on the axis of /-indices is \Si + TI, that is, the order of the
term involving A r .g. for a substitution w' = vz'^. The perpendicular from the
origin for a line through At and Aj is less than for any parallel line through
other points with the same inclination ; and, as this perpendicular is
(X* + *).(!.+ XT*i
it follows that, for the particular substitution w' = vz'*, the terms correspond-
ing to the points lying on the line with coordinate X are the terms of lowest
order, and consequently they are the terms which give the initial terms for
the associated set of quantities w'.
Evidently, from the indices retained in the equation, the quantities X
for the various pieces of the broken line from A m to A are positive and
finite.
Consider the first piece, from A m to Aj say ; then taking the value of A, for
that piece as /* 1} so that we write v^z'^ as the first term of w, we have as the
set of terms involving the lowest indices
Bw'
Sj being the smallest value of s retained ; and then
so that
m s m
* Keference in this connection may be made to Chrystal's Algebra, ch. xxx., with great
advantage, as well as the authorities quoted on p. 197, note.
200 GROUPING OF BRANCHES [96.
Let pfq be the equivalent value of /^ as the fraction in its lowest terms ; and
P
write z = *. Then w' = v l z' < * = v^ p ; all the terms except the above group
are of order > mp, and therefore the equation leads after division by m Pv*j to
an equation which determines ra Sj values for v lt and therefore the initial
terms of m Sj of the w-branches.
Consider now the second piece, from Aj to Ai say ; then taking the value
of \ for that piece as fa, so that we write v 2 z'^ as the first term of w', we
have as the set of terms involving the lowest indices for this value of fa
where s f is the smallest value of s retained. Then
Proceeding exactly as before, we find
A r . g v,J-*i + 22A rg v/-< + A r {g{ =
as the equation determining Sj Si values for v 2 , and therefore the initial
terms of 8j Si of the w-branches.
And so on, until all the pieces of the line are used ; the initial terms of
all the w-branches are thus far determined in groups connected with the
various pieces of the line A m AjAi...A . By means of these initial terms,
the m branches can be arranged for their interchanges, by the description of
a small circuit round the branch-point, according to the following theorem :
Each group can be resolved into systems, the members of each of which are
cyclically interchangeable.
It will be sufficient to prove this theorem for a single group, say the
group determined by the first piece of broken line : the argument is
general.
77 7* 7**
Since - is the equivalent of - and of - and since 5,- < s, we have
q m s m Sj
m - s = kq, m Sj = fyq, kj > k ;
and then the equation which determines v 1 is
Bvffl + ^A rg v*r k) i + A rj9 . = 0,
that is, an equation of degree kj in v-p as its variable. Let U be any root of
i
it ; then the corresponding values of i\ are the values of U<*. Suppose these
q values to be arranged so that the arguments increase by 2?r - , which is
possible, because p is prime to q. Then the q values of w', being the values
of v l z' lil , are
p p P
v u z'v, v l2 z'<i, v n z\...,
96.J GROUPING OF BRANCHES 201
where v la is that value of U? which has -- for its argument. A circuit
round the 2;' -origin evidently increases the argument of any one of these
w -values by 27rp/q, that is, it changes it into the value next in the succession ;
and so the set of q values is a system the members of which are cyclically
interchangeable.
This holds for each value of U derived from the above equation ; so that
the whole set of m Sj branches are resolved into kj systems, each containing
q members with the assigned properties.
It is assumed that the above equation of order kj in v^ has its roots unequal.
If, however, it should have equal roots, it must be discussed ab initio by a
method similar to that for the general equation; as the order kj (being a
factor of m Sj) is less than m, the discussion will be shorter and simpler,
and will ultimately depend on equationa with unequal roots as in the case
above supposed.
It may happen that some of the quantities /j, are integers, so that the
corresponding integers q are unity : a number of the branches would then be.
uniform at the point.
It thus appears that z = a is a branch-point and that, under the present
circumstances, the m branches of the function can be arranged in systems,
the members of each one of which are cyclically interchangeable.
Lastly it has been tacitly assumed in what precedes that the common
value of w for the branch-point is finite. If it be infinite, this infinite value
can, by 95, arise only out of singularities of the coefficients of the equation :
and there is therefore a reversion to the discussion of 95, 96. The dis-
tribution of the various branches into cyclical systems can be carried out
exactly as above.
Another method of proceeding for these infinities would be to take
ww' 1, z = c + z ; but this method has no substantial advantage over the
earlier one and, indeed, it is easy to see that there is no substantial
difference between them.
Note. In the first case considered, a single transformation of the variables
represented by z = m , w = v, was sufficient to discriminate among the m
branches.
In the second case, the number of different directions in the broken line
of fig. 21 is finite (<:ra); to each such direction there corresponds a trans-
formation of the variables which leads to a discrimination among one of the
groups out of the m branches, and therefore the whole number of trans-
formations needed to discriminate among the m branches is finite.
If the m branches are infinite at the point, the corresponding analysis
shews that the whole number of transformations needed to discriminate among
those m branches is finite.
202 EXAMPLES [96.
Moreover m is finite, being < n ; hence the various branches of the
function w are discriminated, at a branch-point, by a finite number of trans-
formations.
Ex. 1. As an example, consider the function determined by the equation
The equation determining the values of z which give equal roots for w is
8z (z -I) 2 = 4 (z -I) 3 ,
so that the values are 2=1 (repeated) and z= 1.
When 2 = 1, then w=0, occurring thrice; and if 2=1+2', then
that is, wf = \z'^.
The three values are branches of one system in cyclical order for a circuit round 2=1.
When z= 1, the equation for w is
4^-30; -1=0,
that is, (w-l)(2w + 1) 2 = 0,
so that w=I, or w= -, occurring twice.
For the former of these we easily find that, for 2 = 1 + /, the value of w is
l+f/+ ...... ,an isolated branch as is to be expected, for the value 4 is not repeated.
For the latter we take w= %+itf, and find
vP
so that the two branches are
and they are cyclically interchangeable for a small circuit round z= 1.
These are the finite values of w at branch-points. For the infinities of w, which may
arise in connection with the singularities of the coefficients, we take the zeros of the
coefficient of the highest power of w in the integral equation, viz., 2=0, which is thus the
only infinity of w. To find its order we take w=z~ n f(z)=yz~ n +. ..... , where y is a
constant and f(z) is finite for 2=0; and then we have
Thus l-3n=-n y
provided both of them be negative; the equality gives n=\ and satisfies the condition.
And 8y 3 = 3y. Of these values one is zero, and gives a branch of the function without
an infinity; the other two are ^V f and they give the initial term of the two
branches of w, which have an infinity of order ^ at the origin and are cyclically
interchangeable for a small circuit round it. The three values of w for infinitesimal
values of 2 are
3 . _i 1 7 73.4 4 275 /3 . $. 4
- - " ~*
/- -i 1 jL /?__!- 275
18 8
96.] ALGEBRAIC FUNCTIONS 203
The first two of these form the system for the branch-point at the origin, which is neither
an infinity nor a critical point for the third branch of the function.
Ex. 2. Obtain the branch-points of the functions which are defined by the following
equations, and determine the cyclical systems at the branch-points :
(i) w 3 w+z=Q;
(ii) M^
(iii) ^
(iv) w 3
44
(v) vfi - (1 - z*) w* - - 6 2 2 (1 - 2 2 ) 4 =0. (Briot and Bouquet.)
Also discuss the branches, in the vicinity of z=0 and of 2 = 00 , of the functions defined
by the' following equations :
( vi) aw 1 + b-it^z + cw^z 4 + dw^ + ewz 7 +fz 9 + gvfl + hw*z* + h w = ;
(vii) w m z n +w n +z m =0.
97. Having shewn how to discriminate at any point among the various
branches of the algebraic function defined by the equation
f(w, z) = h (z} w n + h, (2) w n ~ l + h z (z) w n ~* + . . . = 0,
where the quantities h (z), h^^z), h z (z), ... are holomorphic functions, we
proceed to indicate the character of the various branches near the point. After
the preceding discussions, it will be sufficient to consider only finite values
of z ; the consideration of infinite values can be obtained through the zero
1
values of /, where is substituted for z. It is only for zeros of h Q (z) that
z
an infinite value (or several infinite values) of w can arise : they can be
discussed through the zero values of w', where . is substituted for w.
w
Accordingly, let a denote a finite value of z, and let a denote a finite
value of w for z = a, where a may be a simple root or multiple root of
/(a, a) = 0. Take w = a + y, z = a + a, so as to consider some vicinity of the
point a and the character of w in that vicinity , and let
f(w, z} =/( + y, a + x) = F(y, x),
where F is a polynomial in y of degree not greater than n, and the coefficients
are holomorphic functions of x which are polynomials when all the coefficients
h , Aj, ... are polynomials. We have F (0, 0)= 0, so that there is no term
free from x and y in F (y, x). Also F (y, 0) does not vanish for all values of
y ; for that would imply that some integral power of # is a factor of F(y, x)
and therefore that some integral power of z a is a factor of f(w, z), which
is not the case. Hence there is at least one term in the polynomial F (y, x),
which has a constant for its coefficient, and there may be more than one
such term ; let the term of lowest order in y be By m , and let the aggregate
of such terms be denoted by F (y). Denoting the rest by F l (y, x), where
F r is a polynomial in y that has holomorphic functions of x for its coefficients,
we have
204 THEOREM OF [97.
clearly Fi(y, x) vanishes when #=0 for all values of y, in any vicinity of
y = 0. Hence* we can choose a region in the vicinity of y = 0, x = 0,
such that
| JJT | ^ I rr I .
!*! >\*i\>
but as F vanishes when y = Q, there may be some limit of \y other than
zero, at and below which the inequality does not hold. Accordingly, assume
as the range for the inequality
I Po I < I y \ < P> \as\<r.
For such values we have, on taking logarithmic derivatives of the equation
the relation
iap = Ji^
F dy F dy
Since F is a polynomial in y that is divisible by y m , we have
where G is a converging series of integral powers. Similarly
W *
where the quantities (TA )M are converging series of integral powers of x, each
00 J JP\
of them vanishing with x. As the series 2 - ^r converges uniformly, we
x=i A, -P
may gather together the various terms that involve the same power of y ;
and we then have
oo 1 PA. oo
2 i*fe- 2
X=l A, ^ p = _
where each of the coefficients G p is a converging power-series in x which
vanishes with x. Thus
1 dF m
Vfl
where the only term on the right-hand side in y~ l is .
7
Now let %, ..., rj g denote the zeros of F(y, K), for values of y such that
| y | < p and for a parametric value K of x such that | K \ < r : it might be that
there are no such zeros (though this will be seen not to be the case) : repeated
zeros are given by repetition in the quantities 77. Then
F
1=1
* What follows is a special case of an important theorem, due to Weierstrass, Get. Werke,
t. ii, p. 135.
97.] WEIERSTRASS - 205
is finite for all values of y within the range, and therefore it can be expanded
in a converging series of positive powers, so that
Now choose values of y, still such that y < p, and also such that they give
moduli greater than the greatest of the quantities 17; | ; the fractions on the
right-hand side can be expanded in descending powers of y, and we have
^(y,*) s |
F dy f w -**i*7
where M = %" + 1;./ 4 + . . . + qf.
The parametric value K in this expansion can be replaced by x ; and thus
comparing the two expansions for -= ^- , we have
U
s = m, Sp = pGr-p-
The first of these results shews that there are m roots of F within the
range. The second of them expresses the sums of the positive powers of
?7j , . . . , r)i as converging series of positive powers of x which vanish with x ;
hence the symmetric integral functions of ^j, ..., yi are regular functions of x
in the vicinity of x = and vanish with x. Let
g (y, x) = (y - *?i) (y - vj (y - m)
- =-y m +9iy mr - i + ... + gi,
where g\ are regular functions of x and vanish with x.
A further comparison of the expansions shews that
where F (y, x) is a regular function of y and x, given by
.
Hence
1 dF 4 1 9
^^-=2 ~-i-a-
* 9y z=i y - ^ 9 2/
and therefore
where U is a quantity independent of y. Now when x is zero, U is B; hence
generally
17= 5 (1 + positive powers of x)
206 AN ALGEBRAIC FUNCTION [97.
where is a regular function of x vanishing with x. Writing G ( y, x) for
F (y, x) 4- f , where G (0, 0) = 0, we have
with the defined significance of g (y, x) and G (y, x).
Our immediate purpose is with such values of y, being functions of x, as
make F vanish in the region considered. Clearly the exponential term does
not vanish ; and therefore we have the values of y given by
where g lt g z , . .., g m are regular functions of a; that vanish with x.
Case 1. The simplest case arises when m = 1 ; the root a is then a simple
root of f(a, a) = 0, and we have
that is,
or in the vicinity of the point a, the branch associated with the simple root
of /(a, a) = is a regular function of z a.
The same result holds for each simple root a of the equation /(a, a) = 0.
Case 2. Let m > 1, so that the root a is a multiple root of /(a, a) = 0,
and z = a may be (and generally is) a branch-point. The equation
g(y, x) = y m + giy m ~ l +gzy m ~ 2 + ... +g m =
determines m branches. By 96 these branches can be arranged in groups,
each group corresponding to a particular order j y \ oc | x \ q for sufficiently small
values of | y| and x , and the order being determined by a portion of a broken
line in a Puiseux diagram.
Thus for the first portion of the line, take x=^ q ,y VP; then the equation
becomes of the form
(*>, C) = 0,
where P (v, ) is a regular function of its arguments. When = 0, we have
V s + ^KrV 8 ^ + K S = 0,
rejecting the zero values of v. If v = Vi be a simple root of this equation,
then in the earlier equation we write v = v l + u ; and it then follows, by
Case 1 above, that
where R is a regular function of that vanishes when =0. Accordingly
for every simple root of the equation in v when is zero, we have
97.] IS ANALYTIC 207
shewing that the corresponding branch of the algebraic function is a uniform
i i
function of (z a) . When q is 1, the branch is a regular function of z a.
When q > 1, there is a system of roots of the same form.
It may happen that ^ is a multiple root* of
V g + ^K r tf- r + K S = 0.
This equation is of degree s, being less than m, the degree of the original
equation. To it we apply, for the multiple root, the preceding process : and
so gradually reach the stage in which each of the branches is discriminated
and analytically expressed.
Similarly for the remaining portions of the broken line in the Puiseux
diagram of 96.
It therefore follows that all the branches (if the branches be more than
one) of the function, denned by the equation f(w, z) = Q and acquiring the
value a when z = a, where f(a., a) = 0, can be represented in the analytical
form
where S() is a regular function of its argument which does not vanish when
=0, and where p, q are positive integers not necessarily the same for all
the branches. (As already remarked, we have assumed a. and a to be finite.
It is easy to see that for an infinite value of w when z = a, we have a branch
of the form
where p' is a finite integer; and similarly for infinite values of z.) Conse-
quently the function defined by the equation f(w, z) 0, which is polynomial
in w and uniform in z, has m branches at any point a, each of the branches
i^
being expressible as a uniform analytic function of (z a) q . If f(w, z) is
polynomial in z as well as in w, the non-regular points of the branches are
poles and branch-points: no point in the plane is an essential singularity for
any branch.
COROLLARY. We have the theorem, originally due to Cauchy, as an
inference from the whole investigation:
The roots w of an equation f(w,z) = 0, which is polynomial in w and
uniform in z, are continuous functions of z.
It follows at once from the two relations
Such is the case for the equation
w 6 - 15w*z - 2w 3 z + 15w 2 z 2 + &wz 2 + z 2 - z 3 = 0.
208 SIMPLE BRANCH-POINTS [97.
Note. If v-i be a multiple root of its equation, the form
is still valid : but p and q are then not necessarily prime to each other. (The
equation represented by
is an example.) The condition is that, if the indices in the expression for
w a have a common factor/, then /is not a factor of q.
98. There is one case of considerable importance which, though limited
in character, is made the basis of Clebsch and Gordan's investigations* in the
theory of Abelian functions the results being, of course, restricted by the
initial limitations. It is assumed that all the branch-points are simple, that
is, are such that only one pair of branches of w are interchanged by a circuit
of the variable round the point ; and it is assumed that the equation /= is
polynomial not merely in w but also in z. The equation /= can then be
regarded as the generalised form of the equation of a curve of the nth order,
the generalisation consisting in replacing the usual coordinates by complex
variables ; and it is further assumed, in order to simplify the analysis, that all
the multiple points on the curve are (real or imaginary) double-points. But,
even with the limitations, the results are of great value in themselves ; and
the theory of birational transformation ( 245 252) brings them within the
range of unrestricted generality. It is therefore desirable to establish the
results that belong to the present section of the subject.
We assume, therefore, that the branch-points are such that only one
pair of branches of w are interchanged by a small closed circuit round any
one of the points. The branch-points are among the values of z determined
by the equations
/(,,) = o, >- a
When /= has the most general form consistent with the assigned
limitations, f(w, z) is of the nth degree in z ; the values of z are determined
by the eliminant of the two equations which is of degree n (n 1), and there
are, therefore, n(n 1) values of z which must be examined.
First, suppose that - does not vanish for a value of z, thus
obtained, and the corresponding value of w\ then we have the first case
in the preceding investigation. And, on the hypothesis adopted in the
present instance, m = 2 ; so that each such point z is a branch-point.
* Clebsch und Gordan, Theorie der Abel'schen Functionen, (Leipzig, Teubner, 1866). It will
be proved hereafter ( 252) that any algebraical equation can be transformed birationally into an
equation of the kind indicated. The actual transformations, however, tend to become extremely
complicated ; and, in particular instances, detailed results would be obtained more simply by
proceeding directly from the original equation.
98.] SIMPLE BRANCH-POINTS 209
O^* / \
Next, suppose that J ' - vanishes for some of the n (n 1) values of z ;
oz
the value of m is still 2, owing to the hypothesis. The case will now be still
d 2 f(w z)
further limited by assuming that \ ^ - does not vanish for the value of z
and the corresponding value of w ; and thus in the vicinity of z = a, w = a we
have an equation
= Az' 2 + ZBz'w' + Cw' 2 + terms of the third degree + ...... ,
where A, B, C are the values of ~- , ~r- , ^r for z = a, w = a.
dz 2 dzdw dw*
If B 2 ^ AC, this equation leads to the solution
Cw + Bz oc uniform function of z'.
The point z = a, w = a is not a branch-point ; the values of w, equal at the point,
are functionally distinct. Moreover, such a point z occurs doubly in the
eliminant ; so that, if there be 8 such points, they account for 2S in the eliminant
of degree n (n 1) ; and therefore, on their score, the number n (n 1) must
be diminished by 28. The case is, reverting to the generalisation of the
geometry, that of a double point where the tangents are not coincident.
If, however, B 2 = A C, the equation leads to the solution
Cw' ' ' ' 2 '
The point z= a, w = a is a point where the two values of z interchange.
Now such a point z occurs triply in the eliminant ; so that, if there be tc
such points, they account for 3# of the degree of the equation. Each of
them provides only one branch-point, and the aggregate therefore provides K
branch-points ; hence, in counting the branch-points of this type as derived
through the degree of the eliminant, the degree must be diminished by 2/e.
The case is, reverting to the generalisation of the geometry, that of a double
point (real or imaginary) where the tangents are coincident.
It is assumed that all the n (n 1) points z are accounted for under
the three classes considered. Hence the number of branch-points of the
equation is
fl = n(n-l)-28-2,
where n is the degree of the equation, 8 is the number of double points
(in the generalised geometrical sense) at which tangents to the curve do not
coincide, and K is the number of double points at which tangents to the
curve do coincide.
And at each of these branch-points, O in number, two branches of the
function are equal and, for a small circuit round it, interchange.
p. F. 14
210 FUNCTIONS POSSESSING [99.
99. The following theorem is a combined converse of many of the
theorems which have been proved:
A function w, which has n (and only n) values for each value of z, and \
which has a finite number of infinities and of branch-points in any part of the
plane, is a root of an equation in w of degree n, the coefficients of which are
uniform functions of z in that part of the plane.
We shall first prove that every integral symmetric function of the n
values is a uniform function in the part of the plane under consideration.
n
Let S k denote 2 W{ k , where k is a positive integer. At an ordinary point
i = l
of the plane, S k is evidently a one- valued function and that value is finite ;
Sic is continuous ; and therefore the function S k is uniform in the immediate
vicinity of an ordinary point of the plane.
For a point a, which is a branch-point of the function w, we know that
the branches can -be arranged in cyclical systems. Let w l} ..., w^ be such a
system. Then these branches interchange in cyclical order for a description
of a small circuit round a ; and, if z a = Z*, it is known ( 93) that, in the
vicinity of Z = 0, a branch w is a uniform function of Z, say
Therefore w k = G k + P k (Z),
\6 1
in the vicinity of Z = ; say
wj = A k + 2 B k>m Z~ + 2 C t , m Z.
m=l m=\
Now the other branches of the function, which are equal at a, are derivable
from any one of them by taking the successive values which that one
acquires as the variable describes successive circuits round a. A circuit
of w round a changes the argument of z a by 2-Tr, and therefore gives Z
reproduced but multiplied by a factor which is a primitive /ith root of unity,
say by a factor a ; a second circuit will reproduce Z with a factor a? ; and so
on. Hence
w 2 k = A k + 2 B km ar m Z- m + 2 C km a m Z m ,
w r k =A
99.] A FINITE NUMBER OF BRANCHES 211
and therefore
B km Z~ m (1 + cr m + a~ 2m + . . .
m=\
Now, since a is a primitive /*th root of unity,
1 + ct s + a 2 * + . . . 4- a*<*-D
is zero for all integral values of s which are not integral multiples of /*, and it
is fji for those values of s which are integral values of //, ; hence
- I w r k =A k + B ttVL Z-* + B kt ^Z-^ + B k>3lt Z- s * + ...
+ C^Z* + C^Z* + C^Z* + . . .
=A k + B' k>1 (z - a)- 1 + & k>2 (z - a)~ 2 + B' k>3 (z - a)~ 3 + . . .
+ C' ktl (z-d) + G' kt2 (z-a)*+C' k>s (z-af + ....
ft
Hence the point z = a may be a singularity of 2 w r k but it is not a branch-
r=l
point of the function ; and therefore in the immediate vicinity of z = a the
H
quantity 2 w r * is a uniform function.
r = l
The point a is an essential singularity of this uniform function, if the
order of the infinity of w at a be infinite : it is an accidental singularity, if
that order be finite.
This result is evidently valid for all the cyclical systems at a, as well as
for the individual branches which may happen to be one-valued at a. Hence
n
8 k , being the sum of sums of the form S w r k each of which is a uniform
r=l
function of z in the vicinity of a, is itself a uniform function of z in that
vicinity. Also a is an essential singularity of 8 k , if the order of the infinity
at z = a for any one of the branches of w be infinite ; and it is an accidental
singularity of S k , if the order of the infinity at z = a for all the branches of w
be finite. Lastly, it is an ordinary point of S k , if there be no branch of w for
which it is an infinity. Similarly for each of the branch-points.
Again, let c be a regular singularity of any one (or more) of the branches
of w ; then c is a regular singularity of every power of each of those branches,
the singularities being simultaneously accidental or simultaneously essential.
Hence c is a singularity of S k : and therefore in the vicinity of c, S k is a
uniform function, having c for an accidental singularity if it be so for each of
the branches w affected by it, and having c for an essential singularity if it
be so for any one of the branches w.
It thus appears that in the part of the plane fcnder consideration the
function S k is one- valued ; and it is continuous and finite, except at certain
142
212 FUNCTIONS POSSESSING [99.
isolated points each of which is a singularity. It is therefore a uniform
function in that part of the plane ; and the singularity of the function at any
point is essential, if the order of the infinity for any one of the branches w
at that point be infinite, but it is accidental, if the order of the infinity for
all the branches w there be finite. And the number of these singularities
is finite, being not greater than the combined number of the infinities of the
function w, whether regular singularities or branch-points.
Since the sums of the kth powers for all positive values of the integer k
are uniform functions, and since any integral symmetric function of the
n values is a rational integral function of the sums of the powers, it follows
that any integral symmetric function of the n values is a uniform function
of z in the part of the plane under consideration ; and every infinity of a
branch w leads to a singularity of the symmetric function, which is essential
or accidental according as the orders of infinity of the various branches are
not all finite or are all finite.
Since w has n (and only n) values w lt . .. , w n for each value of z, the equation
which determines w is
(w Wi) (w w 2 ) ...(w w n ) = 0.
The coefficients of the various powers of 'w are symmetric functions of the
branches Wj, ...,;; and therefore they are uniform functions of z in the part
of the plane under consideration. They possess a finite number of singularities,
which are accidental or essential according to the character of the infinities of
the branches at the same points.
COROLLARY. If all the infinities of the branches in the finite part of the
whole plane be of finite order, then the finite singularities of all the coefficients
of the powers of w in the equation satisfied by w are all accidental ; and the
coefficients themselves then take the form of a quotient of an integral uniform
function (which may be either transcendental or merely polynomial, in the sense
o/ 47) by another function of a similar character.
If z = oo be an essential singularity for at least one of the coefficients,
through being an infinity of unlimited order for a branch of w, then one
or both of the functions in the quotient-form of one at least of the coefficients
must be transcendental.
If z = oo be an accidental singularity or an ordinary point for all the
coefficients, through being either an infinity of finite order or an ordinary
point for the branches of w, then all the functions which occur in all the
coefficients are rational expressions. When the equation is multiplied
throughout by the least common multiple of the denominators of the
coefficients, it takes the form
w n h t (z) + w n - l h^ (*) + ...+ wh n ^ (z) + h n (z) = 0,
where the functions h (z), /^ (z), ..., h n (z) are polynomials in z.
99.] A FINITE NUMBER OF BRANCHES 213
A knowledge of the number of infinities of w gives an upper limit of the
degree of the equation in z in the last form. Thus, let a f be a regular
singularity of the function; and let a f) &, 71, ... be the orders of the infinities
of the branches at a; ; then
where \i denotes Of + & + yt + . . . , is finite (but not zero) for z = at.
Let Ci be a branch-point, which is an infinity; and let //, branches w form a
-'.
system for c t -, such that w(z d)v- is finite (but not zero) at the point; then
w l w*...w llL (z erf*
is finite (but not zero) at the point, and therefore also
W 1 ...W n (z-C i f i+ * i+ + i+ -
is finite, where B i} fa, i/r t -, ... are numbers belonging to the various systems;
or, if e; denote t + fa + fa + . . . , then
w 1 ...w n (z- erf*
is finite for z = d. Similarly for other symmetric functions of w.
Hence, if a 1} o^, ... be the regular singularities with numbers X 1} \2, ...
defined as above, and if c l , c 2 , ... be the branch -points, that are also infinities,
with numbers e 3 , e 2 , ... defined as above, then the product
(w-O ...... (w-w n } II (z-ajfr II (z-d) i
;=i i=i
is finite at all the points a { and at -all the points c t -. The points a and the
points c are the only points in the finite part of the plane that can make the
product infinite : hence it is finite everywhere in the finite part of the plane,
and it is therefore polynomial in z.
Lastly, let p be the number for z= oo corresponding to \f for a; or to a
for c i} so that for the coefficient of any power of w in (w w^ . . . (w - w n ) the
greatest difference in degree between the numerator and the denominator is
p in favour of the excess of the former.
Then the preceding product is of order
p
which is therefore the degree of the equation in z when it is expressed in a
holomorphic form.
o
CHAPTER IX.
PERIODS OF DEFINITE INTEGRALS, AND PERIODIC FUNCTIONS IN GENERAL.
100. INSTANCES have already occurred in which the value of a function
of z is not dependent solely upon the value of z but depends also on the
course of variation by which z obtains that value ', for example, integrals of
uniform functions, .and multiform functions. And it may be expected that,
a fortiori, the value of an integral connected with a multiform function will
depend upon the course of variation of the variable z. Now as integrals
which arise in this way through multiform functions and, generally, integrals
connected with differential equations are a fruitful source of new functions,
it is desirable that the effects on the value of an integral caused by variations
of a z-path be assigned so that, within the limits of algebraic possibility, the
expression of the integral may be made completely determinate.
There are two methods which, more easily than others, secure this result ;
one of them is substantially due to Cauchy, the other to Riemann.
The consideration of Riemann's method, both for multiform functions and
for integrals of such functions, will be undertaken later, in Chapters XV.,
XVI. Cauchy 's method has already been used in preceding sections relating
to uniform functions, and it can be extended to multiform functions. Its
characteristic feature is the isolation of critical points, whether regular
singularities or branch-points, by means of small curves each containing one
and only one critical point.
Over the rest of the plane the variable z ranges freely and, under certain
conditions, any path of variation of z from one point to another can, as will
be proved immediately, be deformed* without causing any change in the
value of the integral, provided that the path does not meet any of the small
curves in the course of the deformation. Further, from a knowledge of the
relation of any point thus isolated to the function, it is possible to calculate
the change caused by a deformation of the 2-path over such a point ; and
thus, for defined deformations, the value of the integral can be assigned
precisely.
100.] INTEGRAL OF A BRANCH 215
The properties proved in Chapter II. are useful in the consideration of
the integrals of uniform functions ; it is now necessary to establish the
propositions which give the effects of deformation of path on the integrals
of multiform functions. The most important of these propositions is the
following :
rb
If w be a multiform function, the value of I wdz, taken between two
J a
ordinary points, is unaltered for a deformation of the path, provided that the
initial branch of w be the same and that no branch-point or infinity be crossed
in the deformation.
Consider two paths acb, adb, (fig. 16, p. 181), satisfying the conditions
specified in the proposition. Then in the area between them the branch w
has no infinity and no point of discontinuity ; and there is no branch-point
in that area. Hence, by 90, Corollary VI., the branch w is a uniform
monogenic function for that area ; it is continuous and finite everywhere
within it and, by the same Corollary, we may treat w as a uniform, mono-
genic, finite and continuous function. Hence, by 17, we have
r
J
b ,'a
wdz + (d) I wdz = 0,
b
the first integral being taken along acb and the second along bda', and
therefore
rb ra rb
(c) I wdz = (d) I wdz = (d) I wdz,
J a J b J a
shewing that the values of the integral along the two paths are the same
under. the specified conditions.
It is evident that, if some critical point be crossed in the deformation,
the branch w cannot be declared uniform and finite in the area, and the
theorem of 17 cannot then be applied.
COROLLARY I. The integral round a closed curve containing no critical
point is zero.
COROLLARY II. A curve round a branch-point, containing no other
critical point of the function, can be deformed into a loop
without altering the value of jwdz ; for the deformation
satisfies the condition of the proposition. Hence, when
the value of the integral for the loop is known, the
value of the integral is known for the curve.
COROLLARY III. From the proposition it is possible
to infer conditions, under which the integral jwdz round
the whole of any curve remains unchanged, when the whole
curve is deformed, without leaving an infinitesimal arc
common as in Corollary II.
216 INTEGRATION [100.
Let CDC', ABA' be the curves: join two consecutive points AA' to two
consecutive points CC'. Then if the area CAB AC' DC
enclose no critical point of the function w, the value of
fwdz along CDC' is by the proposition the same as its
value along CABA'C'. The latter is made up of the
value along CA, the value along ABA', and the value
along AC', say \ C / X
rA r r<? Mb
\ wdz + \ wdz + w'dz, Fie 23.
J C J B J A'
where w' is the changed value of w consequent on the description of a simple
curve reducible to B ( 90, Cor. II.).
Now since w is finite everywhere, the difference between the values of w
at A and at A' consequent on the description of ABA' is finite: hence as
A A is infinitesimal the value of \wdz necessary to complete the value for
the whole curve B is infinitesimal and therefore the complete value can be
taken as the foregoing integral wdz. Similarly for the complete value
J B
along the curve D : and therefore the difference of the integrals round B and
round D is
rA re-
I wdz +- w'dz,
Jo J A'
rA
say (w-w')dz.
J c
In general this integral is not zero, so that the values of the integral
round B and round D are not equal to one another : and therefore the curve
D cannot be deformed into the curve B without affecting the value of fwdz
round the whole curve, even when the deformation does not cause the curve
to pass over a critical point of the function.
But in special cases it may vanish. The most important and, as a
matter of fact, the one of most frequent occurrence is that in which the
description of the curve B restores at A' the initial value of w at A. It
easily follows, by the use of 90, Cor. II., that the description of D (as-
suming that the area between B and D includes no critical point) restores
at C' the initial value of w at G. In such a case, w = w' for corresponding
points on AC and A'C', and the integral, which expresses the difference,
is zero : the value of the integral for the curve B is then the same as that
for I). Hence we have the proposition :
If a curve be such that the description of it by the independent variable
restores the initial value of a multiform function w, then the value of fwde
taken round the curve is unaltered when the curve is deformed into any other
curve, provided that no branch-point or point of discontinuity of w is crossed
in the course of deformation.
100.] OF MULTIFORM FUNCTIONS
This is the generalisation of the proposition of 19 which has thus far
been used only for uniform functions.
Note. Two particular cases, which are very simple, may be mentioned
here : special examples will be given immediately.
The first is that in which the curve B, and therefore also D, encloses
no branch-point or infinity; the initial value of w is restored after a
description of either curve, and it is easy to see (by reducing B to a
point, as may be done) that the value of the integral is zero.
The second is that in which the curve encloses more than one branch-
point, the enclosed branch-points being such that a circuit of all the loops,
into which (by Corollary V., 90) the curve can be deformed, restores the
initial branch of w. This case is of especial importance when w is two-valued :
the curves then enclose an even number of branch-points.
101. It is important to know the value of the integral of a multiform
function round a small curve enclosing a branch-point.
Let c be a point at which m branches of an algebraic function are equal
and interchange in a single cycle ; and let c, if an infinity, be of only finite
order, say k/m. Then in the vicinity of c, any of the branches w can be
expressed in the form
00 A
w= 2 g s (z-c} m ,
s=-k
where & is a finite integer.
The value of fwdz taken round a small curve enclosing c is the sum of
the integrals
s^
g s j(z-c) m dz,
the value of which, taken once round the curve and beginning at a point z l} is
ma s L+i
-^- (z, - c) m V - 1],
m + s ^ l L
where a is a primitive rath root of unity, provided m + s is not zero.
If then ra + s be positive, the value is zero in the limit when the curve
is infinitesimal : if m + s be negative, the value is oo in the limit.
But, if m + s be zero, the value is 27rig s .
Hence we have the proposition : If, in the vicinity of a branch-point c,
where m branches w are equal to one another and interchange cyclically, the
expression of one of the branches be
then fwdz, taken once round a small curve enclosing c, is zero, if k<m; is
infinite, if k>m; and is 2-jrigk, if k = m.
218
MULTIPLICITY OF VALUE
[101.
It is easy to see that, if the integral be taken m times round the small
curve enclosing c, then the value of the integral is 2ra7n<jr m when k is greater
than m, so that the integral vanishes unless there be a term involving (z c)" 1
in the expansion of a branch w in the vicinity of the point. The reason that
the integral, which can furnish an infinite value for a single circuit, ceases to
_ A
do so for m circuits, is that the quantity (z^ c) **, which becomes indefi-
nitely great in the limit, is multiplied for a single circuit by a* 1, for a
second circuit by a 2X a x , and so on, and for the mih circuit by a"^ a (w|r ~ 1)X ,
the sum of all of which coefficients is zero.
Ex. The integral \{(z -a)(z b)...(z -f)}~^dz taken round an indefinitely small curve
enclosing a is zero, provided no one of the quantities 6, ...,/ is equal to a.
102. Some illustrations have already been given in Chapter II., but
they relate solely to definite, not to indefinite, integrals of uniform
functions. The whole theory will not be considered at this stage ; we shall
merely give some additional illustrations, which will shew how the method
can be applied to indefinite integrals of uniform functions and to integrals
of multiform functions, and which will also form a simple and convenient
introduction to the theory of periodic functions of a single variable.
We shall first consider indefinite integrals of uniform functions.
Ex. 1. Consider the integral / , and denote* it by /().
The function to be integrated is uniform, and it has an accidental singularity of the first
order at the origin, which is its only singularity. The value of \z~ l dz taken positively
along a small curve round the origin, say round a circle with the origin as centre, is 2ni ;
but the value of the integral is zero when taken along any closed curve which does not
include the origin.
Taking z\ as the lower limit of the integral, and any point z as the upper limit, we
consider the possible paths from 1 to z. Any path from 1 to z can be deformed, without
crossing the origin, into a path which circumscribes the origin positively some number of
times, say m,i, and negatively some number of times, say m^, all in any order, and then
leads in a straight line from 1 to z. For this path the value of the integral is equal to
'*
I ,
J i z
that is, to
where m is an integer, and in the last integral the variation of z is along a straight
line from 1 to z. Let the last integral be denoted by u; then
and therefore, inverting the function and denoting f~ l by <, we have
Hence the general integral is a function of z with an infinite number of values ; and z is a
periodic function of the integral, the period being 2jri.
* See Chrystal, ii, pp. 288 297, for the elementary properties of the function and its inverse,
when the variable is complex.
102.] OF INTEGRALS 219
h\i\ 2. Consider the function I 2 ; and again denote it by f(z).
The one-valued function to be integrated has two accidental singularities i, each of
the first order. The value of the integral taken positively along a small curve round i is
IT, and along a small curve round i is -TT.
We take the origin as the lower limit and any point z as the upper limit. Any path
from to z can be deformed, without crossing either of the singularities and therefore
without changing the value of the integral, into
(i) any numbers of positive (m lt w 2 ) and of negative (ra/, m 2 ') circuits round i and
round i, in any order, and
(ii) a straight line from to z.
Then we have
dz
f(z) m^ir + MI ( - TT) + m 2 ( - TT) + m 2 ' {(-)}+ I
r '
nir + ]o
1+Z*
dz
1+s 2
nn + u,
where n is an integer and the integral on the right-hand side is taken along a straight line
from to z.
Inverting the function and denoting f~ l by $, we have
Z = (
The integral, as before, is a function of z with an infinite number of values ; and z is a
periodic function of the integral, the period being TT.
Ex. 3. Denoting by 7 the value of the integral
dz,
['
tori J i
(z a)(z- b) (z c}
taken along a straight line from ZQ to z on which no one of the points a, b, c lies, find the
general value of the integral for a path from z that goes I times round a, m times round 6,
n times round c.
What is the form of the result, when a and b coincide ?
103. Before passing to the integrals of multiform functions, it is con-
venient to consider the method in which Hermite* discusses the multiplicity
in value of a definite integral of a uniform function.
Taking a simple case, let < (z) = | ^ ^
and introduce a new variable t such that Z = zt ; then
rl zdt
When the path of t is assigned, the integral is definite, finite and unique in
value for all points of the plane except for those for which "L +zt = Q; and,
according to the path of variation of t from to 1, there will be a 2-curve
which is a curve of discontinuity for the subject of integration. Suppose the
* Crelle, t. xci, (1881), pp. 62 77; Cours a la Faculte des Sciences, 4' me ed. (1891),
pp. 76 79, 154 164, and elsewhere.
220 HERMITE'S [103.
path of t to be the straight line from to 1 ; then the curve of discontinuity
is the axis of x between 1 and oc . In this curve let any point be
taken where f > 1 ; and consider a point z 1 = ^ + ie and a point z 2 = f ie,
respectively on the positive and the negative sides of the axis of x, both
being ultimately taken as infinitesimally near the point . Then
2ie " 2l ' e '
Let e become infinitesimal ; then, when t is infinite, we have
t
tan" 1 - - = i-Tr,
e
for e is positive ; and, when t is unity, we have
t
tan" 1 - - = i-TT,
e
for f is > 1. Hence <f> (zj) <f> (z 2 ) = 2-jri.
The part of the axis of a; from 1 to oo is therefore a line of discon-
tinuity in value of < (*), such that there is a sudden change in passing from
one edge of it to the other. If the plane be cut along this line so that
it cannot be crossed by the variable which may not pass out of the plane,
then the integral is everywhere finite and uniform in the modified surface.
If the plane be not cut along the line, it is evident that a single passage
across the line from one edge to the other makes a difference of ^iri in the
value, and consequently any number of passages across will give rise to the
multiplicity in value of the integral.
Such a line is called a section* by Hermite, after Riemann who, in a
different manner, introduces these lines of singularity into his method of
representing the variable on surfaces f.
When we take the general integral of a uniform function of Z and make
the substitution Z = zt, the integral that arises for consideration is of the form
We shall suppose that the path of variation of t is the axis of real quantities :
and the subject of integration will be taken to be a general function of t and
z, without special regard to its derivation from a uniform function of Z.
* Conpure ; see Crelle, t. xci, p. 62. t See Chapter XV.
103.} SECTIONS 221
It is easy, after the special example, to see that 3> is a continuous function
of z in any space that does not include a z-point which, for values of t included
within the range of integration, would satisfy the equation
G (t, z) = 0.
But in the vicinity of a 2-point, say corresponding to the value t = 6 in
the range of integration, there will be discontinuity in the subject of
integration and also, as will now be proved, in the value of the integral.
Let Z be the point I', and draw the curve through Z corresponding to
t = real constant ; let NI be a point on the positive side and N 2
a point on the negative side of this curve positively described,
both points being on the normal at Z ; and let ZN-^ = ZN 2 = e,
supposed small. Then for N^ we have
#, = e sin -dr. 11-1 = T? + e' cos ^Ir,
Fig. 24.
so that z 1 = +ie (cos-^r + tsini/r),
where -^ is the inclination of the tangent to the axis of real quantities. But,
if da be an arc of the curve at Z,
do~ , t \ d? df) dfc
dt dt dt dt
for variations along the tangent at Z, that is,
COS
Thus, since -T- may be taken as finite on the supposition that Z is an
Guv
ordinary point of the curve, we have
where
. P
Similarly 2 2 = + * e /y
Hence $> ( Zl} =
222 HERMITE'S [103.
with a similar expression for <I> (z 2 ) ; and therefore
f* 1
<*0- fo), fcj^e
The subject of integration is infinitesimal, except in the immediate vicinity
of t = 6 ; and there
.
powers of small quantities other than those retained being negligible. Let
the , limiting values of t, that need be retained, be denoted by 6 + v and
6 /j, ; then, after reduction, we have
Swftw'
in the limit when e is made infinitesimal.
Hence a line of discontinuity of the subject of integration is a section
for the integral ; and the preceding expression is the magnitude, by
numerical multiples of which the values of the integral differ*.
Ex. 1. Consider the integral
dZ
\+Z*
zdt
l+Z 2 t 2 '
Wtfl f\
We have
so that TT is the period for the above integral.
Ex. 2. Shew that the sections for the integral
t a sinz
where a is positive and less than 1, are the straight lines x (2k+l)7r, where k assumes all
integral values ; and that the period of the integral at any section at a distance 17 from the
axis of real quantities is 2*r cosh (arj). (Hermite.)
* The memoir and the Cours d' Analyse of Hermite should be consulted for further develop-
ments ; and, in reference to the integral treated above, Jordan, Cours d' Analyse, t. ii, pp. 293
296, may be consulted with advantage. See also, generally, for functions defined by definite
integrals, Goursat, Acta Math., t. ii, (1883), pp. 170, and ib., t. v, (1884), pp. 97120; and
Pochhammer, Math. Ann., t. xxxv, (1890), pp. 470 494, 495 526. Goursat also discusses
double integrals.
103.] SECTIONS 223
Ex. 3. Prove that the function defined by
has a logarithmic singularity at x=l and no other finite singularity. If the plane be
divided by a cut extending along the positive part of the real axis extending from 1 to oo ,
shew that in the divided plane the function defined by the above series and its con-
tinuations is one-valued, and that, at corresponding points on opposite sides of the cut, its
values differ by Znilogx. (Math. Trip., Part II., 1899.)
Ex. 4. Shew that the integral
o
where the real parts of ft and y ft are positive, has the part of the axis of real quantities
between 1 and +00 for a section.
Shew also that the integral
>-/:
where the real parts of ft and 1 a are positive, has the part of the axis of real quantities
between and 1 for a section : but that, in order to render $ (z) a uniform function "of z,
it is necessary to prevent the variable from crossing, not merely the section, but also the
part of .the axis of real quantities between 1 and +co . (Goursat.)
(The latter line is called a section of the second kind.)
Ex. 5. Discuss generally the effect of changing the path of t on a section of the
integral ; and, in particular, obtain the section for I - ^. when, after the substitution
Jo 1 + 4
Z=zt, the path of t is made a semi-circle on the line joining and 1 as diameter.
Ex. 6. Shew that, for the function f(z) defined by the definite integral
oo e 2nirzi , . Znirzi v+l / g~2ir
-co e Znnt + e -2n^ e -2,t~^rzi
where n is a positive integer and the integration is for real values of t, while z=x+iy, the
sections are the lines
*=0, 1, 2,...,
and that the increment of f(z) in crossing the section #=0 in the positive direction is z n .
(Appell.)
Note. It is manifestly impossible to discuss all the important bearings of theorems
and principles, which arise from time to time in our subject; we can do no more than
mention the subject of those definite integrals involving complex variables, which first
occur as solutions of the better-known linear differential equations of the second order.
Thus for the definite integral connected with the hypergeometric series, memoirs by
Jacobi* and Goursatt should be consulted; for the definite integral connected with
Bessel's functions, memoirs by HankelJ and Weber should be consulted; and Heine's
Handbuch der Kugelfunctionen for the definite integrals connected with Legendre's
functions.
* Crelle, t. Ivi, (1859), pp. 149 165 ; the memoir was not published until after his death,
t Sur Vequation differentielle lineaire qui admet pour integrale la serie hypergeometrique,
(These, Gauthier-Villars, Paris, 1881).
J Math. Ann., t. i, (1869), pp. 467501.
Math. Ann., t. xxxvii, (1890), pp. 404416.
224
EXAMPLES
[104.
104. We shall now consider integrals of multiform functions.
Ex. 1. To find th 
