Introduction To The Theory Of Linear

u< OU_1 60954 >[g

OSMAN?A UNIVERSITY LIBRARY

Call No. $ VI ' VrStffc*."* Accession No. ($ > 3 3,

Author ^-oiyio ^ g -

Title

l.hft ^turned orfor before the date last marked below.

INTRODUCTION TO THE

THEORY OF LINEAR
DIFFERENTIAL EQUATIONS

OXFORD UNIVERSITY PRESS

AMEN HOUSE, E.G. 4

LONDON EDINBURGH GLASGOW NEW YORK
TORONTO MELBOURNE CAPETOWN BOMBAY

CALCUTTA MADRAS

HUMPHREY MILFORD

PUBLISHER TO THE UNIVERSITY

INTRODUCTION TO THE

THEORY OF LINEAR

DIFFERENTIAL

EQUATIONS

BY
E. G. C. PODJLE

Fellow of New College, Oxford

OXFORD

AT THE CLARENDON PRESS
193d

Oui, 1'oeuvre sort plus belle

D'une forme au travail

Rebelle,

Vers, marbre, onyx, email

TH. GAUTIEB
iZmaux et Camdes

PRINTED IN GREAT BRITAIN

PREFACE

THE study of differential equations began with Newton and Leibnitz,
and most of the elementary methods of solution were discovered in
the course of the eighteenth century. Where a problem could not be
solved in finite terms, expansions in power-series were tentatively em-
ployed by Newton. But the theory was not placed on a satisfactory
logical basis until about a century ago, when Cauchy distinguished
between analytic and npn-analytic systems, and constructed rigorous
existence -theorems appropriate to each type.

Ordinary linear equations, with" which this book deals, have always
attracted particular attention by their comparative tractability and
their numerous practical applications. Extensive monographs have
been devoted to many separate branches of the theory, such as
spherical and cylindrical harmonics, expansions in series of ortho-
gonal functions, oscillation and comparison theorems, the Heaviside
calculus, polyhedral, elliptic modular and automorphic functions.
While some branches arose out of physical problems, others were
created by the progress of the theory of functions and of the theory
of groups. Many important ideas were first worked out in connexion
with the hypergeometric equation by Euler, Gauss, Kummer, Rie-
mann, or Schwarz, and were then generalized by Fuchs, Klein,
Poincare, and many other writers of the highest distinction.

The present Introduction is based on lectures to senior under-
graduates at Oxford, and is designed for students who have already
taken an elementary course of differential equations, but have not
yet specialized in one of the more advanced branches. It is not a
compendium of this vast subject (to which no single author could
do justice), but a selection of investigations of moderate length and
difficulty, illustrating those aspects of it which are most familiar
to myself. The first five chapters deal with properties common to
wide classes of equations, and the last five are devoted to a more
detailed examination of the hypergeometric equation, Laplace's
linear equation, and the equations of Lame and Mathieu. I have not
discussed systematically the equations of Legendre and Bessel, as
there are so many admirable accounts of them in English suitable
for students of every grade. On the other hand, I have thought it
well to devote a chapter to equations with constant coefficients. I find

vi PREFACE

that candidates in university examinations have great difficulty in
constructing the solution of such equations which takes assigned
initial values, even when they can write down the complete primitive.
A very slight sketch of Heaviside's method should enable them to
make short work of this problem, which is of great practical impor-
tance. Again, the theory of simultaneous equations with constant
coefficients gives an excellent opportunity of introducing in an easy
context the notion of invariant factors, which is of fundamental
importance in the Fuchsian theory.

The short bibliography and the footnotes serve both to acknow-
ledge my debt to the authorities and to guide the more ambitious
reader. Besides some of the great classical memoirs and the systematic
treatises of Forsyth, Heffter, and Schlesinger, the books from which I
have learnt most are Klein's lectures on the icosahedroii and on
the hypergeometric function, the masterly summaries of the general
theory in the works of Goursat, Jordan, and Picard, and the studies
of particular equations in Whittaker and Watson's Modern Analysis.
Those ^vho wish to learn more about existence-theorems should con-
sult the recent work of Kamke.

While I am solely responsible for the shortcomings of this book, I
gladly avail myself of this opportunity of expressing my profound
indebtedness to my former tutor, Mr. C. H. Thompson, whose lectures
at Queen's College first aroused my interest in differential equations,
and later to Professor A. E. H. Love, who inspired and directed my
first efforts at research. My interest in everything connected with
conformal representation and Schwarz's equation was greatly stimu-
lated by discussions with Mr. J. Hodgkinson. I have received
valuable references and information on particular points from
Mr. W. L. Ferrar, Dr. F. B. Pidduck, Professor G. Polya, Professor
G. N. Watson, and Professor E. T. Whittaker, not to mention many
others who have courteously sent me offprints of their papers. My
pupil, Mr. G. D. N, Worswick, Scholar of New College, has been of
great assistance to me in reading the proofs. Last but not least,
I desire to thank the Delegates of the Clarendon Press for accepting
the book, and the Staff for their unfailing skill and courtesy in
printing it. E Q Q p

CONTENTS

I. EXISTENCE THEOREMS. LINEARLY INDEPENDENT

SOLUTIONS

1. The Method of Successive Approximations . . .1

2. Solutions in Power-Series . . . . .4

3. Linearly Independent Solutions . . . .6

4. Wronskian Determinants . . . . .10

5. The General Linear Equation . . . .12

EXAMPLES I . . . . . .15

II. EQUATIONS WITH CONSTANT COEFFICIENTS

0. Heaviside's Solution of Cauchy's Problem . . .18

7. Operators in D and 8 . . . . .22

8. Simultaneous Equations. Invariant Factors . . 27

EXAMPLES II . . . . . . .30

III. SOME FORMAL INVESTIGATIONS

9. Linear Operators . . . . . .33

10. Adjoint Equations . . ... 36

11. Simultaneous Equations with Variable Coefficients . . 39

EXAMPLES 111 . . . . . . .41

IV. EQUATIONS WITH UNIFORM ANALYTIC COEFFICIENTS

12. Group of the Equation . . . . .45

13. Canonical Transformations . . . . .47

14. Hamburger Sots of Solutions . . . .53

15. Fuchs's Conditions for a Regular Singularity . . 55

EXAMPLES IV . . . . . .59

V. REGULAR SINGULARITIES

16. Formal Solutions in Power-Series . . . .62

17. Convergence . . . . . .65

18. Apparent Singularities . . . . .68

19. The Method of Frobenius . . . . .70

20. The Point at Infinity. Equations of the Fuchsian Class . 74

EXAMPLES V . . . . . .79

VI. THE HYPERGEOMETRIC EQUATION

21. RiemamVs P-Functioii . . . . .83

22. Kummer's Twenty -four Series . . . .87

23. Group of Riemann's Equation . . . .88

vhi CONTENTS

24. Recurrence Formulae. Hypergeometric Polynomials . 92

25. Quadratic and Cubic Transformations , . .97

26. Continuation of the Hypergoometric Series . . .100

27. Hypergeometric Integrals . . . . .104

EXAMPLES VI . . . . . . .109

VII. CONFORMAL REPRESENTATION

28. Schwarz's Problem . . . . . .118

29. The Reduced Curvilinear Triangle . . . .122

30. Symmetrical Continuation . . . . .125

31. Some Special Cases . . . . . .129

EXAMPLES VII . . . . . . .133

VIII. LAPLACE'S TRANSFORMATION

32. Laplace's Linear Equation . . . . .136

33. The Confluent Hyporgeometric Equation . . .139

34. Integral Representations of Kummer's Series . .143

35. Bessel's Equation . . . . . .147

EXAMPLES VIII . . . . . . .150

IX. tAM^'S EQUATION

36. Lame Functions . . . . .154

37. Introduction of Elliptic Functions . . . .159

38. Oscillation and Comparison Theorems . . .164

39. The General Equation of Integral Order . . .167

40. Equations of Picard's Type . . . .170
EXAMPLES IX ....... 175

X. MATHIEU'S EQUATION

41. Nature and Group of Mathieu's Equation . . .178

42. The Methods of Lindstedt and Hill . . . .182

43. Mathieu Functions . . . . . .187

44. The Methods of Lindemann and Stieltjes . . . 193
EXAMPLES X ....... 197

SHORT BIBLIOGRAPHY . . . . . .200

INDEX OF NAMES . . . . . . .201

EXISTENCE THEOREMS. LINEARLY INDEPENDENT
SOLUTIONS

1. The Method of Successive Approximations

Standard Forms. AN ordinary differential equation is said to be in
the canonical form when the highest derivative of the unknown
function is given by an explicit formula

j^ )= F(x;y,y\y(,...,y(-V). (1)

This is evidently equivalent to a system of n simultaneous equations
of the first order

('= 1.2 ..... n-i),

Dy n = F(x; yi ,y z ,...,y n ) (D = d/dx).

More generally, it may be shown that every well-determined system
of simultaneous differential equations is equivalent to a normal
canonical system

Dy t = Fifoy^,...^) (*' - l,2,...,w). (3)

Consider, for example, three equations O^ = (i 1,2,3), in
three unknowns (u, v, w) and their derivatives up to (u^\ v ( &, uW). In
a domain where the Jacobian

0(0^,*,) ( .

-^ ' w

the equations can be solved for the three highest derivatives and
written in the canonical form

and, on introducing auxiliary variables as in (1), we obtain a normal
canonical system of order (ot+p+y). If, however, J = 0, the three
relations connecting (u ( *\ v ( P\ w/^) are not algebraically independent,
and can be replaced by two equations involving the highest deriva-
tives, and one where they do not appear, say Q = 0. We then have
three possible cases.

(i) If D = 0, the system is indeterminate, the three given equations
being algebraically equivalent to not more than two.

(ii) If Q does not vanish identically, but does not involve any of

4064 B

2 EXISTENCE THEOREMS Chap. I, 1

the unknowns or their derivatives, the system is incompatible, or
else the problem is incorrectly formulated.

(iii) If D involves the unknowns and their derivatives of order
not higher than (u (OL ~ K \ v$~"\ trf?-**), suppose u (OL ~ K ^ is actually present.
Differentiating K times the relation O = 0, we may eliminate all
derivatives of u above u^"^ from the system, without introducing
any new derivatives above v& or w&\ The apparent order of the
system will thus be reduced by K, and the process may be repeated
if necessary until a well- determined system has been obtained.

Real Linear Systems. The most general normal canonical system
which is linear in the unknowns may be written in the standard form

(8 n ) Dy, = f ,,(*)%+/*(*) (' = ] > 2 ""> )

j=i

the coefficients (a^) and (/ 7 ) being any functions of a:.

We shall assume in the first instance that the independent variable
x is essentially real; the coefficients and the unknowns may be com-
plex; but, if they are, we ca"h separate real and imaginary parts and
obtairi a purely real system of order 2n-, accordingly we may assume
them to be real also, without essential loss of generality. The following
fundamental theorem is a particular case of Cauchy's general exis-
tence theorem for non-linear systems.

THEOREM. If the coefficients of the real linear system (S n ) are con-
tinuous in the finite interval (x f ^ x ^C x"), the system is identically
satisfied by functions with continuous derivatives

y< = &(*) (=l,2,...,w),
which are uniquely determined by the arbitrary initial values

&() = i?i (* = l,2,...,w)
at a point x = f of the interval.

The simplest proof of Cauchy's existence theorem for the most
general system is by Picard's method of successive approximations,
which is particularly suitable for linear systems. It is closely con-
nected with methods used by Liouville (1837) and Caque (1864), and
with the 'calculus of integral matrices' developed by Peano, Baker,
and Schlesinger.f

We take as a first approximation

$(*) = ,, (i=l,2,...,n), (6)

f L. Schlesinger, Vorlesungen uber lineare Differentialgleichungen (1908).

Chap. I, 1 LINEARLY INDEPENDENT SOLUTIONS

and construct further sets of functions according to the rule

We shall show that the required solution is

h(x) = lira $() ( = 1, 2,...,n).

k-K*>

To examine these limits, let us write them as infinite series

where

(8)

(9)

(10)

Since (77^) are constants, and since the coefficients of the system
(S n ) are continuous, we can choose positive numbers A and M such
that

(ii)

in the interval (x f ^ x ^ a;"). Now {(/^(a;)} are defined successively
as integrals of continuous functions, and are therefore themselves
continuous; and (10) and (11) give

\UV>(x)\ ^ MnA \x\ (x' ^ x ^ x").
Suppose that, for a certain positive integer k, we have

\U((x)\ ^ Mn k A k \x-\ k /kl
Then the recurrence formulae (10) give the inequalities

(12)

(13)

3-1

JU

J A.Mn k A k \t-\ k dt/k\

(14)

and since (13) is true for k = 1, it is true in general, by induction.
We now have uniformly in the entire interval

| U<f >(x) | < Mn k A k (x"-x') k /k ! ; (15)

and so the terms of the series (9) are dominated by those of a series
of positive constants M exip[nA(x"~ x')], which is convergent. Hence

4 EXISTENCE THEOREMS Chap. I, 1

the series {^(x)} converge uniformly in the interval and their sums
are continuous. Making k -> oo in (7), we have

= -m+ J [

.

By putting x , we verify the initial values <^-() = 77^; and,
because the integrands (16) are continuous, we may differentiate
and so verify that y i = ^(x) is a solution of the system (S n ).

Uniqueness. If there were two sets of functions satisfying all
the conditions, their differences <&i(x) ^ &(#)<*(#) (* 1? 2,...,n)
would satisfy the relations

.1 J

Since {^(a?)} are continuous, we can find an upper bound

|O,(aO|< (s'<z<aO. (18)

By the same argument as in (14) we can then prove by induction
that \Q>i(x)\ < Bn k A k \x-\ k lk\ < Bn k A k (x"-x') k /kl (19)

for all positive integers k. By making k -> oo we have |O^(^)| < e,
uniformly in the interval, where is an arbitrarily small constant.
Hence ^(x) = 0, or the two solutions are identical.

2. Solutions in Power -Series

Analytic Linear Systems. The system (S n ) is said to be analytic
if the coefficients (a^) and (f t ) are analytic functions of a real or
complex variable x, in the sense of Cauchy. A point x = g where
every coefficient is holomorphic is called an ordinary point; all other
points are called singular. At an ordinary point the coefficients can
be expanded as Taylor series:

which are convergent when \x \ < d($), where d() is the shortest
distance between x = and the nearest singular point.

The following existence theorem consists of a particular case of
Cauchy's existence theorem for analytic non-linear systems, together
with a rider by Fuchs specifying the radius of convergence of the
solution, when the system is linear.

Chap. I, 2 LINEARLY INDEPENDENT SOLUTIONS 5

THEOREM. At an ordinary point x = the analytic linear system
(S n ) is identically satisfied by n power-series

y f = &(*) = i<f >(*-)* (i = l,2,...,n), (2)

fc =

which are uniquely determined by the initial values <f>i() = -rj^ These
series are convergent when \x\ < d(), where d() is the shortest
distance between x and the nearest singular point.

To obtain a formal solution we substitute the power-series (2) in
the initial conditions and in the system ($J, and equate coefficients
of like powers of (x f) on both sides. The initial conditions give

^ cj> = % (i = l,2,...,n), (3)

and then we have

(*+l)cJ*+ = I I atf->cj+/[ (i = 1, 2,...,n; k = 0, 1, 2,...). (4)

^ = 1 s =o

It is evident, by induction with respect to k, that every coefficient
is uniquely determined.

Let x be any point where \x f | r < d() = d, and let
72 = i( r -\-d) < c?. The series (1) for the coefficients are absolutely
convergent when \x\ ^ E\ so that we can choose a positive
number M such that

I#W \ff>\R k <M. (5)

The relations (4) then yield the inequalities

(6)

If we put c (k) = 2 Icj^l, we obtain from (6) by adding

(Jfc+ \}&+v ^ nMR~ k \ 2 J8^>+ ll (7)

L s== o J

We shall clearly obtain a dominant power-series

gcF(x-tV ^c<\x-t)* <gCP>(x-W (8)

if we put (7<> = c<>, and then

(4+ 1)C< A + 1 > = nJf JB-*f 2 jR 8 C^+ ll. (9)

L-o *

But this gives

(10)

EXISTENCE THEOREMS Chap. I, 2

r r

and hence lim ^ = hm y^-ry^r = '

fc->oo C (lc) &->oo|_ (tf-f-l)./t J jft;

Hence the dominant series 2 C (k \xg) k and, a fortiori, the r& series
(2) are absolutely convergent if \x\ = r < R < d. The series
2 <7 (A:) (a; f )* is determined by the differential equation

DY = nJfi--+l), (12)

and the initial value Y = c (0) , when x . By separation of the
variables we have the solution

A~l~ w3fJB

and this explains the binomial recurrence formula (10).

Tins investigation shows that the solutions of a linear analytic
system are holomorphic at all finite points of the plane where the
coefficients are holomorphic. Accordingly the singularities of the
solutions are known a priori by inspection of the coefficients. This
fundamental property was pointed out by Fuchs.

/
3. Linearly Independent Solutions

Combination of Solutions. If y i = <f>i(x) an( l Vi = 4>t( x ) are an y
two solutions of the linear system ($), then y i <f>i(x) </>*(%) is a
solution of the corresponding homogeneous system

(SJ) Dy i =

We therefore require only one particular integral of (S n ), together
with the complete primitive of ($*).

If we have m solutions of the homogeneous system ($*)

yi = fajW (* == l,2,...,w; j/ - l,2,...,m), (1)

another solution is

m

< = 2>^(*) (=l,2,...,n), (2)

J = l

where (c^) are any constants. The solutions (1) are said to be linearly
connected if constants (not all zero) exist such that

m

.I<V<M*) = (=1,2,...,). (3)

THEOREM. The necessary and sufficient condition that m solutions
(1) of the homogeneous system (S*) should be linearly connected is that
the rank of the matrix (^(x)) should be less than m.

Chap. I, 3 LINEARLY INDEPENDENT SOLUTIONS 7

The condition is necessary; for, if the relations (3) hold, every
w-rowed determinant of the matrix vanishes. Suppose now that the
condition is satisfied and that the rank is k (k < w, k ^ n). We can
number the variables and solutions so that the determinant

A*H= |<M*)I^O (*J=l,2,...,i). (4)

Corresponding to any solution y i = <f> is (x) of the set, we can find a
unique set of multipliers (Uj 8 ) such that

k

<t>is( x ) =2, u js<l>ij( x ) (*' = l,2,...,n; s = l,2,...,m), (5)

j-i

these relations being compatible, because every determinant of
(fc-f 1) rows of the matrix vanishes. On introducing this solution in
($*), the terms multiplied by (Uj 8 ) cancel, leaving

-0 ( = 1, 2,..., n); (6)

and, because A^ ^ 0, we must have

DUj S = 0, Uj 8 = constant, (7)

so that the condition is sufficient.

At an ordinary point x we can construct n solutions, taking
arbitrary initial values such that the determinant A() l^() I 7^ 0,
and these solutions are necessarily linearly independent. Since k ^ n,
every other solution can be written in the form

Vi = J>^ w (*) (i - 1,2,. ..,n), (8)

where (GJ) are constants. Any such system of n linearly independent
solutions is called a fundamental system.

Jacobi's Determinant. The determinant A = |^#(#)| f an y n
solutions of the homogeneous system ($*) can be evaluated by means
of an auxiliary differential equation. In an interval or domain where
{a>ij(x)} are continuous we have

aA

8 EXISTENCE THEOREMS Chap. I, 3

If we introduce the Kronecker deltas

8 - 1, 80 - (i ^j\ (10)

and use the property that a determinant with two identical rows
vanishes, we may write

(11)

This is integrated by separating the variables and gives Jacobi's
result x

A(z) - A()exp[ V f a u (t) dt\ (12)

L<-i/ J

Hence, either A (a;) = 0, and the n solutions are linearly connected;
or else A(#) ^ in the entire interval of continuity, the exponential
factor being neither zero nor infinite. Thus the necessary and
sufficient condition for a fundamental system is that A(f ) 7^ at
any on^ point of the interval.

Method of Variation of Parameters. If a fundamental system
of solutions of the homogeneous system ($*) is known, the non-
homogeneous system (S n ) is soluble by quadratures. Since
A = \<f>ij(%)\ 7^ 0, a set of multipliers (u^) is uniquely determined
by the relations

^=.I%^W (<=1,2,...,). (13)

On substituting these expressions in (S n ) and cancelling terms multi-
plied by (Uj) 9 there remain the relations

/<(*) (i = l,2,...,n), (14)

j 1

which give uniquely

For the principal solution y t = ^(a;;^), determined by the condi-
tions y i = at x = , we must have also u^ = at x |; and
(15) accordingly gives

Chap. I, 3 LINEARLY INDEPENDENT SOLUTIONS
and (13) now becomes, with a change of suffixes,

y k = 0,(x; fl = 7 V f Jfi ^#<*)/<
,4liy A() 8ft,

We can write this in the form

(k= l,2,...,n), (18)

where #w(*;0 = T T,, *) (,* = M,-,*)- (19)

If t is regarded as a constant, y k = (j> ki (x\t) is a solution of the
homogeneous system ($*), taking at a; = t the values <f) k i(t\t) = 3 ki .
Since the determinant of these values is unity, the solutions
Mb = tfrkifa'ft) (i, k = 1,2,...,%) are a fundamental system, which
can be constructed directly from the initial conditions at x = t and
then introduced in (18).

The method can also be used to reduce by m the order of the
homogeneous system (8%), when m linearly independent solutions
are known. Let the variables be numbered so that the determinant
A m == \</>ij(x)\ ^ (i,j = 1, 2,...,ra), and let us put

These relations give (Uj) and (1^) uniquely. Substitutuig, and can-
celling the terms in (uj)> we have

a ik (x)Y k (t=l,2,...,m),
*-+!

n

= 2 a .(a;)y fc (i

j A:=m4-l

The first set gives, since A /n 7^ 0,

1 PA

(22)

and when these values are substituted in the second set, we get
a homogeneous system of order (nm) of the type

Z>J f = i A ik (x)Y k (i = m+ !,...,). (23)

10 EXISTENCE THEOREMS Chap. I, 3

If this system can be solved, (uj) can then be obtained from (22) by
quadratures.

4. Wronskian Determinants

Linearly Connected Functions. The functions {<f>t(x)} are said to
be linearly connected if constants (c t ) not all zero exist such that
^Ci<l>i(x) = 0. If there are n functions, which are differentiate
(nl) times, we have the relations

and on eliminating (c t ) we obtain the determinantal relation

= 0. (2)

The vanishing of this expression, which is called a Wronskian
determinant, is thus a necessary condition for the functions to be
linearly connected. The converse is true only with qualifications.
We suppose that W(<j>i, $&..., <f> n ) = 0, and that there is no sub-set
of (n 1) functions whose Wronskian vanishes identically. For, if
there were, we could reason on the sub-set. We can then prove the
following

THEOREM. // n functions {(/>i(x)} are differentiate (nl) times,
and if W(<f> v <f> 2 ,...,<l> n ) = 0, but TF^i^-^n-i) ^ in a certain
interval, then the functions are connected by a linear relation with
constant coefficients, which is valid in the entire interval.

We can determine uniquely multipliers (u t ) such that

^-tynfr) =*|V ^-W*) (3 = 1, 2,..., n). (3)

By combining the jth relation with the derivative of the (j l)th,
we have n _ 1

o O'=i.2,...,-i); (4)

and because ^F(^i,^2.->^i*-i) ^ ^^ we mus ^ have Du t 0, and so

n-l

<l>n( x ) = 2 c ^i(^) ( c i = constant). (5)

i = l

The relation can be extended to any interval where the derivatives
exist and where all the Wronskians of sub-sets of (nl) functions

Chap. I, 4 LINEARLY INDEPENDENT SOLUTIONS 11

nowhere vanish simultaneously. The necessity for the condition
was indicated by Peano, and may be simply illustrated. Consider
two functions with continuous derivatives defined as follows:

Then ^(x) = 2<f> 2 (x) (x > 0), but ^(x) = %(/> 2 (x) (x < 0). The
theorem is inapplicable to any interval containing x 0, because
<^ 1 (0) = = < 2 (0)> so that the subsidiary condition is not satisfied,
although W (<f> v </> 2 ) EZ 0. If, however, the functions are analytic, and
if 2 c i<t>i( x ) == in any finite interval, the relation holds everywhere,
by the principle of analytical continuation.

The General Linear Homogeneous Equation. If the n functions
(</>i(x)} are differentiable n times, and if W^,^ %,..., <f) n ) ^ in a
certain interval or domain, then the expression y = 2 c i <l>i( x )> where
(c^) are constants, must satisfy the relation

=

^ >

The form of this relation is unchanged if {(/>i(x)} are replaced by n
linearly independent combinations with constant coefficients

*<(*) = I cM*) (' = l > 2,...,n), (8)

j = l

where the determinant \c tj \ ^ 0. For, by the rule for multiplying
determinants, we have

and hence

,...,^ n ,y)

( '

Conversely, consider any homogeneous equation
(E*) Dy+ Pl (x)D-iy+...+p n (x)y = 0, (11)

and suppose that {<f>i(x)} are n known linearly independent solutions,
so that W((f) v (/>2f^t^ n ) 7^ in a certain interval.

Eliminating {p r (x)} from (11) and the equations satisfied by
{&(*)}, we have JF Wl ,^,...,^,y) = 0; (12)

and, since we assumed that T^(^i,^ 2 >---^n) ^ ^> Wronski's theorem

12 EXISTENCE THEOREMS Chap. I, 4

shows that the most general solution of (11) is of the form

n

y J Cifafe), where (c t ) are constants.

Every solution of (E%) is accordingly expressible in terms of any
set of n linearly independent solutions, which are called a funda-
mental system.

5. The General Linear Equation

The Complete Primitive. We shall now specialize some of the
preceding results for the particular system

%* = y*u (*'= 1,2,...,% 1), ]

} (1)

which is equivalent to the general linear equation
(E n ) D"y+ Pl (x)D-iy+...+p n (x)y - f(x).

If x is real and all the coefficients are continuous in the interval
(x r ^ x < x"), there is a solution with n continuous derivatives,
which, is uniquely determined by the arbitrary initial values
D l y T) (V > (i = 0, 1,..., n 1) at any point x = g of the interval. If
x is real or complex and the coefficients are analytic, the solution is
analytic and has no singularities in the finite part of the plane,
except at singularities of the coefficients.

The difference of any two solutions of (E n ), say y = (f>(x) <f>*(x),
satisfies the corresponding homogeneous equation

(E*) D"y+Pi(x)D n - 1 y+...+p n (x)y - o.

We therefore require only one particular integral of (E n ), together

with a complementary function which is the complete primitive of

The Abel-Liouville Formula. The analogue of Jacobi's deter-
minant for the system (1) is the Wronskian W(^i,^ z ,.-->^) of an Y
n solutions of the homogeneous equation (E*). By a double applica-
tion of the rule that determinants with two rows identical vanish,
we have

3W

Chap. I, 5 LINEARLY INDEPENDENT SOLUTIONS 13

= -P^W. (2)

Separating the variables and integrating, we have

0)

a result due to Abel for n 2 and to Liouville in general. In a finite
domain where PI(X) is continuous, J^(<i,< 2 > ><) vanishes either
identically or not at all. The necessary and sufficient condition that
n solutions {&(%)} should form a fundamental system of solutions of
(E*) is that their Wronskian should not vanish at one point x = ,
chosen at random in the domain.

Method of Variation of Parameters. To solve the non-homogeneous
equation (E tl ), knowing a fundamental system of solutions of (E*),
we put w

W- l y - I UtDI-ifaW (j - 1, 2,..., w). (4)

t -i

These relations give unique values for (?^), since W (</>!, < 2 v > <l>n) ^ 0-
By combining (4) and their derivatives we now find

= ^Du.D^^x) (j =- 1,2,..., 7i-l), (5)

1-1

and Dy^u t D^ t (x)+Du t D-^ i (x)]. (6)

When the expressions (4) and (6) for the unknown and its deriva-
tives are substituted in (E n ), terms multiplied by (u f ) cancel, leaving

From (5) and (7) we now get

If the principal solution at x = f , say ?/ = O(a;;|), is determined by
the conditions D^-^y = (j = 1, 2,...,n) when # = , we find that

14 EXISTENCE THEOREMS Chap. I, 5

Ui = (i = 1, 2,...,tt) when x = f . Hence

If /is a constant, the expression

is that solution of the homogeneous equation ($*), which is deter-
mined by the initial values

Dl-iy - 0, when x = t (j = 1, 2,...,n-l); j (12)

D n ~ l y = 1, when # . I

With this definition of </>(x; t) we obtain a formula of great impor-
tance for the principal solution, which is due to Cauchy, namely,

*(*;) = jt(x;t)f(t)dt. (13)

*

We can also apply the method of variation of parameters to reduce
a homogeneous equation (E%), when we know m independent solu-
tions (m < ri)] a simpler method will be given later, so that the
details will be left as an exercise. If TF^,^,...,^) ^ 0, we write

in the normal canonical system (1), where now /(a:) EZ- 0.
We have (n1) relations

m

2 DUfDl-VAz) = (j = 1, 2,...,m-l),

and also the relation

(16)

Chap. I, 5 LINEARLY INDEPENDENT SOLUTIONS 15

From (14), (16), and (E*) we find

n nm

2 Lu { D*-V 1 (x)+DY n + 2 Pr(*)Y n+1 -r = 0. (17)

i-1 r=l

On eliminating (Du^) between (15) and (17), we have a canonical
system of order (nm) in (Yj), which can be replaced by a single
homogeneous equation of order (nm) for Y m+1 ; while (14) gives

V - l"->n no\

*m+l W/JL JL " A \ ' * '

^ (<PV 92 > ,9m)

so that 1^+! (regarded as an expression in y) is the left-hand side of
the equation of order m admitting the given solutions.

EXAMPLES. I

1. Reduce to canonical systems the pairs of equations

[D 3 y+D*z+y = 1, D*yD*zz = 0].

2. The system

Dy = t\(x, y, z), D z y = JF 2 (x, y, z, Dy, Dz)

is in general of the second order. Show by examples that it may be (i) indeter-
minate, (n) incompatible, or (hi) of order lower than the second.

3. Show by the method of successive approximations that the equation
Dy = a(x}y is satisfied by

y -
and that the equation Dy = ct(x}y-\-f(x} is satisfied by

x x _

= f/(s)exp f a(t)dt Ids.

4. Solve from first principles the systems:
(i) Dy ~- xy; y = 1, when x = 0.

(li) Dy = ay/(l+x); y 1, when x ~ 0.
(iii) Dy = z, Dz = y; y A, z = B, when x 0.
(iv) Du ~ v, Dv = w, Dw = u; u = A, v = B, w = C, when a; 0.

5. If y { ~ c/>ij(x) is a set of m solutions (m < n) of the homogeneous system
), and if constants (c,) riot all zero exist such that

for a particular value , then 2 Cj</>q(x) ^ 0.

.7 = 1

6. If (^ w (a^)) is a matrix of n 2 arbitrary differentiate functions, whose
determinant does not vanish identically, then there is a unique homogeneous
system (S%) admitting the n solutions y i = ^^(x] (i,j l,2,...,n). If the
determinant vanishes identically, the system (S$) does not exist unless the

16 EXISTENCE THEOREMS Chap.I,Exs.

functions are connected by relations with constant coefficients
.2e,#(*)-0 (*= l>2,...,n).

If the rank of the matrix is k, any (k -hi) solutions must be connected by rela-
tions with constant coefficients. When these conditions are satisfied, the system
(S%) is indeterminate.

7. The homogeneous system ($*) cannot admit m linearly independent

re-
solutions y i = <y(#) an d also a solution y^ 2j^ v (aO, wnere ( u i) aro n t

3^1

constants.

8. If (y % ) is a solution of the homogeneous system (S%) and the dotormiiiant

n

\oLij(x)\ -7^ Q, then Y } s= 2 a y(^)2/ satisfy another homogeneous system of
t=i

?&
order n. The expression Y ^- 2 c^M^ with coefficients difTcrcntiable n times,

i-l

satisfies a linear homogeneous equation of order n or less.

9. SYSTEMS WITH CERTAIN KNOWN SOLUTIONS. If the homogeneous system
(S%) has (n 1) known solutions y^ = <j>y(x), which are linearly independent,
show that for any other solution

2/l 2/2' > #n

(f> n , ^ 21 , ..., ^ wl

i \a n (t}dt\
i=ii J

Complete the solution of the system by quadratures, using the method of
variation of parameters.

10. The homogeneous equation (E%) has (nl) known independent solu-
tions. Show that any other solution satisfies the equation of order (nl)

-- Ccxp[

which is soluble by quadratures.

11. If </>i(.r) is a known solution of the homogeneous equation (J*) show
that the equation can be simplified by d'Alembert's substitution y ^>^x]u\
and that it is reduced to an equation of order (nl) by Fuchs's substitution

y = &() J * dx -

12. REMOVAL or A TERM. If the equation (E*) is transformed by writing
y wexp -- I pi(x) dx\, the equation satisfied by u has no term in D n ~ l u.

If in (J^*) we take a now independent variable z = ^(^), determined by the
relation n(n l)i/j"(x)-{-2p 1 (x)i/j'(x) ~ 0, the new equation satisfied by y has
no term in d n ~ i y/dz n ~ 1 .

13. Every solution of the non -homogeneous equation (E n ) satisfies the
homogeneous equation of order (n+1)

*y+...+p' n y]-
-f(x)[D n y+p 1 D n - 1 y+...+p n y'\ = 0.

Chap.I,Exs. LINEABLY INDEPENDENT SOLtJTIOHS 17

14. Solve xD 2 y (2x-{-l)Dy-\-(x-{-l}y = 0, given that the quotient of two
particular integrals is x 2 .

15. Solve (x*+x*)D z y -\-xDy (x-\-l)*y = 0, given that the product of two
particular integrals is constant.

1 6. C AUCH Y 's FORMULA FOB A REPEATED INTEGRAL. Show that the principal
solution of D n y =f(x), determined by the conditions D l ~ l y = (i 1, 2,..., n)
when x , is x

Hence obtain Cauchy's form of the remainder in Taylor's theorem

17. If F Q (x) -~f(x), F n (x) - J F^t) dt, prove by integration by parts that

o
18. Give another proof of Cauchy's formula by moans of the relation

J dt, J cft a ... | f(t n ) dt n -= | d* n J ^ n _! ... ff(t n ) d^.

II

EQUATIONS WITH CONSTANT COEFFICIENTS

6. Heaviside's Solution of Cauchy's Problemf

Introduction. THE simplest class of linear differential equations are
those with constant coefficients, which can be solved with the aid
of elementary functions. But the systems occurring in mechanical
and electrical problems may be very complicated, and give scope for
labour-saving symbolical methods. If we require a particular solution
taking assigned initial values, the operational calculus of Heaviside
affords the most appropriate shorthand for the method of successive
approximations. If we require the complete primitive with the
arbitrary constants of integration displayed in the simplest form, but
without reference to any particular set of initial values, the most
effective method is the symbolical calculus of Boole.

Solution in Power-Series. The homogeneous system with constant
coefficients n

% (t=l,2,...,n) (1)

is analytic and free from singularities in the finite part of the plane.
In the neighbourhood of a typical point x the solutions can be
expanded in power-series, which will converge for all finite values of
x. Let us write these in the form

Then the solution taking the values y i = 7] t at x is given by the
relations n

cf> = n, #+ = i lcf>. (3)

Let (a$) be the kth power of the matrix (a^), and let (affi) denote the
unit matrix (8^), whose elements are Kronecker deltas. By the multi-
plication rule we have

f J. R. Carson, Electrical Circuit Theory and the Operational Calculus (New York,
1926); H. Jeffreys, Operational Methods in Mathematical Physics (Cambridge, 1927);
E. J. Berg, Recknung mit Operatoren (Munich and Berlin, 1932); P. Humbert, Le
calcul symbolique (Paris, 1934).

Chap. II, 6 EQUATIONS WITH CONSTANT COEFFICIENTS 19

Then the relations (3) give

(=l,2,...,n;t = 0,l,... > oo); (5)

and

m" s*>k

<

7-x /t-0

where y i = (f>a(x) is the solution with the initial values <^-(0) = 8^- at
the origin.

If every element of the matrix (a^) has the upper bound \a^\ ^ A,
then \of$\ ^ n k ~ l A k for positive integers k. Hence the series (6) are
absolutely convergent for all finite values ofx, and at least as rapidly
as $xp(nAx). To solve the non -homogeneous system

Dy i ^ fyji/j+f^x) (^ 1,2,..., w), (7)

we observe that y i = <f>^(x t) is the solution of the homogeneous
system (1) which takes the initial values y i = 8^- at any given point
x t. Applying the method of variation of parameters as in 3, we
see that x

y i. s I T"ii\^~~~ ) J i\ I V^ 1 ~~~~ > > > / \ /

7 =s 1 J

is the principal solution of (7), given by y i = when a; = 0.

Heaviside Operators. To confirm these results by the method of
successive approximations, we introduce the notation

f(x) = J/() *. (9)

"

By Cauchy's formula for a repeated integral, we have, for positive
integers m, , x ti

o
If f(x) ~ 1, we may omit the operand and write simply

- s .

P m ml

20 EQUATIONS WITH CONSTANT COEFFICIENTS Chap. II, 6

Proceeding exactly as in 1, we write the solution as a uniformly con-

00

vergent series y i 77 ^+ ]T Ui( x )> where

(12)

Since the operation of integration is commutative with that of multi-
plication by a constant, we find that (12) are satisfied by

and the solution is

Using the rules (10) and (11) to interpret these operators, wo find,
exactly as in (6) and (8),

^ I M^i+ I <t> l3 (x-t)fj(t) dt. (15)

*-i ^-i^

Summation of the Series. Consider the auxiliary series

which are certainly convergent if |A| > nA. By reason of (3), these
are found to satisfy the relations

X(w i rf i )=J f a ii w t (j= l,2,...,w), (17)

il

which may be written

2 (AS y ,-a^K = A % y - 1, 2,...,n). (18)

i=0

If A(A) = |AS^a^| is the determinant of this system of linear
algebraic equations and A^(A) the minor of the element [AS^ o^],
we have n

' =l..-..,). (10)

Chap. II, 6 EQUATIONS WITH CONSTANT COEFFICIENTS 21

We can now write (14) symbolically in the form

Since &(p) is of degree n and {A^(p)} are at most of degree (n 1), the
operators can be expanded formally in negative integral powers of
p and interpreted term by term.

Heaviside's Partial-Fraction Rule. The expansions of the operators
may be found by resolving {^j i (p)/j\(p)} into partial fractions, which
are interpreted as follows.

p ^ (m+r~ 1)1 of

(poc) m Z^(ml)\r\p m + r - 1

* x __. x _ e ocx. /2i)

X

rv _

__ _______ /VA dt

Jlv / rv r( T _ /\m+r-l

y ^__Jl _______

ZJ (m-l)!H

(x-tr-

(m-f)!
o

(22)

]f A^^!^) is the H.C.F. of all the first minors (A^(^)}, and if
&(p)/& n -i(p) is divisible by (p a) r , then the solution involves terms
of the types x s e* x (s = 0, 1, 2,...,r 1).

Non-Commutative Operators. The operator p, of which only nega-
tive integral powers have been defined, is not identical with D, nor
are the two operators commutative. For we have

22 EQUATIONS WITH CONSTANT COEFFICIENTS Chap. II, 6

ift") = =/(*) < w>m )

(24)

p

< 26)

= D m ~ n f(x) (m>n);

--^-^/""-"(O); (25)

Summary. The practical rule for solving the system (7) with given
initial values is first to integrate between the limits and x, and
write

(=l,2,...,n). (27)

This is a system of integral equations equivalent to the given differ-
ential equations plus the given initial conditions. We now solve (27)
formally} as linear equations in (yj), as though p were a number. This
gives the formulae (20), which are to be interpreted by Heaviside's
partial-fraction rule.

7. Operators in D and 8
Linear Operators. If we write

F(D)y ~ \ I a, />-*] y = | a,D-<y, (1)

H=0 J 1=0

where (a^) are constants, the expression F(D) is called a linear operator
with constant coefficients. Since the differential operator D is commu-
tative with constants, we have

F(D)G(D)y =

= G(D)F(D)y, by symmetry. (2)

Hence such operators are commutative with one another.

Chap. II, 7 EQUATIONS WITH CONSTANT COEFFICIENTS

Again, by Leibnitz's theorem, we have

23

= V l\

=* \ T '

> w y, (3)

and so, for any polynomial in D,

F(D)[e* x y] = e iX F(D+oi)y. (4)

Homogeneous Equations. The algebraic equation F(X) = is called
the cfiaracteristic equation of the differential equation F(D)y == 0. If
its roots are known, we may write F(D) EE a Q JJ(Z> AJ, with the
factors in any order. If A = A is a root of multiplicity %, we may
write the equation as

F(D)y =
and this is certainly satisfied if

or if

= 0,

(5)

(6)
(7)

A particular integral with n t arbitrary constants is therefore
y = e** x F* n t- l) (x), where P (n ^- l \x) is an arbitrary polynomial of degree
(Ti^1). Taking all the roots in turn, we have a solution with n arbi-
trary constants = y e A a: P^- 1) (a;). (8)

(0 l

We can prove that this is the complete primitive by evaluating the
Wronskian determinant of the n solutions corresponding to the n
constants; if this is not zero, the solutions must be linearly inde-
pendent. In the first place we have

11.1

which is an identity in (AJ as well as in x. This expression does not
vanish if (A t -) are all unequal.

If we differentiate with respect to A 2 , and then put A 2 = A 1} we have

XJ*. JJ (A r -AJ. (10)

24 EQUATIONS WITH CONSTANT COEFFICIENTS Chap. II, 7

If we now differentiate twice with respect to A 3 , and then put A 3 = A 1?
we have

and the Wronskian corresponding to any combination of equal roots
of the characteristic equation may be evaluated in the same way.

Non-Homogeneous Equations. To solve F(D)y f(x), we construct
the algebraic partial fraction identity

i nt n(r)

-:=y y -^ . (12)

EI/\\ / / (\ \ \r v

jP (A) 4J 4S (A A:)

(i) r=l

This gives an identity between operator polynomials

or , l-=2Jl(fl)0,(D) say, (13)

(0

where {jP t (Z))} are defined as in (5). We can accordingly write

/(*) = 2 F,(D)^(D)f(x) = I ^(^(a?) ; (14)

<i) (0

and the equation F(D)y ~ f(x) can now be satisfied by putting
y = ly t , where F(D)yi = Fi(D)fi(x)> (16)

or ^.(^[(^-Ai)"'^-/^*)] = 0- (16)

These relations are certainly satisfied if

and so the problem is reduced to the solution of 'simple equations'
of the type (17).

To solve (D\) k y = f(x), we write it as

D k [e~ Xx y] = e- Xx f(x). (18)

The solution is now given by k integrations, which may be reduced
to a single integration by Cauchy's formula, if we ignore constants
of integration. The latter yield only terms duplicated in the com-
plementary function. We write symbolically

~

Chap. II, 7 EQUATIONS WITH CONSTANT COEFFICIENTS 25

The practical rule derived from this discussion is to resolve formally
into partial fractions the inverse operator

M' (20)

and to interpret each symbolical fraction by the rule given in (19).

Equations Soluble without Quadratures. For many important appli-
cations the right-hand side is of the form

*^(s), (21)

where (/^-) may be complex, and where {tf*j(x)} are polynomials. It is
sufficient to solve the typical equation

F(D)y = e^(x). (22)

If ifj(x) is of degree (m 1), the right-hand side is annihilated by
(D~ ju,) w ; and so the solution of (22) is included in that of

(D-^F(D)y = 0. (23)

If we write down the complete primitive of (23), and then omit terms
annihilated by F(D), we find that there is a particular integral of the
form y e^xfe), where %(x) is a polynomial of degree (m 1), if
F(IJL) 7^ 0, or of degree (m+k 1), if A - ju, is a fc-tuple root of
F(\) - 0.

We could, of course, find x( x ) by ^ ne method of indeterminate
coefficients; but this is unnecessarily tedious. The required particular
integral is to satisfy both the equations

F(D)y -^ e^iff(x) 9 (D { j,) m + k y = 0. (24)

If F(D) = (Dp)*G(D), let us construct by the H.C.F. process the
identity between polynomials

A(D)Q(D) + B(D)(D-p,)>* ~ 1, (25)

where A(D) is of degree (ra 1) in />. If we operate with A(D) on the
first of the equations (24) and with B(D) on the second and add, we

find

, (26)

so that, if y = e^ x x(x), we get

D* x (x) = A(D+p,W(x). (27)

The operator A(D+fi) can be found without calculating B(D)\ for

26 EQUATIONS WITH CONSTANT COEFFICIENTS Chap. II, 7

the identity (25) corresponds to a relation between rational functions:

(28)

Hence A (p+h) is identical with the first m terms of the Taylor expan-
sion of l/6r(/x+/0 in ascending powers of h. Since A Q l/G(fJi) is
neither zero nor infinite, the expression

[A +A l D+AsD*+...+A m _iD*-*]t(x) = +*(x) (29)

is a polynomial of the same degree (m 1) as 0(#). The required poly-
nomial %(x) ~ D~ k i()*(x) may be written down by inspection, or as
the result of k successive integrations. The constants of integration
correspond to terms already accounted for in the complementary
function. The practical rule for solving (22), where if/(x) is a poly-
nomial, is to write

v = -m^^ x ^ ^ fi^^ x) ' (30)

and to ; expand the operator in ascending powers of D, omitting all
terms which annihilate /(#),

(*) = 0). (31)

Euler's Homogeneous Equation. The equation

a Q x n Dy+a l x- l D n - l y+...+a n y =/(*), (32)

where (a { ) are constants, can be reduced to an equation with constant
coefficients by putting x = ef. If 8 = d/dt = xd/dx, we can prove by
induction the well-known identity

x m D m y = 8(8 l)...(8w+l)y; (33)

and so (32) can be transformed into

b Q Z"y+b l $- l y+...+b n y =/(*). (34)

Corresponding to (4), we have the rule F^x^y) x^Ffi+aty.
If (8 c^-) 71 * is a factor of the operator with constant coefficients
jP(S), then a solution of F(8)y = is given by y = x^Pfr-VfLogx),
where Pfr~ l) (logx) is an arbitrary polynomial of degree (^1) in
logrr.f

t The reader should investigate directly the Wronskian determinant of the solu-
tions of F(8)y = 0.

Chap. II, 7 EQUATIONS WITH CONSTANT COEFFICIENTS 27

As an exercise in these operators, we observe that the equation
D^y __ ^j.) ma y a i so b e written in the form

8(8-l)...(8-n+l)y - **f(x). (35)

By resolving into partial fractions the expression

f(x) (36)

and interpreting the inverse operators, we obtain another proof of
Cauchy's formula for a repeated integral

X

j / / x _ s) 71 " 1

J(x) = ~ - T\r/(*) ds + a polynomial of degree (nl). (37)
JJ J (ft i)-

8. Simultaneous Equations. Invariant Factorsf

Characteristic Equation. Consider the homogeneous system

(=l,2,...,n), (1)

J 1

where (F^(D)) is a matrix of linear operators with constant coefficients
whose determinant A(Z>) = \F^(D)\ is of degree N. If we add the
results of operating on the equations with the minors of the elements
of the kill column of the determinant, we eliminate all the unknowns
except one, which is found to satisfy the equation

&(D)y k = (*=l,2,...,n). (2)

The corresponding algebraic equation A (A) = is called the charac-
teristic equation of the system. Suppose first that the roots (A,.) are
unequal. Then we must have

9* = Zc*r** (i=l,2,...,n), (3)

r=l

and, on substituting these expressions in (1), we get

f fi^A) c ;>l e ^ = (=l,2,...,n). (4)

r^lLj^i J

The coefficients of (e* rX ) must vanish separately, since these functions
are linearly independent; and so we have for each root n relations

I^Uc^O (i = l,2,...,n). (5)

3 = 1

These are compatible and determine uniquely the ratios (c lr : c 2r : . . . : c nr ) ;
for we have, on the one hand, A(A r ) = |jP^(A r )| = 0; while, on the

t C. Jordan, Coura d'anctlyse, 3, 175-9 ; M. Bdcher, Higher Algebra, 262-78 ; H. W.
Turnbull and A. C. Aitken, Canonical Matrices, 19-31, 176-8.

28 EQUATIONS WITH CONSTANT COEFFICIENTS Chap. II, 8

other hand, at least one first minor of the determinant is not zero.
For suppose that every first minor of A(A) were divisible by (A A,.);

0A(A)

then the reciprocal determinant

== A W - X (A) would be divisible

by (A A r )", and so A = A r could not be merely a simple root of
A(A) 0. Thus each root of the characteristic equation gives a
solution with one arbitrary constant. Solutions of the type y i c l e^ x ,
where the unknowns remain in a fixed ratio to one another, are called
in dynamics normal solutions.

The reader should now verify, by the method of 1, that the
system (1) is equivalent to a normal canonical system whose order is
equal to the degree of A(D). If the determinant vanishes identically,
the system may be replaced by a smaller number of equations and so
is indeterminate.

The method of indeterminate coefficients is equally applicable
when the characteristic equation has equal roots, but the discussion
of the linear equations corresponding to (5) becomes very laborious.
In certain cases the equation (2) can be simplified; for if A / ,_ 1 (/>) is
the H.O.P. of all the first minors of A(D) - \F 13 (D)\, we may divide
those minors by ^ n _ 1 (D) before we operate on (1), and so we have

r,~-0 (i=l,2,...,w). (6)

But the problem can be very much more clearly treated by
simplifying the given system (1). The process is a direct application
of H. J. S. Smith's canonical reduction of a matrix of polynomials.

Equivalent Systems. We employ two kinds of transformations,
both of which are reversible. We may replace two equations
((|> a = o O^) by an equivalent pair {O a +J^(/>)O^ = ^}, where
F(D] is any linear operator with constant coefficients, since either
pair implies the other. This process is called reduction by rows of the
matrix {F^ (D)}. Or we may replace two variables (y a , y^} by

since either pair can be written in terms of the other. This is called
reduction by columns.

Our object is to reduce the system to be solved to as few equations
as possible, and, failing this, to reduce to a minimum the degree of
the lowest operator in the matrix. If any operator is a constant other
than zero, the accompanying variable can be explicitly written in

Chap. II, 8 EQUATIONS WITH CONSTANT COEFFICIENTS 29

terms of the others and eliminated from the system. Every such
opportunity of elimination and reduction of the system is to be
seized. Suppose now that every operator actually involves D. We
pick out the operator of lowest degree, and look for another operator
in the same column not exactly divisible by it. For example, if
F U (D) is the lowest operator and F 2l ^ QF U + ^*i where F^(D) is of
lower degree than F n (D), but not identically zero, we replace O 2 =
by [^> 2 Q(D)^> 1 ] = 0; the new matrix will have at least one operator
of lower order than before. If the lowest operator is a factor of every
operator in its column, we make all of them identical and obtain a
system of the type

Al(D) yi + J o G l] (D)y j = (i = l,2,...,n), (7)

where no operator is of lower order than M(D).

Suppose now that one of the operators is not exactly divisible by
M(D), say G 12 (D) = M(D)Q(D) + Gf 2 (D), where *>(>) is of lower
degree than M(D) but not identically zero. Then the matrix can be
reduced by columns on taking [2/i+$(% 2 ] as a ncw variable instead
of y v If, however, every operator in (7) is divisible by M(D), we
eliminate y l from all equations except the first and write

- 0, \

I (8)

= ( = 2,3,...,n).

j-2

The first equation need not again be disturbed. The remaining set
may be reduced in the same manner as before, and we shall finally
obtain a system of the type

E l (D)z l - 0, E 2 (D)z 2 - 0, ..., E tt (D)z n - 0, (9)

where each operator E k (D] is divisible by the one preceding it. The
old variables (y^ can be expressed in terms of the new variables
(z t ) and vice versa. If k of the original unknowns have been eliminated,
the number of equations in the system (9) will be only (n k}\ or we
may suppose that E t (D) = 1 (i 1, 2,..., k), so that the first k
equations give the explicit formulae for the eliminated variables.

We have now only to write down the complete primitives of the
equations (9) for (z { ) and to introduce these expressions in the
formulae giving (y t ] in terms of (z t ). The method of indeterminate
coefficients is not required.

30 EQUATIONS WITH CONSTANT COEFFICIENTS Chap. II, 8

Invariant Factors. At each step of the reduction, we have merely
added to one row or column of the matrix (F^D)) a certain multiple
of another. This leaves invariant the determinant A(D) = |^(Z))|;
but much more than this is true.

Let A A .(Z)) be a polynomial in D (with the highest coefficient unity)
defined as the H.C.F. of all &-rowed determinants of the matrix
(F { j(D)). If A*(Z>) is the corresponding H.C.F. after an elementary
transformation, it is easily seen that every new A: -rowed determinant
is divisible by A fc (Z>); hence A*(Z>) is divisible by A^(/>). But the
process is reversible, so that A A .(Z>) is divisible by A(jD), and therefore
A(D) == A A .(Z)) (k = l,2,...,7i). But these invariants (A A .(/>)} can
be written down by inspection of the reduced canonical matrix

^(D) ... \

E(D) ... \

. (10)

... E n (D)l

Since ^ i+l (D) is divisible by E^(D], every Arrowed minor which

k

does not vanish identically is divisible by JJ E { (D). If the highest
coefficients in (E^D)} are reduced to unity, we have therefore

A fc (Z>) = E l (D)E 2 (D)...E k (D) = ^ k _^(D)E k (D). (11)

Hence {E k (D)} are likewise invariants of the original matrix; these
expressions (E k (D)} are called its invariant factors. They can be
found by rational operations, without solving the characteristic
equation A(A) = 0.

EXAMPLES. 11
1. HEAVISIDE OPEBATORS. Verify the formulae

P * P 2 np

____ = e inx t __* __ cosn.r, ----- 9 --

^fW - \cwn(x-t)f(t)dl,
n J

o

2. (i) The solution of F(D}y --- determined by the initial conditions
y = 0, Dy = 0, ..., D n ~ 2 y - 0, D^-^ - 1,

when x = 0, is y =

Chap.II,Exs. EQUATIONS WITH CONSTANT COEFFICIENTS 31

(ii) The solution of F(D)y = f(x) determined by the initial conditions
y = 0, Dy = 0, ..., D n ~ l y = 0,

when x = 0, is y -77

3. The solution of F(D)y determined by the initial conditions

.D^ -_- ^(), when x - (* = 0, l,...,n 1),

where P^)

4. Verify that

a; a;

= ( f(s)da (<b(x

where

and also that

5. Express as integrals

i _ _!

i . i

6. If l-t(p) is a rational function, whose denominator is of degree not lower
than the numerator, show that

Hence prove by induction that

<2n-l)!p _ _ sin2 n-i r

(? + l 2 )(p 2 + 3 a )...{^ + (2n-l)2}

(2n)!

___ ______ _ v _ __ _____ ___ _~ aiii-''.^;

(p* + 2*)(p* + 4*)...{p* + (2n)*} '
[H. V. Lowry, Phil. Mag. (7) 13 (1932), 1033-48, 1144-63.]

7. Verify Lowry's formulae by showing that y = sin 271 " 1 ^ satisfies the
equation

(I> 2 +l 2 )(^ 2 + 3 2 )...{X>H(2/i-l) 2 }2/ - 0,
with the conditions

D { y = (* = 0, l,...,2n-2), Z)^- 1 ^ = (2/i 1)!,
when = 0.

32 EQUATIONS WITH CONSTANT COEFFICIENTS Chap. II, Exs.

8. A sequence of Legendre trigonometric polynomials is defined by the
relations P (cosa:) = 1, P^cosa;) = cos#,

(n -f- 1 )P n+1 (cos x) ( 2n + 1 )eos a?P w (cos a?) -f nP n _ x (cos x) = 0.

Verify the formulae

p , . __ ^ 2 4
H-I* >

fB. van dor Pol, Phil. Mag. (7) 7 (1929), 1153-62; 8 (1929), 861-98.]
9. BOBEL'S RELATION. If tho Hoaviside operators (FJip)}, generating
ordinary power-series {f % (jc)} t are connected by the relation

then f^x) - J/ifc

10. Show that

? sin 2w r

fP aw {cos(a? 0}sin 2w -^<tt = -^ '. [LowRY.J

o 2n

11. CAUCHY'S METHOD. If F(z), G(z) are polynomials of degree n, m re-
spectivel^ (m < n), and if C is a contour enclosing all the zeros of F(z), show

[A. L. Cauchy, (Euvres (2) 6, 252-5, 7, 40-54, 255-66; C. Hermite,
Bulletin des sc. math. (2) 3 (1879), 311-25.]
12. Solve tho simultaneous equations
(1) D~u -- 2uvw

Ill

SOME FORMAL INVESTIGATIONS

9. Linear Operators

Operators with Variable Coefficients. IF we write

F(x,D)y = [l o Pr(x)D n - r ]y = J>r(z)-D' l - r 2A (1)

the expression F(x, D) is called an ordinary linear differential operator.
If {(f>i(x)} arc functions of x, differentiate sufficiently often, and (c^
are constants, we have

21 2 ' [ (^)

By combining such relations, we have for any operator F = F(x,D)
of the type (1) ^/ / , , ^ ^ p, _L^ )

^(Tc.^o -=" ycFct* 2 ' I (3)

But it is not in general true that ^(7<^ is the same as (rF(f>, for distinct
linear operators F, G.

Division and Factors. Operator polynomials in D have many
analogies with ordinary polynomials in a variable. Consider two

operators F _ po ( x )D +pl ( x )D-i+...+p(x),

(4)

G = - '

where m ^C n. Let us form the expression

fy-[A tl (x)D*-<+A 1 (x)D*^- 1 +...+A H _ m (x)]Gy, (5)

and let ns determine successively

(6)

so that the expression (5) shall have no term in (D n y t D H - l y,...,D m y).
We see that {A^(x)} are uniquely determined, and we have a relation

Fy - QGy+Ry, (7)

where Q = J ^ t D n - m ~ i . > and 7? is an operator of order not exceeding
(m 1). These are analogous to the ordinary quotient and remainder
of two polynomials.

4064

34 SOME FORMAL INVESTIGATIONS Chap. Ill, 9

If we know m (< n) linearly independent solutions of Fy = 0, let
the equation satisfied by them be

/ 8 \

l ;

If we construct the identity (7) and substitute for y the solutions
(0!, ^ 2 ,. ..,^J, we have

Rfa = B D-i<j> l +... + B m _ 1 <j> i == (i = l,2,...,m); (9)
and since TF(<i,^ 2 -"^w) 7^ 0, these give

.B, HZ (i = 0,1,..., w 1).
Hence the relation becomes

Fy = QGy, (10)

so that G is an tuner or right-hand factor of the operator F. We can
thus break up the equation into the pair

Qz^O, Gy = z. (11)

The first equation is of order (n m); and, when it is solved, the
second is soluble by quadratures, by the method of variation of
parameters.

Highest Common Factor. To discover whether two given equations
have any common solutions, we use a process of Brassine analogous
to the extraction of the H.C.F. of two polynomials. Let (F lt F 2 ) be
two given operators, of which the former is of the higher order; we
construct a sequence of operators (F t ) of steadily decreasing order
% > n 2 > n 3 > ... > n k , in accordance with the scheme

After a finite number of steps we must have F k+l ^E 0; and the last
operator which docs not vanish identically is an inner factor of all
the preceding ones, since

**-! = *-!**. J*- = (ft-2^-1+1)^... (13)

But, on the other hand, each of these operators can be written as
J; for we have

From the identity

(15)

Chap. Ill, 9 SOME FORMAL INVESTIGATIONS 35

we see that every solution common to (F t y =. 0, F 2 y 0) must also
satisfy F k y = 0. Hence F k is the highest common inner factor of

(V 2 ).
Consider now two non-homogeneous equations

*\y==Mx)> F*y=f*(*)-
This system is in all respects equivalent to the pair

^x) =f B (x), say, (17)

where jP 3 (FiQiF 2 ) as ^ n U^); for we can deduce either pair from
the other. Thus the original system can be replaced successively by
the pairs

f t y=f t (x), ^y =/!(*) ( = 2,3,...,i). (is)

But, since F k+l is identically zero, the last equation necessitates that
fk+i( x ) should be identically zero, for otherwise the equations are in-
compatible. This means that

f k+1 (x) - A- + i/i(*) + *+i/2(*) = 0, (19)

is a necessary condition for the compatibility of the system. The
condition is sufficient; for, in the kth equivalent pair (18), one equation
is merely = 0, so that the system is effectively equivalent to the
single condition I\y = f k (x). (20)

Least Common Multiple. Let the operators (F 19 F 2 ) have no common
inner factor; and let us expand ay linear homogeneous functions of
(D l y) the (n^n.y+2) expressions

(j-0,l,...,%). (21)

These (n l j rn 2 j r 2) linear forms, homogeneous in (^ 1 +^ 2 + ^) quantities
(y y Dy,..., D Hl \ ~ n *y), must be connected by an identity

My == f a t (x)DV\ y = f ^(x) ZW, y, (22)

-

-4=0

or My = ^F iy = %F t y, (23)

where (Tj^,^) are operators of order (n^n^ respectively.

The equation My = is satisfied by any solution of either F l y =
or F 2 y = 0; so that the operator M is analogous to the least common
multiple of two polynomials. The relation (23) is equivalent to the
one expressing that the last remainder in the H.C.F. process is

36 SOME FORMAL INVESTIGATIONS Chap. Ill, 9

Again, if the H.C.F. of (1\, F 2 ) is of order n k = v > 0, we have

\

where the operators (G l9 G 2 ) are of order (n l v,n* v). By the same
method, we now obtain an equation My 0, of order only (n l +n 2 v),
which is satisfied by every solution of either equation. Conversely,
if we have an identity (23), where (^, X K) are only of order (n 2 v,
n^v), the given equations have v common solutions. For let
{$i( x )} be T&I linearly independent solutions of F l y ~- 0. We have
then V 2 [^ a ^)] = (1=1,2,...,^), (25)

and so {F 2 <f>i(x)} are all solutions of an equation of order (n^v}. Since
not more than (n^v) of the expressions can be linearly independent,
we must have v relations of the type

*i&(*)= 2 cJW*) ('=l,2,...,i'), (20)

j- 1> M

where (c <; -) are constants. Hence the simultaneous equations
(Fy ~ t O = F 2 y) have v common solutions

-*/(*)- I c .^(-r) (i=l,2,...,v). (27)

J = V + 1

If we erase the last (i>+l) derivatives in each set (21), we have
(n l -\-n 2 ~2v) expressions linear and homogeneous in z ^ F k y and
its first (n 1 -{-n 2 ~ 2v~ 1) derivatives. If these expressions are not
linearly independent, we can construct an identity showing that
(G^z = ~ G%z) have a common solution other than zero, contrary
to our hypothesis that F k is the H.C.F. of the given operators.
We can therefore solve for z from the (n l -\-n 2 ^v) linear expressions,
and write z ^ Ak o^z+B k G 2 z, (28)

where the operators (A k , B k ) are of order not higher than (n 2 v-l,
n 1L -vl). This is equivalent to the identity (15), and indicates the
order of the operators concerned.

10. Adjoint Equations!

Equations of the First Order. We can readily solve the equation

Dy+p(x)y = f(x) (1)

by an appropriate use of the identity

D(yz) = z [Dy+p(x)y}+y[Dz-p(x)z\. (2)

f G. Darboux, Thtorie gtndrale des surfaces, 2, 112-34; M. B6cher, Lemons sur lev
mtthodes de Sturm, 22-42 ; F. B. Pidduck, Proc. lioyal Soc. A, 117 (1927), 201-8.

Chap. Ill, 10 SOME FORMAL INVESTIGATIONS 37

Let y be indeterminate, but let z be chosen so that

Dz = p(x)z, or z exp| j p(x) dx\ = eW say.
Then we have, on multiplying (1) by z,

D[^(^y] - e"Wf(x), (3)

and so e m(x) y = ( e mM f(x) dx + C. (4)

But, conversely, we may leave z indeterminate, and choose y, so that
Dy p(x)y, or y = e- m(x >. We can then solve any equation of the

form Dz~p(x)z = g(x) 9 (5)

by writing it as D[e- mM z] ~ e- OT(<r) </(#), ( 6 )

so that e~ m ^z = ( e~ w(x ^(x) dx +C'. (7)
There is complete reciprocity between the pair of equations

Dy+p(x)y = 0, Dz-p(x)s - 0, (8)

the solutions of either being integrating factors of the other. Such
equations are said to be mutually adjoint.

Adjoint Canonical Systems. The more general identity

D \ i>.*<] ~- i 4Dyt- i " ti (x)y i ]+ I y t \Dz t + % a^z,], (9)

L i-=l J 1 = 1 L j--l J i-l L 7 = 1 J

suggests a similar reciprocal relationship between the two homo-
geneous systems

If (^.) are indeterminate, but (s ) satisfy (2*), the first group of terms
on the right in (9) becomes an exact derivative; conversely, if (z t ) are
indeterminate, but (?/,-) satisfy ($*)> ^ ne second group becomes an
exact derivative.

Lagrange's Adjoint Equation. Consider the identity

- z[p Dy lt + Pl y n
n-l

(10)

38 SOME FORMAL INVESTIGATIONS Chap. Ill, 10

Let us write y i r=~ D l ~ l y (i = 1, 2,...,n) and also put

so that only y and z remain indeterminate. If we put

Fy ~ p*D' n y+p l D n -iy + ...+p n y, (12)

the operators F, F* are said to be adjoint to one another, and we
find the identity

zFy-yF*z = DY(y,z), (14)

where x F(y, 2;) is the bilinear concomitant

22)+- + (-) ll - 1 ^ B - 1 (Po2)]y- (IB)

The same method of interpretation shows that every solution of
F*z = is an integrating factor of Fy 0, and vice versa. To show
that F**, the operator constructed from F* by the same rule as F*
was formed from F, is identical with F, we need only write down the
identities

zf y-yl*z ^ ,,

which show that

z(Fy-F**y) = D[Y(y,2)+Y*(2,y)] (17)

is an exact derivative, identically in y and 2. But this is impossible
unless both sides vanish identically.

Composite Operators. If (F^F*) and (F 2 , F*) are pairs of adjoint
operators, we have

hence z (F 1+ F 2 )y-y(F*-{- F*)z = D[^ 1 (y,z)+^ 2 (y,z)]. (19)

Thus the operators (J^+J^, Ff +F$) are also adjoint to one another,
and similarly for the sum of any number of operators.

Again, write F s y instead of y in the first identity, and F*z instead

Chap. Til, 10 SOME FORMAL INVESTIGATIONS 39

of z in the second identity of (18); adding the results, we have the
identity

zF^y-yFSFfz == D[^(F 2 ij,z) + ^(y,FJz)l (20)

showing that the products (F^^F^Ff), with the factors reversed,
are adjoint operators; and similarly for any number of factors.

When two equations (F^y = 0, F 2 y = 0) admit v linearly inde-
pendent common integrating factors, their ad joints have v common
solutions ; hence they can be written

Ff z = G*H*z = 0, F*z : - G*H*z = 0, (21)

where H* is of order v. The original equations are therefore
(HG l y = 0, HG 2 y = 0). By a process parallel to Brassine's, it
would be possible to extract directly the highest outer common factor
of two operators.

11. Simultaneous Equations with Variable Coefficients

Reduction to a Diagonal System. The procedure of 8 can be
applied to the reduction of systems with variable coefficients, pro-
vided that we take account of the presence of non-commutative
operators. Consider the system

*< = i* T ,,.y;-- (t=l,2,...,w), (1)

j=i

where (F f j) arc operators of the type

F ~ ai(x)D"+a 1 (x)D"-*+...+a m (x), (2)

whose coefficients are difTerentiable as often as may be necessary in
the course of the reduction. Our object is to reduce the system to be
solved to as few equations as possible. If there is any operator F^
which does not involve D and is not identically zero, we can solve
<E> a = for yp and eliminate yp from the remaining equations. It will
be assumed that every opportunity of thus reducing the system is to
be taken.

If every operator present involves D, we pick out the one of lowest
degree, say F n , and look for any operator in the same column which
does not admit F n as an inner or right-hand factor. Suppose
F 21 ^^ QFn+F&y where F 2l is of lower degree than F n but not identi-
cally zero. Then on replacing O 2 by [O 2 QOx] = 0, the matrix
will be reduced by rows to one having an operator of lower degree in
D. If every operator in the first column is divisible on the right by
F 1V we look for any operator in the first row which does not admit

40 SOME FORMAL INVESTIGATIONS Chap. Ill, 11

F n as an outer or left-hand factor . Suppose F 12 = F n Q+J?* 2 , where
F* 2 is of lower degree than F u but not identically zero. Then on
taking [j/i+Qf/o] as a new variable instead of y ly the matrix will be
reduced by columns to one having an operator of lower degree. If an
operator of degree zero appears at any stage, we at once eliminate
a variable.

If the lowest operator is an inner factor of every operator in its
column and an outer factor of every operator in its row, the system
is of the type

(3)
= ( = 2, 3,..., ).

The first equation may be written as F n z l = 0; and we can eliminate
y l from the others by taking the combinations

<D.- t <D 1= :0 (i= 2, 3,. ..,T?,).

Proceeding with the reduction of the latter set, we ultimately obtain
a diagonal system

lZl =0, 2 * a = 0, ..., G n z n = 0, (4)

where the old variables (?/,) are expressible in terms of the new
variables (z f ) and vice versa.

Analogues of the Invariant Factors. The reduction (4) is not so
complete as that of a system with constant coefficients to the
canonical form. But it forms a convenient intermediate stage, after
which the equations may be examined two at a time. Any pair
(Q 1 z l = = G 2 z 2 ) can be further reduced, unless the lower operator
G is both an inner and an outer factor of G 2 , or unless

this need not imply that G 2 is of the form G^HG^ as is obvious by
considering operators with constant coefficients.

If G l is not an outer factor of G 2 , we can write

1 2 1 +G 2 2 2 -0, G 2 z 2 = 0; (5)

this system is reducible by columns.

If G-L is not an inner factor of (7 2 , we can write z 2 _ z^z*, and the
system Q ^ = ^ O^+O^ - 0, (6)

will be reducible by rows.

After a finite number of steps and eliminations we must arrive at

Chap. Ill, 11 SOME FORMAL INVESTIGATIONS 41

a system where every pair of operators fulfils the conditions. Arrang-
ing the operators in order of ascending degree, the system will be of
the type

^^ = 0, # 2 w 2 -0, ..., H n w n = 0, (7)

where H k+l == U^H k -= H k H^ In other words, the (+l)th
equation is satisfied by every solution of the kth, and admits every
integrating factor of the kth equation.

Generally speaking, two equations taken at random would have
no common solution. We could then construct the H.C.F. identity
[K^G-^K^G^ ^ 1, and the equation satisfied by any solution of
either equation, Lz = 0, where L =^ f\G l ^ P 2 G 2 . The general
solution of Lz = is z = Zi+z 2 > where G 1 z 1 = and G 2 z 2 0. But
we can also express z l and z 2 in terms of z, for we have

*i = (K^+KM^z, = KM.Zi - K.G^+z.) - K 2 G 2 z, (8)
and similarly z 2 K l G z.

Conclusion. In certain investigations there is a gain in symmetry
and clarity in considering a normal canonical system; and we have
seen in 1 how every equation or system can be reduced to that form.
But in studying the permutations of the solutions of an analytic
system, it is obviously simpler to consider a set of n analytic functions
(and as many derivatives as we please) than an array of n 2 functions,
the former being a fundamental system of solutions of a single
equation, and the latter that of a normal canonical system, both
of the same order n. Knowing how to replace any analytic system
by single equations, each involving one unknown, we need only
examine directly the properties of the solutions of a single typical
equation with analytic coefficients.

EXAMPLES. Ill
1. WRONSKIAN DETERMINANTS. If <f>,(x) = <7M

2. If A is any function of #, provo that

W x (X^\^...^ n ) -

Hence show that, if (j) l ^ 0,

42 SOME FORMAL INVESTIGATIONS Chap. Ill, Exu.

3. If m <n and Gy = D m y+qiD m ~ l y+- + q m y> show that the elements
of a column of W(<j> lt <f> 2 ,... ,<) may be replaced by

show that

W(4 v <f> t ,...,<l> H ,y) = W(<f> 1 ,<f> 2 ,...,<f> m )W(G<f> nH . v G<f> m+2 ,..., <ty w> Gy).
4. SYMBOLIC FACTOKS OF FROBENIUS. (i) By considering the minors of the
determinant W(^i^2> ^n>2/)> show that

(ii) If wj, -= ^i(:r), 7i r - - TFrTI^^i x deduce from the above that

_

. j - 71 n J

2-9> n n-1 2 1

(iii) Show that the equation

_L.olj>...!0 = o

<Xn+i n 2 a A

admits the solutions

a- jr a a-r-t

J ^ r () = i(- r ) | ^2(^2) ^2 J ^3(^3) ^3 J OrfaY) ^V-
(iv) Verify that W^,^,...,^) aJaS- l ...a /t ,

^ ,J

ar ^.i ~ L
5 - lf ^^^L^ID...!!)^

a r+1 a r a 2 atj
/ \r I i ^

JF*2 = ( } Z> -- D...D~,

a n r-fl a -r-f-2 a a n+l

show that F n y --- and F*z ~ are adjoint equations, and that their bi-
linear concomitant is w _j

m S) = 2 (F n -r-iy}(F* *). [DABBOUX.J

=

6. A self -adjoint equation of even order may be written

Vl V2 Yn Yn+l Y>n

and one of odd order

[FKOBENIUS; DAEBOUX.J

7. Verify that the following equations are self -ad joint (except as regards
sign) 1 j

(i) T.D j = 0, (ii) D n /D n y = 0,

Chap. Ill, Exs. SOME FORMAL INVESTIGATIONS 43

(iii) D n fD n+1 y+D n + l /D n y = 0, (iv) FyF*y = 0,

(v) FfF*y = 0, (vi) FfDfF*y = 0,

where (<,/) are functions of #, and (F,F*) are adjoint operators.

8. If *F(y,2) is the bilinear concomitant of a self-adjoint equation of order
n, Fy 0, then *(y,y) 0, if w is even, and Y(2/, 2/) = ZyFy, if ft is odd. By
writing (y+Az) for y, show that, if yFy is an exact derivative, n is odd and
Fy is self-adjoint. [DAiiuoux.J

9. If Fy -=P D 2n yi-p 1 E> 2n - 1 y-}-...-{-p 2n y = is self-adjoint, p^ -- nDp Q ,
and [Fy D n p D n y] is also self-adjoint. Hence show that the most
general self -ad joint equation of even order is of the form

D n *lj Q D n y+D n -^ 1 D n - l y-\-...+il, n y = 0,
and that of odd order is of the form

2 [D n - i i/f i D n - i + l y-}-D H - i + l i/j i D n -''y] = 0.
1=0 [JACOBI; DAKBOUX.]

10. If F - 2 KjcfiD^^and^* - 2 f?)^,^^-* are adjoint operators, show

i-O^/ '4-U^* 7

that _

11. If 2A = ^(^) (z.y = l,2,...,w) is a fundamental system of solutions
of a normal canonical homogeneous system, and A = |^ tj (^)! then a funda-

mental system of solutions of the adjoint system is z t = O i3 (x) p , and

A ^<pi;
there is complete reciprocity between the two systems.

12. If {<f> % (x)} is a fundamental system of solutions of a homogeneous equa-
tion of order n, and \V W(^ lt ^ 2 ,...,^ M ),thon0 t (.*;) - ~ - 7^-^ are integrating
factors. Verify the relations

2 OMD'hW ----- 0' = 0, l,...,n-2), 2 OMD^-^x) -= 1,

i=l i=l

and by means of these and their derivatives prove that

and hence that (O^x)} are linearly independent.

13. Show how the order of a system or of an equation can be depressed by
w, when m independent solutions of its adjoint are known.

71

14. In a self-adjoint normal system, ^^(x^-l-aj^x) - 0, and 2 2/i == con '

i= I

stant, for all solutions.

15. Solve by means of an integrating factor

x(l~x z )D 2 y-\-(2-5x*)Dy-4xy - 0.
10. If /3,A are constants, show that the equation

A0(a:)]y - 0,

44 SOME FORMAL INVESTIGATIONS Chap. Ill, Exs.

with the initial conditions y = A, Dy -~ B, when x = 0, is equivalent to the
integral equation x

j-i )i r

y = Acospx-\- sin/xE-}- - I smp(xt)i/j(t)y(t)dt.

P r J

lf\ifj(x) is continuous and small compared with p in the interval (0 < x < X),
show that the equation can bo satisfied by a series

?/ = U (x)+Xu 1 (x)i-\ 2 U 2 (x) J r ... . [LlOUVILLE.]

17. APPELL'S THEOREM. If {^(x)} is a fundamental system of solutions of
a homogeneous equation, any polynomial in (D^^x)} which is merely multi-
plied by a constant, when {^(x)} is replaced by any other fundamental system,
is of the form [Wfy^^ %,..., <f> n )] k P, where P is a function of the coefficients of
the equation and their derivatives.

[E. Picard, Traite d'analyse, 3, 541; L. Schlesinger, Handbuch, 1, 40.]

IV
EQUATIONS WITH UNIFORM ANALYTIC COEFFICIENTS

12. Group of the Equation

Analytical Continuation. Let us now suppose that the coefficients
of the equation

(E*) D-y+p^(x}D^y+...+p n (x)y -

are uniform analytic functions, having only isolated singularities in
the complex plane. By 2, every solution y = <f>(x) is analytic, and
its only singularities in the finite part of the plane are at singularities
of the coefficients; but <f>(x) is not in general single-valued.

To examine this question, we draw any closed circuit F of finite
length, beginning and ending at an ordinary point x , and not
passing through any singularity. We can apply the process of ana-
lytical continuation simultaneously to the coefficients and to the
solution </>(x), which will continue to satisfy (-"*) identically. For if
we expand the solution and the coefficients in Taylor series of powers
of (x ), convergent in a circle C, and if x = ' is any point in C, we
can rewrite <f>(x) and [pi(x)} as Taylor series in powers of (x '), con-
vergent in a circle C'. In the region common to the circles (C, C')
the two sets of expansions take the same values at every point, so
that the second set of power-series satisfy (E*) identically; and they
will continue to satisfy the equation in the entire circle of convergence
C", which will in general extend beyond C.

After completing the circuit, the coefficients [Pi(x)} resume their
original forms as power-series in (x ), being by hypothesis uniform
functions; but c/)(x) need not resume its original form, though it will
be some solution, <D(#) say, of the equation. To trace these changes,
the process of analytical continuation must be simultaneously applied
to all the solutions of a fundamental system {<f>t(x)}, which assume
the new forms

*<(*) = iW*) (=l,2,...,n), (1)

J = l

where (a^) are constants.

Determinant of the Transformation. The determinant A = \a i} \
cannot vanish; for this would imply a linear relation ^ c tt( x ) =
between the new forms; and, on returning along the same path, we

46 EQUATIONS WITH UNIFORM Chap. IV, 12

should have 2 C ;^( X ) > contrary to the hypothesis that {^(#)}
are linearly independent. The determinant A was evaluated by
Poincare. We obtain from (1) by differentiation

. (' = 1, 2,..., n), (2)

so that the derivatives of the set {<^(#)} undergo cogredient transfor-
mations. By the multiplication of determinants, we now have

W(^ v ^...^ n ) = AW(^^...^J. (3)

But Liouville's formula gives

W(x) - fF(< l3 >,..., ) - W)exp|~- j Pl (t) dt\ (4)

and after a complete circuit W(x) is multiplied by

A expj jpi(#)e?#|, (5)

the contour integral being taken once round F. If Pi(x) is holo-
morphic inside the contour, we have A = 1 .

Group of the Equation. Let 1^ and F 2 be two closed circuits start-
ing from )the same base-point x = ; and let (a^) and (by) be the
matrices of their respective transformations, for the same initial set
of functions {^(x)}. If the functions are continued around the com-
bined circuit in the order P x !!>, the matrix of the new transformation
is the product (c,-), where

n

C tj r -~- 2 a ik b kj (*,j = 1, 2,...,7i). (6)

fc=i

This product of matrices is not commutative, for we must take the
rows of the matrix belonging to the first circuit with the columns
of that of the second. All the linear transformations belonging to
every possible closed circuit form a group, called the group of the
differential equation.

If there are m winding points in the finite part of the plane, every
closed circuit can be deformed without crossing a singularity into a
sequence of standard loops, each starting from the base-point and
encircling one singularity. The m transformations belonging to these
loops are a set of generating operations of the group, in terms of
which every matrix can be expressed.

Riemann's Converse Theorems. A set of n linearly independent
analytic functions with isolated singularities, which admit a group
of linear transformations with constant coefficients for all closed

Chap. IV, 12 ANALYTIC COEFFICIENTS 47

circuits, must be a fundamental system of solutions of a linear
differential equation with uniform coefficients. For by virtue of (1)
and (2), every ?i-rowed determinant of the array
fa, Dfa, ..., Dfa

fa, Dfa, ..., D-fa
is multiplied by the same constant A ^ 0, after continuation about
a closed circuit. Hence quotients of these determinants are uniform
functions of x. But in the equation formally satisfied by the given

W(fa,fa,...,fa,y)

- ~

functions

every coefficient is a quotient of this kind; so that the coefficients of
the equation (8) are in fact uniform.

Every set of functions {^(#)} which undergo cogredient transforma-
tions with the set {^(x)} for all closed circuits can be written in the
form

where (u t ) are uniform analytic functions. For since we assume that
{(/>i(x)} are linearly independent, we have

except at isolated points. Hence (u t ) are uniquely determined by the
equations (9), each of them being the quotient of two ?i-rowed deter-
minants of the array

fa, Dfa, ..., D'^fa, v
,, Dfa, ..., D'^fa, v- a (u)

p Dfa, ..., D'^fa, fa
By hypothesis, the elements of every column of this array are co-
gredient; and so each n-rowed determinant is multiplied by the same
factor A ^ 0, corresponding to a given circuit; therefore the quotients
(u t ) are uniform analytic functions.

13. Canonical Transformations

Characteristic Determinant. With a view to simplifying the trans-
formation n

A i ij j

48 EQUATIONS WITH UNIFORM Chap. IV, 13

corresponding to any selected circuit F, we may look for solutions

V = Cl<t>l(x) + C 2<i>2( x ) + ~' + Cn<t>n( X )> ( 2 )

which are merely multiplied by a constant. Since we are to have

ff ***<(*)] = A[2c, &(*)], (3)

-i~l J L i=l J

we find from (1) and (3) the identity

[1 i<6(*)]-A[ic y 0,(*)l; (4)

L l=lj-=l J L j=l J

and, because {$,(#)} are linearly independent, this implies that

Ao; (j=l,2,...,). (5)

If this system of equations for (GJ) is compatible, A must satisfy the
equation

A,, (A) =

(6)

which is called the characteristic equation of the matrix (a^).

Evidently no root can be zero; for we saw in 12 that the deter-
minant A \a t j\ 7^ in the present problem. Each root gives at
least one linear form (2), and k such forms (y\,y^"^yk) belonging
respectively to unequal roots (Xi,X 2 ,...,X k ) are linearly independent.
For suppose they were connected by a linear relation

(8)

(9)

After s complete circuits this relation is transformed into

CiA 8 1 y 1 +c 2 A|y 2 +-..+c A: A^ fc == ( = 1,2,...,*-!).
But since (X t ) arc unequal, we have

1, 1, -., 1

A 2 ,

k-l \k-l
\! , /\ 2 , ...,

A,

so that the relations are incompatible and so the coefficients (c t ) are
all zero.

Accordingly, when the characteristic equation has n distinct roots

Chap. IV, 13 ANALYTIC COEFFICIENTS 40

we get n linearly independent combinations of {<^(#)}, forming
a new fundamental system, which undergoes the canonical trans-

formation

r< = A,y, (t=l,2,...,n). (10)

Invariants. The numbers (A t -) must be independent of the choice
of the original fundamental system {<^(#)}. For a canonical solution
y.f has the same multiplier A,,-, whether it is expressed in terms of
{(f>i(x)} or of some other fundamental system {&(%)}, which under-
goes the transformation

TO = iW) (i=l,2,...,n). (11)

J-l

Hence A ; - must be a root of the characteristic equation of the matrix
(b ij ) also.

To prove this algebraically, let us suppose that the two systems
are connected by the relations

h(*)=2<>ij<l>,W (=l,2,...,w), (12)

2 = 1

whose determinant C = \c^\ ^ 0. In terms of the first system,
(11) may be written

j=*l j = l fc-1

and by (1) we have

s-*<iM*) ^ i

j=l

These n relations between the independent functions {<f> k (x)} must
be identically satisfied, and so

a jk = J,b {j c jk (i, fc = 1 , 2, . . . , w) . (15)

Thus the matrices (a^-), (6^.), (c^) are connected by the relations

Let us write the two characteristic determinants in the forms

ll = I-ASI, (17)

\ Vii \ = |6y-A8y|, (18)

50 EQUATIONS WITH UNIFORM Chap. IV, 13

with the usual notation for the Kronecker deltas. Using (15), we
write n

tf

j = l

(19)

We thus have () = (c y )( ) = ( )(c w ). (20)

Now the determinant of the product of two matrices is equal to the
product of their determinants; so we have

Kl = |c| - || = \v,j\ . \c t} \, where \c tj \ J= 0; (21)

hence | -A8 W | = A n (A) = |6 -A8 W |. (22)

Again, each s -rowed minor of |^.| is a linear homogeneous function

of the s-rowed minors of \u^\, or of \v^\, and vice versa. For instance,

(23)

1 n ^ n

and the method of proof is general. The converse follows from the
relations

where (c^-) is the inverse matrix. The elements of a row in (c^) are
proportional to the first minors of those of a column in (c^); and we
have the relations

c = 7^ F~> where C ^ l c 'v

(25)

From the relations (23), the H.C.F. of all s-rowed minors of
A rt (A) = \Uy\ divides every 5-rowed minor of \Wy\\ hence the H.C.F.

Chap. IV, 13 ANALYTIC COEFFICIENTS 51

of all s-rowed minors of \w^\ is also divisible by that of all s-rowed
minors of \u tj \. But, from the converse relations (24), we see that
the H.C.F. of all s-rowed minors of \u^\ is divisible by the H.C.F. of
all s-rowed minors of \w^\. Hence the two expressions are identical.
But a similar connexion holds between \v {j \ and \w^\. If therefore
A S (A) is the H.C.F. of all s-rowed minors of the characteristic deter-
minant AJA), it has the same form, whether it is calculated from
|a# A8y| or from \b^ A8 y |.

It is obvious that A S (A) must be divisible by A S _ 1 (A), the corre-
sponding H.C.F. of the (s l)-rowed minors. We have already met
with the rational invariant factors

E S (X) = A.W/A^A) (s = 1, 2,..., n), (26)

in connexion with systems of simultaneous equations with constant
coefficients. These can be found by elementary algebraic operations
without solving the characteristic equation. If, however, the roots
of A^(A) are known, and we write

^(A) = (A-A,)(A-A;)4.., (27)

where (A s ,Ag,...) are unequal, the irrational invariants {(A X s ) e *} are
called elementary divisors of A 7i (A).

Following a less complete reduction by Jacobi, Jordan reduced
the most general linear transformation to a canonical form, which
is the analogue of Weierstrass's reduction of a pair of quadratic forms.
The canonical variables are divided into sets ; each set is transformed
independently of the others, and corresponds to one elementary
divisor of A, ? (A).

Jordan's Canonical Form. Every linear transformation (whose
determinant is not zero) is equivalent to one or more mutually inde-
pendent ones of the type

(28)

where A t is a root of the characteristic equation and (A A x ) a an
elementary divisor.

If there is only one variable, we must have Y l = \ l y 1 as the only
possible type of transformation, and the theorem is true. Assuming
its truth for n variables, we will prove it for (n-\-l). Consider the
transformation

(29)

52

EQUATIONS WITH UNIFORM

Chap. IV, 13

and let A' be any root of A 7l (A) 0. We can construct a linear form
z == 2 c ;2/i> suc h that ^ = A'z. Without loss of generality, let us
suppose that c n+l ^ 0. We may then eliminate y n+l and obtain an
equivalent transformation in the independent variables (z, y^ y r 2 ,..., y n )

Z = A'z, \

(30)

If we ignore z and apply the theorem, assumed true for n variables,
to the (i/i), we obtain a number of sets of the type

(31)

Now let
so that

(i = 1,2,. ..,<*),

(32)

(33)

(i) If A 7 7^ A 15 we can choose (^jj^g,...,^^ in turn so that z shall
disappear from these formulae (33). The set (U^) is then of the
canonical type.

(ii) If A' A 1? A a does not appear in (33); but since, of course,
A r z 0, we can choose (A l) A 2 ,...,A OL _ l ) so that z appears only once, in
the first equation of the set, whose form is now

U' 2 =

(iii) If, by chance, % 0, the set (34) is canonical and no further
reduction is needed.

(iv) If there is just one set with the multiplier A' and a v ^ 0, we
put a^z = A'^o, and adjoin u' Q to the set, which is now canonical in
(a+1) variables

[), ..., Uy = A'^^+^a). (35)

Chap. IV, 13 ANALYTIC COEFFICIENTS 53

(v) If there are several sets with the multiplier A', where the
coefficient corresponding to a is not zero, we select the one with the
greatest number of variables and reduce it to the canonical form (35).
If ft ^ a and another such set is (after eliminating z) of the form

Fi = A'(l+6 1 tii). F; = AX+i), ..., F = A'(/J-i+i8). (36)
we have only to write

vl-vl-b^ (=1,2,...,/S) (37)

to obtain the canonical set of ]8 variables

F; = AX, F; = A'K+tS), .... ^ = A'(^_ X +^). (38)

We have thus completed the reduction for (n+1) variables, and the
theorem follows by induction.

14. Hamburger Sets of Solutions

Euler's Homogeneous Equation. The interchange of solutions
around a circuit is well illustrated by Euler's equation

= o (S = *z>), (i)

which has an isolated singularity at x 0, and which is soluble by
elementary methods. If all the (p r ) are unequal, a fundamental system
of solutions is given by y r = xP r (r = 1, 2,...,n). After a simple
positive circuit about the origin, these undergo the canonical trans-
formation Y r ^\ r y r {\ r = exp(2^ r )}. (2)

IfF(8) has a multiple linear factor, (8p^ say, the corresponding
solutions are known to be

x? 1 , x? 1 log x,..., xP^log x) 71 !- 1 . (3)

These solutions are linearly transformed among themselves, but the
transformation is not of Jordan's canonical type. But we can easily
write down an equivalent system which does undergo a canonical
transformation. For this purpose let us introduce the notations

r ... , / _L T _L(L-\)

L = l ' L I = i? ' L * ^ 2 ' -' L

After the circuit, L becomes (L-\- 1) and the polynomials L s , whose
form is suggested by the calculus of finite differences, become

(L+l).= i.-i+^. (6)

as is immediately verified. Since (L , L lt ..., L s ) arc respectively of

54 EQUATIONS WITH UNIFORM Chap. IV, 14

degree (0, !,...,$) in L, they are linearly independent; and the set of
solutions (3) may be replaced by the equivalent set

y s = xP*L s ^ (8=1,2,...,^). (6)

These undergo the canonical Jordan transformation

Such a set of solutions is called a Hamburger set.

Solutions at an Isolated Singularity. Let x = be a typical isolated
singular point of a linear differential equation, whose coefficients are
uniform analytic functions; and let P be a simple positive circuit
enclosing only this singular point. We shall suppose that the corre-
sponding linear transformation of the solutions has been reduced
to the canonical form, and that one of the canonical sets undergoes
the transformation (7).

Let p 1 be any finite root of the equation exp(2^7rp 1 ) A t , and let
us put y r = xP l z r (r = 1, 2,...,7i 1 ). Then the corresponding trans-
formation of the (z r ) takes the form

Z l = z 1 , Z 2 = z l +z. 2 , ..., Z tti '--= z tli ._ l +z fh . (8)

Now we ''can write down n v sets of functions which are transformed
in this manner, namely

L Q , L v L 2 , ..., Zf /?i _ 1

0, 0, 0, ..., LQ.

Since the determinant of this system is L% 1 1, we can find n L
functions (iv r ) satisfying the equations

z =

If we substitute these forms in (8) and use (5), we find the relations

which are equivalent to

(H)

(12)

Chap. IV, 14 ANALYTIC COEFFICIENTS 55

and imply that (w r ) are single-valued in the neighbourhood of x = 0.
They can be expanded as Laurent series

w r =II r (x)= | c rs x* ^=1,2,...,^) (13)

K~ 00

converging in a ring (0 < e < \x\ < R), whose inner radius is as
small as we please. f The solutions of a canonical Hamburger set can
therefore always be expressed in the form

y r = x*. fL ni _ s I1 s (x) (r - l,2,..., Wl ), (14)

s=l

where (II s (a;)} ai'e locally uniform near the isolated singular point
x -- 0.

Regular Solutions. It may happen that each of the Laurent series
has only a finite number of negative powers of .r, which can be re-
moved by an adjustment of the value chosen for p^ the auxiliary
functions {ll r (#)} will then be holomorphic. A solution of this type is
said to be regular, and a singular point where all the solutions are of
this type is called a regular singularity.

15. Fuchs's Conditions for a Regular Singularity

THEOREM. The necessary and sufficient conditions that an isolated
singularity, say x 0, of the, homogeneous equation (E*) should be
regular are that it should be at most a pole of order r of the coefficient
p r (x) of J) n ~ r y, in the canonical form where the highest coefficient is
Po (x) = 1.

Following Thome, we proceed by induction. If the equation of the
first order

\ * / 1 \

)y = U (1)

has the regular solution

y = fax) - Ax[l+c 1 x+w*+...] 9 (2)

where 2 c n x>l converges when \x\ ^ R l9 say, then we have

^ p l I'c 1 +2c 2 a;+3c 3 a; 2 +...1

X [^ 1 ~y~ C-t X ~i~Co X ~Y~ , . . J

- -^-V-&i*~&2* 2 -> (3)

X

f The same method can be applied if the coefficients of the differential equation
are single-valued and holomorphic in a ring (r < \x\ < R) whose inner boundary is
finite and encloses several singularities, and if P is a circuit enclosing the inner
boundary.

56 EQUATIONS WITH UNIFORM Chap. IV, 15

where 2 b n x n converges in some circle \x\ ^ jR 2 > where < E 2 < R v
Hence PI(X) has at most a pole of the first order.

Conversely, if the coefficient^^) of (1) is of the form (3) the solu-
tion exp[ J Pi(x) dx] is regular and has an expansion of the form (2).
Thus Fuchs's theorem holds for equations of the first order.

Now the general equation witli uniform coefficients (E%) has at
least one solution, which is merely multiplied by a constant after a
circuit about the origin. Let this be chosen as the first solution of a
certain fundamental system, which undergoes the transformation

- f * (4)

whose determinant is not zero. Then the expressions

undergo a corresponding transformation with a determinant not equal
to zero : / n

They therefore satisfy an equation of order (n 1), whose coefficients
are uniform at x = 0,

W(t a> t...,h,z) = .

IK^^,,...,^) ' { '

or Z)- 1 2+ ?1 (a;)7)"-2 2 +...+ ? , t _ 1 ( a ;) 2 = 0. (8)

We can pass from the given equation (E*) to (8) by putting

y = fax) J z dx, (9)

and the coefficients are connected by the relations

(10)

-Vl , /W

A + i
0i \ l

Chap. IV, 15 ANALYTIC COEFFICIENTS 57

to which we may add the identity

i + ...+ Pn (x), (11)

<Pi 9i

expressing that (E*) is satisfied by ^(x).

First, suppose that the solutions (<f> r (x)} are all regular. Then the
expansion of the leading solution ^(x) is of the form (2); its reciprocal
has a regular expansion

l/<f>i(x) = x~P^+c(x+c^+...l (12)

and so (-> r <i)/<i has at most a pole of order r at the origin. Moreover,
the (n 1) functions {*j* r (x)} will also be regular; if the conditions are
assumed to be necessary for equations of order (n 1), the coefficients
{q r (x}} of (8) will have respectively poles of order r at most at x = 0.
The relations (10) and (11) taken in turn then show that (p r (x)} will
have respectively poles of order r at most at x = 0. Thus the condi-
tions will also be necessary for equations of order n. But we know
them to be necessary for equations of the first order, and so it follows
by induction that they are necessary in general.

Conversely, suppose that {p r (x)} have respectively poles of order
r at most at x = 0. We shall see in the next chapter how to construct
a fundamental system of n regular solutions; but let us assume pro-
visionally that there is one regular solution </>i(x) of the form (2).
Then the relations (10) show that {q r (x)} have respectively poles of
order r at most at x = 0. If the conditions are assumed to be
sufficient for equations of order (n 1), all the solutions {*/i r (x)} of (8)
will be regular. By (9), we then see that all the solutions {<f> r (x)} of
(E*) will be regular; hence the conditions will also be sufficient for
equations of order n. But we know them to be sufficient for equations
of the first order, and so it follows by induction that they are sufficient
in general.

Alternative Method. f A direct proof of the sufficiency of the con-
ditions has been given by Birkhoff, on the lines of an investigation by
Liapounoff . If the conditions are satisfied, we may write the equation
in the normal form

xDy+**-*P 1 (x)D-*y+...+P n (x)y = 0, (13)

where {P r (x)} are holomorphic at x = 0. If we put

yi = xt-W-iy (i = 1, 2,...,), (14)

f G. D. Birkhoff, Trans. American Math. Soc. 11 (1910), 199-202.

40G4 j

58 EQUATIONS WITH UNIFORM

this equation is equivalent to the system

This is a system of the type

Chap. IV, 15

(15)

x) yj (i = l,2,...,n), (16)

where {A^x)} are holomorphic and admit an upper bound
\Ajj(x)\ < M in the circle \x\ ^ R (say).

Now the equation (13) has at least one solution, which is merely
multiplied by a constant after a circuit about the origin; and this

GO

can be written in the form x? l H(x), where 11 (x) = 2 c m xm a

m~ -co

Laurent series convergent in a ring (e ^ \x\ ^ R), whose inner radius
may be as small as we please. The system (15) will have a correspond-
ing solution y i = xf )1 ll i (x) (i = 1,2,..., n). On the circumference
\x\ = r of a circle lying within the ring, 11 (a:) or (ll^a:)} are single-
valued and bounded. If we can find a real number K such that
|IJ t (#)| = Q(\x ~ K ) as x -> along a fixed radius vector, we can show
that c m = 0, if m < K. For if P is the inner boundary of the ring,
we have by Cauchy's integral

I

^~ I

(17)

and, if (K+W) is negative, this tends to zero as e is made arbitrarily
small.

Since (y t ) are analytic functions of x, or of logo:, we have

x-j ~ r -j~* where r = \x\. Hence (16) gives

dr

(18)

If (yi) are complex numbers conjugate to (y t ), let

then we have from (18)

r-

Chap. IV, 15 ANALYTIC COEFFICIENTS 59

(19)
Along a fixed radius vector (r > r > 0), we have by integration

Iogrlogr

2nMi

and so, as r -> 0, S(f) = 0(r ^ nM)> (21)

provided that $(r ) is bounded. Now, if p 1 is complex, the factor
xPi can become infinitely large as am(#) increases or decreases in-
definitely. But, if am(o;) is bounded, $(r ) remains bounded; we then
have, a fortiori, \y^\ 0(r~ nM ) as r -> 0. After removing the factor
x?i, we can choose K so that we have uniformly

H(x), ll t (x) = 0(r-) (\x\ = r->0). (22)

It then follows from (17) that the number of negative powers of # in
these Laurent series is limited. The particular solution ^(x) is thus
regular and the proof may be completed by induction as before.

EXAMPLES. IV

1. JACOBI'S FORM. Show that every linear transformation is equivalent to
one of the form

2, ELKMENTAKY DIVISORS, (i) If A g (A) is the H.C.F. of all 5-rowed minors
of A n (A), show by differentiation that every root of A^^A) = is a root of
higher multiplicity of A n (A) 0.

(ii) If A^(A 1 ) = 0, A g _ 1 (A 1 ) -/= 0, tlien A -- A r is a root of multiplicity (n s-\- 1 )

n
at least of A W (A) = 0; and the system 2 ( (t tj ~-^i$ij) c i U =- l,2,...,?i) has

1-1
exactly (n $ \ 1) linearly indeponcl(nt solutions (M).

3. Show that the elementary divisors of the canonical characteristic deter
mimmfc A,, o, o ..... o

o,

A.-A,

Al,

o,

...,

o,

o,

o,

o,

..., A,

o,

o,

o,

o,

..., Aj-A

60 EQUATIONS WITH UNIFORM Chap. IV, Exs.

4. If tho characteristic determinant is of the form

where (A, J5, (7,...) are canonical subdetcrminants belonging to the elementary
divisors {(A AJ)*, (A A 2 )^, (A A 3 ) v ,...} and where tho asterisks represent blocks
of zeros, show that:

(i) There is a first minor Aj~ 1 A n (A)/(A Aj) 01 ,

a second minor A- 1 A?- 1 A n (A)/(A-A 1 )'XA-A 2 )^, etc.

(ii) If (A Aj)* is tho only elementary divisor belonging to the root A - X lt
it is a factor in E n (X).

(iii) If (A Ai)*, (A A^ 01 ', (A Ai)**,..., all belong to tho same root and
a: > <x f > a* > ..., then (A A].)* is a factor in E n (X) 9 (A AJ)' is a factor in
-& 7 /t _i(A), (A Xi)*" is a factor in E n _ z (X), etc.

(iv) Show that E n (X) is divisible by AVi(A), E n _^(X) by E^X), etc.

[Cf. M. Bocher, Higher Algebra, 262-78; H. Hilton, Linear Substitutions,
1-33, 50-9; H. W. Turnbull and A. C. Aitken, Canonical Matrices,
64-73.]

5. Two LIMITING CASES, (i) An analytic function has singular points at
x -- o 1 ,a 2 ,...,a m , oo, and it is merely multiplied by a constant after any closed
circuit. Show that its logarithmic derivative is a uniform function, and that
it is of the form

f(x) == (x-a 1 )Pi(x-aJp*...(x-a m )

where ll(x) is uniform.

(ii) An analytic function has singular points at x = 0, oo only, and has n
linearly independent branches undergoing an arbitrarily assigned linear trans-
formation after a circuit about x = 0. Show that its branches are cogredient
with those of an equation of Euler's type F($}y = (S = xD).

6. By repeated use of Fuchs's substitution, show that a fundamental system
{<f> r (x)} at a regular singularity can bo written in the form

X

) J

where (ot^x)} are regular and free from logaritluns. Discuss how logarithms
make their appearance in the solutions.

7. LINEARLY INDEPENDENT SOLUTIONS. If (X p ) are unequal, if A vq are con-
stants, and if tho expression

tw =f i^Aj^

V-l -0

vanishes when x = 1, 2, 3, ..., show that

A pg = (p ~ l,2,...,m; q -= 0, l,...,n).

Chap.IV,Exs. ANALYTIC COEFFICIENTS 61

8. If (p p ) do not differ by integers, and if (H^x)} are single-valued near the
origin, show that if the expression

F(x) = f 2 o^log^n^')
p-i a=o

vanishes identically, then {H pq (x)} all vanish identically.

[Consider the form of F(x) after s complete circuits about the origin.]

9. If {Tl^x)} are single-valued near x = 0, and if x? 2 (k>% x ) % N- M satisfies a

i-o
linear equation with uniform coefficients, then so does the coefficient of each

power of A in the expression x p 2 (A H-log-)*Ut(#)'

t=o

1 0. If [n v (a;)} are single-valued near x -- 0, and (pj do riot differ by integers,
show that if an equation with uniform coefficients is satisfied by

it is satisfied by each group of terms

REGULAR SINGULARITIES

16. Formal Solutions in Power -Series

Indicial Equation and Recurrence Formulae. LET x be a typical
singularity where Fuchs's conditions are satisfied. To expand the
solutions we use the normal form of Fuchs and Frobenius

*Dy+P l (x)ar-iD-*y+...+P n (x)y = 0, (1)

where {P t (x)} are holomorphic at x = 0. Let 8 EE xD, so that
x m D m ~ 3(8 1)...(8 ra+1), and let (1) be written in the form

F(x 9 8)y = 8*y+Q l (x)8-*y+... + Q n (x)y = 0. (2)

The coefficients (PJ and (Q t ) are mutually expressible as linear func-
tions of one another with constant coefficients, so that both sets are
holomorphic together. We now expand the coefficients in Taylor
series oo

I G<(s) = 2C3' (<=l,2,...,n), (3)

j=o
and write

/n

(4)

...,, J ..
The equation may now be written

and on substituting as a trial regular solution

y = xf[c Q +c 1 x+c 2 x*+...], (6)

we have m m

= 0. (7)

This vanishes identically if each coefficient is zero, or if (c t ) satisfy the
relations k

I**-<(H-)e<=0 (i = 0,l,2,...). (8)

Chap. V, 16 REGULAR SINGULARITIES 63

In particular, if c ^ 0, p must be a root of the equation

F ( P ) = p*+Q w p*- 1 +...+Q n0 = 0. (9)

This is called the indicial equation and its roots are called the expo-
nents of the given singularity.

Sets of Exponents. The number of linearly independent regular
solutions (6) cannot exceed the number of distinct roots of the
indicial equation and may fall short of it. For if two independent
solutions belong to the same exponent p, we can form by subtraction
a solution belonging to a higher exponent p', which must itself satisfy
the indicial equation. Thus one distinct power-series (6) at the most
is associated with each distinct exponent.

If F Q (p) = 0, F (p+k) ^ (k = 1,2,...), we may choose c ^ 0,
and every subsequent coefficient (c k ) is then uniquely determined,
But if several exponents differ by integers, the relations (8) may or
may not be compatible, and the number of distinct series may be
smaller than the number of distinct exponents.

Whenever there is a shortage of regular solutions of the type (6), on
account of multiple roots of the indicial equation or roots differing by
integers, the deficiency is supplied by solutions involving logarithms.
In practice these may be constructed either by a direct method, which
has been fully worked out by Heffter,f or by an artifice such as the
method of Frobenius.J We shall give modified versions of both
methods in this chapter.

Let all the roots of the indicial equation which differ by integers be
collected in sets, and let each set be arranged in order of ascending
real parts. We shall show how to construct all the solutions corre-
sponding to a typical set of h distinct exponents (/o t -) of multiplicity
(/^). All these solutions are associated with the same multiplier
A = exp(2i7r/>), which is a root of multiplicity ]T Ki of the character-
istic equation belonging to a circuit about the origin.

Heffter's procedure consists in arranging the solution as a poly-
nomial in log x of degree less than n, with coefficients which are
regular series of the type (6); he constructs first all solutions not
involving log x, then those of the first degree, and so on. Instead of
this we assume at once a trial solution of the form

y = x p[u Q +U l X+U 2 X 2 +...] 1 (10)

f L. Heffter, Einleitung in die Theorie der linearen Differ entialgleichungen (Leipzig,
1894), 20-34, 104-32.

t G. Frobenius, Crelle'a J.fur Math. 76 (1873), 214-33.

64 REGULAR SINGULARITIES Chap. V, 16

where (u t ) are polynomials in logo: of degree less than n. We then
have co oo

**(*,% = 2 2

t=0 j=

= J,

(11)
which vanishes identically if (w ) satisfy the relations

2^(8+/)+X = (i = 0,l,2,...). (12)

1 =

This may be regarded as a system of linear differential equations with
constant coefficients in the independent variable t = logo:. We do
not require the complete primitive, but only the most general solu-
tion in polynomials. We can accordingly simplify the system as
follows.
The Auxiliary Systems. The first equation may be written

Ffo) K+^oW 8X+- = o, (13)

and is the generalized indicial equation. If u is a polynomial in t not
identically zero, this expression is a polynomial of the same degree
unless F (p) = 0. For an effective solution, p must satisfy the
indicial equation; and if we identify it with the lowest root of the
set p p l9 the series belonging to the higher roots will appear in
due course. Since ^J)(8+pi)t* = G f 1 (S)8 lfx w , where 6^(0) ^ 0, the
polynomial U Q must satisfy the reduced equation

S*IM O = 0. (14)

Again, suppose that the polynomials (u Q9 u l9 ... 9 u k ^ 1 ) have been
found. If F (p 1 -\-k) ^ 0, u k is uniquely given as a polynomial, whose
degree does not exceed the highest degree of any earlier coefficient,
by the symbolic formula

, k-l

i=0

= L k (u ,u l ,... 9 u k _ l ). (15)

But if k PIPI, we have

F 9 (8+ Pl +k) = Ftf+Pt) = Gi(*P {GiW^O}, (16)

Chap. V, 16 REGULAR SINGULARITIES 65

and then we have instead of (15) the relation

The structure of the solution is completely determined by the first
(N-\-l) relations, where N is the difference between the highest and
lowest exponents of the set. For every subsequent relation can be
reduced to the standard form (15). We may distinguish the h critical
polynomials U t =: u p p > and express all the others explicitly in

where (U^ satisfy a system of equations of the form
8*0; - 0,

: = >

Invariant Factors. Hamburger Sets. To each exponent p of multi-
plicity K correspond solutions beginning with

xP'(logx) (* = 0,l,...,*c'-l),

which may involve higher powers of logo: in the later terms. To
resolve the aggregate of solutions into Hamburger sets, we have only
to eliminate as many (UJ as we can express explicitly in terms of
the rest, and to replace the reduced system of equations by an
equivalent diagonal system exhibiting the invariant factors of the
matrix of operators (19),

Se jF ._o ( = 1,2,..., A' < h', 2>* = 2*i)- ( 2 )

The (%) are explicit linear functions of (U t ) and their derivatives,
and vice versa; neither set can contain a polynomial of higher degree
than all the other set. Hence each invariant factor 8 e ' yields a solution
whose degree in logo; is (e r 1). If we take this as the highest solution
of a Hamburger set, the other solutions of the set are constructed by
taking successive differences. Two different invariant factors yield
distinct solutions. For we cannot make (u t ) all identically zero unless
(U t ) and (PJ) are also zero, and so no combination of (Ify yields a
nul solution except (y t ~ 0).

17. Convergence

Rearrangement of the Series. When the critical terms at the
beginning of the series have been found and the degree s of the solu-
tion in logo; has been determined, we can calculate as many more

66 KEGULAR SINGULARITIES Chap.V,17

coefficients as we require by means of recurrence formulae. We shall
suppose that the lowest exponent of the set has been made zero, by
putting y = xPiy*, to simplify the notation. We introduce binomial
coefficients and write

, (I)

where u t N| . j c^(log )-', ^ = J c^a'. (2)

j=o ^' i ~

By Leibnitz's theorem, we have

(3)

and hence, for any linear operator F = ^(a;, 8) of order TI,

F[uv] = V i{jp(%}8^, (4)

i =

where ^^ is an operator of order (n i), formally defined by differen-
tiation as though 8 were a variable

=^0(35, 8) = ^,8). (5)

Now F, = l

= 22

We must therefore have

F, = (4 = 0,1,...,*); (7)

and, by re-combining these relations, where s does not appear expli-
citly, we find that the expressions

7 , Klogs+rj, [7 (loga;) 2 +27 1 (lo g a;)+7 2 ], etc., (8)

are each a solution of Jfy = 0.

Chap.V,17 REGULAR SINGULARITIES 67

We now arrange (7) in powers of x, and find

Sinoe each coefficient vanishes, we have

(* = 0,1,...,*; ?==0,l,...,oo), (10)
and in particular, for q = 0,

= (4 = 0,1,...,*); (11)

* /4\

these relations are compatible because [J^(0)]* +1 = 0, since zero is a
root of the indicial equation.

If p = N is the highest exponent of the set, we assume that
coefficients of all powers of x up to X N have been determined by the
methods of 16. If q > N, F Q (q) ^ 0, and the coefficients of x q are
uniquely given by (10) in terms of the earlier coefficients.

Convergence. Let d be the shortest distance between x = and any
other singularity. Then the series {(2*0*0} are a ll convergent when
\x\ < d\ the convergence of (Yj) may then be proved by means of the
following lemma. Let 2 b k x k be any series convergent when \x\ < d,
and such that the equation

f 6 t ** (12)

is formally satisfied by a power-series Y = ^c k x k \ then that power-
series is convergent when \x\ < d.

Assuming for a moment the truth of this lemma, we first establish
the convergence of Y Q) by putting zero on the right-hand side of (12).
Then, if we have proved the convergence of (lo>^if"'5V-i) we can
reduce the equation (7) giving 7^ to the above form (12); and another
application of the lemma establishes the convergence of Y k . The
convergence of the complete set of power-series follows by induction
with respect to k.

68 REGULAR SINGULARITIES

Now (12) gives the recurrence formulae

Chap. V, 17

b k (i = 0,l,2,...), (13)

which are (by hypothesis) algebraically compatible. Let x be any
number such that \x\ = r < d\ and let R = \(r-\-d) < d. Then
because ^ Qtk xk an( ^ 2 b k x k are absolutely convergent when \x\ R,
their terms have upper bounds

\Q ik \R*, \b k \R<M. (14)

Hence, for positive integers j, we have

(15)
1, we can

(16)

(k > N). (17)

c k x k by putting

(18)

Again, since -F (&) 7^ (& > N), and since

fe-

find a positive 6 such that

^(fc) > ^~ (Jfc > JV
From (13), (14), (16) we get the inequality

Ok n \c k \

We now construct a dominant series

= 0(k-l) n E k - l C k _ l +Mk n - l R k - l C k _ l (k >
Since lim (C r A ./(7 A ._ 1 ) 1/jR < 1/r, the dominant series

(19)

absolutely convergent for the value of x in question, and a fortiori
the series 2 c k xk - The lemma has accordingly been proved; and the
convergence of all the series (Y^) follows.

18. Apparent Singularities

Example. Consider the equation

xD*y-(l+x)Dy+y = 0, (1)

which has a regular singularity at x = 0. If we substitute
y ~ 2 c n % p+n > we have the indicial equation and recurrence formulae

P (p-2)c = 0,
(p+-2)[(p+w)c n -c n _ 1 ] = (n = 1, 2,...).

Chap. V, 18 REGULAR SINGULARITIES 69

As the exponents (0, 2) differ by an integer, we must examine whether
the solution belonging to the lower exponent is free from logarithms.
In fact, on putting p = 0, we find that c and c 2 are both arbitrary,
and that _ >

Cl ~ C ' (3)

= <>n-i (w = 3,4,...). /

The solution is y = A(l+x)+Be x ) as may easily be seen by elemen-
tary methods. Now this is always holomorphic at x = 0, so that we
are led to ask why this should be a singular point. The reason is that,
when x = 0, we have (y ~ A-\-B, Dy = A-\-B], so that arbitrary
values cannot be assigned to y and Dy at the origin.

Conditions for an Apparent Singularity. A singular point where
the complete primitive of the differential equation is holomorphic is
called an apparent singularity. If x = is an apparent singularity of
an equation of order n, there are n linearly independent ordinary
power-series (y t ) satisfying the equation, and we can arrange by sub-
traction that no two of them shall belong to the same exponent.
Accordingly the exponents of the singularity are n unequal non-
negative integers (p t ). Now the Wronskian determinant

is also regular, and belongs to the exponent

a = Pi+Pz+'"+Pn \n(n 1). (4)

This exponent cr is a positive integer for every admissible set of ex-
ponents except (0, 1,2,..., n 1), when we have a = 0. We then have,
by Liouville's formula,

= --+b +b l x+b 2 x*+.... (5)

x

If a = 0, PI(X) has a pole at x and the point cannot be an ordinary
point. But if a = and if, as we are assuming, every solution is
holomorphic, every coefficient of the equation

2 ,...,y n9 y) = Q m

... 9 y n ) { }

is holomorphic at x = 0, and the point is an ordinary point of the
equation.

We can of course have a singular point where the exponents are
(0, 1,2,. ..,7i 1), but in that case the solution involves logarithms.

70

REGULAR SINGULARITIES

Chap. V, 18

As an example, the reader may examine the equation

xD*y-y - 0. (7)

If the roots of the indicial equation at x = are unequal non-negative
integers, we must examine the most general holomorphic solution
y 2 c n x n . If N ~ (p n pi) is the difference between the highest
and lowest exponents, we must examine the compatibility of the
(N+I) recurrence formulae satisfied by the coefficients of (o^ 1 ,
x^ 1+1 ,...,^ n ). The necessary and sufficient condition that the point
should be an apparent singularity is that the n critical coefficients of
the powers (x^) should be arbitrary, or that the rank of the system
should be (N n+I). We can always write the coefficients of the
non-critical powers of x explicitly in terms of these n critical ones,
and obtain a reduced system of relations between the latter only :

= 0,

(8)

The necessary and sufficient condition for an apparent singularity
is that all the coefficients (A fJ ) should vanish. Further details of the
procedure will be found in HefTter's treatise. f

It may happen that all the exponents at a regular singularity are
unequal, but differ only by integers, and that the solution is found
to be free from logarithms. In that case the solutions are rendered
holomorphic by a transformation of the type y = (x)P l y*, and the
point is said to be reducible to an apparent singularity.

19. The Method of Frobenius

D'Alembert's Method. Suppose we know that Euler's equation
F($)y = is satisfied by y = 2 C^o^, where (p r ) are the roots of the
indicial equation F(p) = 0. If the roots are unequal, this solution is
the complete primitive; but if there are multiple roots, fresh solutions
of a different type must be found. The latter can be obtained by
evaluating the limits of such expressions as

o o

pi

Pi
1

P2
1

P3

I

P\

Pi
I

Pi

P2
1

I

(1)

f Loc. cit. 20-34.

Chap.V,19 REGULAR SINGULARITIES 71

when P! -> p 2 -> p 3 . The required solutions are, as we know, of the
type xP^logx, xP*(logx) 2 , etc., when p l is a double or triple root.
D'Alembert's method consists in introducing a parameter into the
equation, and studying the behaviour of the complete primitive as
that parameter approaches some critical value. Frobenius has
elaborated a method of this kind for obtaining all the solutions of a
linear equation at a regular singularity, when the exponents become
equal or differ by integers. We retain the same notation as before.
The Auxiliary Equation. Frobenius's plan is to consider an equa-
tionoftheform F(x,8)y == f(a)x, (2)

where a is a parameter at our disposal. We shall choose /(a) slightly
differently from Frobenius, so that the coefficients of the solution
shall be integral functions of a. If the indicial polynomial at x = is

tfo(p) = (p~Pl}(p~~P^-"(p~~Pn)y (3)

we write x F(z) = T(zp l }Y(z~p 2 }...r(zp n }, (4)

1 r/ \ -i

where - - - = & z z 1 T (l+-)e~ 2!/r , (5)

1 \Z) 1 JL I \ Ti I

v ' r=l L v ' J

and y is Euler's constant. We now consider the equation

whose right-hand side vanishes when a = p^ m, where m is zero or
any positive integer.
We substitute formally

and obtain the relations

(+)c,() = (4=1,2,3,...).

i=0 /

Since we know that

r(z+i) = zr(z), Y(Z+I) = F 9 (zye(z), (9)

we find that c (a) = l/ l F(cx+l), (10)

and we observe that c Q (p h ) ^ 0, if a = p h is the highest of a set of
roots differing by integers of the indicial equation j^ (a) = 0. We
have further

c (a} = _ - _ - t , .

fcV ' - ( '

72 REGULAR SINGULARITIES Chap. V, 19

by (9) and (10), where h k (a.) stands for the determinant

...

(12)

**-s(+2) - F Q (*+k-l)
^fc( a ) F k-\

**() =

Since this is a polynomial in a and l/\F(a++l) is an integral func-
tion, c k (oi) is an integral function, for which we require a uniform
upper bound in a circle |a| < K. To obtain a dominant polynomial
for Afc(a), we use the relations 17, (15),

'i(a) < o^+Mfa+I) 71 - 1 a n +(/>((*) say, \
^(a) < R-vMfa+I) 11 - 1 = R-v</>(oc) (q > 0), /

(13)

where E is any fixed number smaller than the distance from x =
to any other singularity, and M depends only on E. We now observe
that in the expansion of the determinant

a n , 6 15 0, ...,

(14)

every term is positive. Hence

(+l), -,

Now we can choose a positive number A such that

hence we have, uniformly in the circle || ^ K,

(15)

(16)

(17)

Chap. V, 19 REGULAR SINGULARITIES 73

Again, when 2 a = 2 & we nav e> fr m tne definition (5) of F(z)
as a product,

J

Hence, as Jc -> oo, we have

l '
From (11), (17), and (19) we have now, when |a| ^ K,

\c k (oi)\ < R- k H(K)r(2nK+B+k)/r(k), (20)

where H(K) does not depend on Jc.

The integral functions (c /c (a)} may be differentiated as often as we
please; and upper bounds for the derivatives when |a| < K' < K
are given by Cauchy's integral

< 21i

-

taken around the circle |z| = IT; we have accordingly

\-f*- -L\- ) 1 \fC f

The expressions (20) and (22) suggest dominant series of the type

x \-2nK-B

- - J for 2 c k (oi)x k and its derivatives with respect to a.

Since R is any number we please smaller than d and K is any finite
constant, the series ]T c$(oi)x k converge uniformly with respect to
a in any finite domain, provided that \x\ ^ d e <C d. The solution
(f>(x, a) is given by multiplying this series by # a , which is an integral
function of a for any value of x ^ 0.

Solutions given by a Set of Exponents. In a circle \x x \ < 77,
which lies entirely within \x\ < d but excludes the point x = 0, the
relation ,

^ ^ ^^(o,) (23)

is an identity between integral functions of a, which may be differen-
tiated as often as we please. If p h is the exponent with the highest
real part in our typical set of exponents differing by integers, we see
from the definition (4) that a = p h is a zero of multiplicity K h of the

74 REGULAR SINGULARITIES Chap. V, 19

right-hand side; but from (10) we know that c Q (p h ) ^ 0. We can thus
obtain K h solutions of F(x, 8)y from the relations

- (a = 0,1,...,^-!). (24)

PA

These solutions are necessarily independent, since they are respec-
tively of degree (0, 1,...,*:^ 1) in logx, which appears when we
differentiate # a .

The second highest exponent of the set p h _ t is a zero of multiplicity
( K h-i~i~ K h) f kh e right-hand side of (23), and yields (^_i+^)
solutions

= (s = Q, 1,..., **_!+**-- !) (25)

But a = p h _ 1 is also a zero of multiplicity K H of c (o:), so that the first
K h of the solutions (25) turn out to belong to the exponent p h and to
be combinations of the solutions already given by (24). The last
K h _ 1 solutions are new and linearly independent.

In the same way, the third exponent yields (
solutions

of which the first (/c^^+ic/J are combinations of the solutions (24)
and (25) and the last /c^_ 2 are new, and so on. Every solution can
thus be deduced from the expression (f>(x, a) by suitable operations.

20. The Point at Infinity. Equations of the Fuchsian Class

Change of the Independent Variable. If the equation (E*) has m
isolated singularities (x a l9 a 2 ,...,a m ) in the finite part of the plane,
they can all be enclosed in a circle \x\ R, outside of which the
coefficients are holomorphic, except at x oo, which is an isolated
singular point or an ordinary point. To examine its character, we
put x = 1/2, and we say that x = oo is an ordinary, regular, or
irregular point (as the case may be) of the given equation, according
as z = is an ordinary, regular, or irregular point of the transformed
equation. Thus x = oo is placed on exactly the same footing as any
other point; this is graphically illustrated when we make a stereo-
graphic projection of the plane of the complex variable x (in the
usual Argand diagram) upon the surface of Neumann's sphere.

Chap.V,20 REGULAR SINGULARITIES 75

A convergent series of the form

+... (c ^0) (i)

is said to belong to the exponent p at infinity.

Conditions for a Regular Singularity. If we use the form

x)y = o, (2)

and put x = ~, 8' = 2- #- = 8, (3)

2 dz do:

we have at once

= o. (4)

The necessary and sufficient conditions that 2 = (or # oo) should
be regular are that {$;(oo)} should be finite. In the canonical form,
this means that {x'pi(x)} must remain finite as x -> oo, or that

fl(x) = 0(x~i).

Conditions for an Ordinary Point. To transform the canonical form
(E*), we put

D k y ~ x- k S(S-l)...(S~

= (-) k z k + l D' k (z k ~ l y); (5)

We now put k (n s) and change the order of summation,

76 REGULAR SINGULARITIES Chap. V, 20

The leading term here is () n z 2n D' n y t so that the canonical form
corresponding to the independent variable z is

F*y = D'y+p*(z)D'^y+...+p*(z}y, (8)

where

.. \n-s)(n-s-l)\ 'W {9)

P *(Z) = (-)"Z- Z "Pn h

Now z is an ordinary point of the new equation, if all these
expressions are holomorphic; hence the necessary and sufficient
conditions that x = oo should be an ordinary point of the original
equation (.#*) are that the analytic functions

--r~l)! (8-12 n-l]

X Pr() I*- 1 ' 2 '""" l >>

and (-Y& n p n (x) (Po(x) = 1},

should remain finite there.

The Indicial Equation. If infinity is a regular singularity, we find,
on introducing an expansion of the form (1), that the coefficient of
the dominant term gives us the indicial equation in either of the

forms / ,_Q 1 ( 00 )p-i+Q,(oo)p--... + (-) n (oo) = 0,
or

+(-)P n (oo) = 0,

(11)
where P r (x) ~ x r p r (x) as before.

Equations of the Fuchsian Class. If the only singularities are m
isolated regular singularities in the finite part of the plane and a
regular singularity at infinity, the equation is said to be of the
Fuchsian class. If we put

ifj(x) EEE (xa 1 )(xa 2 )...(xa m ), (.12)

where (a t ) are the affixes of the singularities at a finite distance from
the origin, the necessary and sufficient conditions that these should
be regular are that {$ r p r (x)} should be holomorphic for all finite
values of x. These expressions are accordingly integral functions of
x\ but (x r p r (x)} are finite at infinity, so that we have

WWfPrW = n (m _ 1)r (z) (r = 1, 2,...,n), (13)

where n a (#) means a polynomial of degree a.

Chap.V,20 REGULAR SINGULARITIES 77

We can now write the equation in the form

^D-y+E m _ 1 ^- 1 D>^y+...+U mn _, l y = 0, (14)

with polynomial coefficients', the advantage of working with this form
in practice, in preference to the canonical one, is that the recurrence
formulae have only a finite number of terms.

Fuchs's Relation between the Exponents. If we put

>-< - sr

the indicial equation at x a t takes the form

... = 0, (16)

so that the sum of the exponents at x = a t is [\n(n 1) A t ]. The
indicial equation at infinity is

p(p+l)...(p+n^l)-(ZA t )p(p+l)...(p+n-2)+... =-- 0; (17)

so that the sum of the exponents at infinity is [J^ |w(ft 1)].
Hence we have the result that the sum of all the (ra+ l)n exponents is

Equations of the Second Order. The most general equation of the
Fuchsian class and of the second order is

Sy = 0, (19)

{m A \ / m T) in s~i \

Y-^MAH- V -- ,+V 2/ = o, (20)
Z^-a,)/ J ^\2 l (x-a l )*^2 l (x-a t )l y V ;

where 2, C t ~ 0, as we see by expanding the last coefficient in
descending powers of x. Now, at x = a t , the indicial equation is
p(p l)-{-A i p-\-B i = 0. If the roots of this are p p t and p ~ p' it
we have

A i =l- Pt -p' i , B i = p iP ' i (i = l,2,...,m), (21)

which gives the canonical form

= 0. (22)

{ x -i

The indicial equation at infinity is

(m \ m

2^]P+I(S t +C ( a t )^0, (23)

i==l ; t = l

78 REGULAR SINGULARITIES Chap. V, 20

and Fuchs's relation (18) may be written in the easily remembered

form m

(24)

If infinity is an ordinary point of the equation of the second order,
we find from (10) that

or, with the form (20) of the coefficients,

m m m m

2 A t = 2; 2 C t = 0; 2 (Bi+C t a,) = 0; (25,0,+ C; a?) = 0. (26)

i=l i1 i 1 i = l

Reduced Forms. If we put y = <f>(x)u, we have

) + ) . = 0. (27)

We can choose <j>(x) in 2 m ways, so as to reduce to zero one exponent
at every singularity in the finite part of the plane ; one choice is

I

ftx) = IT (x-a t )f, (28)

t-1

and the others are obtained by interchanging (p^ />'J. If we introduce
the exponent-differences 8^ = (p^ p t ), we find that (27) takes the
form

u = 0, (29)

exhibiting the exponent-differences (8 t ). We may call this the first
reduced form.

We can also choose <j>(x) so as to make the middle term of (27) dis-
appear; this requires that

<f>(x) - exp[-| J Pl (x) dx], (30)

and leads to the second reduced form

D 2 u+Iu=Q, (31)

where / is the invariant

)-\{ Pl (x)}\ (32)

/ s = + .. (33)

Chap.V,20 REGULAR SINGULARITIES 79

The necessary and sufficient condition that the equations

)y - 0,

(34)
D*z+ qi (x)Dz+q 2 (x)z = 0,

should be transformable into one another by putting y zx(x) is
that they should have the same invariant, or that

p 2 -lD Pl -lpl = / = q,-\Dq^lq\. (35)

EXAMPLES. V

1. BESSEL'S EQUATION. If v is not an integer, the solution of

x z D z y -\~xDij-\- (x*-v*)y =
is y AJ v (x)-\-BJ_ v (x), where

r/ r \ _

Jv(x] ~

m~0

2. BesseFs equation of order zero

xD*y+Dy+xy =

OO

can be satisfied by y = 2 U zm x2m > where (u 2m ) are polynomials of the first

m

degree in log x, determined by the relations

3X - 0, (S-l~2m)Xm+w 2w _ 2 = o.
The reduced equations for polynomials are of the form

1

The solution J (^) corresponds to w 1. If we put u ^ 2 log( jx) -f- 2y,

where y is Euler's constant, we have Hankel's function

(ml

Since \jj(m-\-l) ~ logm, the series converges for all finite values of x, except
x = Q.

3. If n is a positive integer, J n (x) = ( ) n J^ n (x). The second solution

[00 T

2 M 2TO a? i wne re (w 2 m) are polynomials of the first degree in logo:,
m=o J

80 REGULAR SINGULARITIES . Chap. V, Exs.

is determined by the relations

, = 0, (S-f 2n)8w 2n -fw aw _ 2 0,
2n)u 2k -\-u 2k _ 2 (k =7^ 0,n).
Show that the equation is satisfied by

w-i

(n w-

2/ Y
^n\\

4. KUMMEB'S EQUATIONS. The doubly confluent hypergeometric equation

xD*y+cDy-y =
is satisfied, when c is not an integer, by
y^A^(

where ^F^c^x) ^ ]

Show that the equation is connected with Bessel's by the relation

5. The simply confluent hypergeometric equation

' xD 2 y+(c~x)Dyay =

is satisfied, when c is not an integer, by

y = A 1 F 1 (a;c;x)-\-Bx l ~ c 1 F l (a~c-}-I;2c;x),

where i^i( a > c > ) === l~f~; #4- *~n 1 ^ 2 -f-"-

If c is an integer, the condition that the solution should bo free from
logarithms is

(i) c> 1, (a-c+l)(a-c+2)...(a-l) = 0,

or (ii) c < 1, a(a-j-l)(a-f 2)...(a c) = 0.

6. If c = 1, the logarithmic solution is in general

m
Examine the form when a is a negative integer or zero.

7. If c is an integer greater than unity, the solution involving logarithms is
in general

c-2 c _ m

y = 2 ^"^r^ x*-*+

y[logcc+0(a-fw) i/j(c-}-m)~ 0(m-|-l)].
m^b

By putting y x l ~ c y f or otherwise, obtain the corresponding form when c is
an integer less than unity.

Chap.V,Exs. REGULAR SINGULARITIES 81

8. HYPEBGEOMETBIC EQUATION. If c is not an integer, the equation

x(l x)D 2 y+[c-(a+b + l)x]Dyaby =
is satisfied by

y =

i *> N T , ,

where F(a,b;c;x) ,, 1 + *+ -

9. If c is a positive integer, the equation is in general satisfied by

c-2

_ ^V / \

ml

~^p^^

10. If c is an integer less than unity, the equation is in general satisfied by
= V (_)o + I

Z-i m<

11. (i) Tho exponents of the hypergeometric equation at x = are (0, 1 c).
If c is an integer other than unity, the singularity will bo free from logarithms if

c-i
c > I and n [( a - r )(b r)] = 0,

r^l

or c < 1 and fl [(<*+*)(*> +r)] = 0.

r =

(ii) The exponents at x = 1 are (0, c a 6). If (c ab) is an integer
other than zero, the singularity will be free from logarithms if

a+b-c
(a + b c) > and ff [(a r)(b r)] = 0,

r = l
c a 61

or (a + 6 c) < and H [(a+0(6+r)] = 0.

r =

(iii) The exponents at x = oo are (a, 6). If (a 6) is an integer other than
zero, the singularity will be free from logarithms if

a-6-l
a > 6 and YI [(b+r)(b c+r)] = 0,

r =
6-a-l

or a <b and J3 [(a+r)(a c+r)] = 0.

r =

12. Examine the singularities of the associated Legendre equation

- 0.

4064

82 BEGULA& SINGULARITIES Chap, V, Exs.

13. Show that Laplace's tidal equation

has regular singularities at x = 1, apparent singularities at x = i/, and an
irregular singularity at a; = oo.

[It is simplest to use Prof. A. E. H. Love's transformation

=_
(f*-x*)dx 9 py dx

which gives (1 x z )D*z+p(f 2 x*)z = 0.

See H. Lamb, Hydrodynamics (5th ed.), 313.]

14. Construct a linear differential equation of the second order satisfied
by y = (x l) p and y = (cc-fl) . If p ^ q, show that there are regular
singularities at x = 1, oo, and an apparent singularity at x = (p+q)/(qp)-

15. Show that the exponents of an equation of the Fuchsian class are un-
changed by a linear transformation x' (Ax+B)/(Cx-\-D). Hence show that
the most general equation with only one regular singularity can be trans-
formed into D n y 0, and that with two into Euler's homogeneous equation
**(% = 0.

16. Verify the expansions in the preceding examples by the method of
Frobenius, where it is applicable.

VI
THE HYPERGEOMETRIC EQUATION

21. Riemann's P-Functionf

Definition. THE most celebrated equation of the Fuchsian class is the
hypergeometric equation, and it is instructive to begin by showing
how it is determined by certain quite general properties of the solu-
tion. Following Riemann, we denote by

a b c

any branch of a certain many -valued analytic function of x with the
following properties.

(i) Every branch is finite and holomorphic, except at the three
singular points x = a, 6, c.

(ii) Any three branches are linearly connected.

(iii) At x = a there are two principal branches (P< a) , P< a ')) which
are 'regular' and belong to the exponents (a, a')- Similarly there are
two regular branches (P ( >, P ( '>) belonging to the exponents (ft, ft')
at x 6, and (P (y) , PW) belonging to the exponents (y,y') at x = c.

(iv) The exponent-differences (a' a), (ft' ft), (y' y) are not
integers; and the six exponents are always connected by the relation

<*+<*' +ft+ ft' + 7 +y' = I. (I)

It is evident that the meaning of the P-symbol is unaltered if we
permute the first three columns, or if we exchange the two exponents
(a, a') in the same column, and similarly (ft, ft') or (y, y').
Linear Transformation. If we put

-BC^O), (2)

we obtain a P-function of the new independent variable x', with
singularities at the points x' = a', 6', c', corresponding to x = a, 6, c,
and with the same exponents, so that

(abc\ fa' b f c' \

ft y a[ = P ft y x'\. (3)

ft' y } U P V )

f B. Riemann, Mathematiwhe Werke (1892), 66-83,

84 THE HYPERGEOMETRIC EQUATION Chap. VI, 21

In particular, we can make the singularities coincide with x' = 0, oo, 1
by putting _ (x -a)(b-e)

_

~

(x-b)(a-c)'

and we observe that this is one of the anharmonic ratios of the set of
numbers (a, b,c,x). For the P-function with the singularities in the
standard position we may write more simply

R \

''

Q' '

p y

By permuting (a, 6, c) the transformation (4) may be effected in six
different ways, the new variables being connected by the relations

/ . 1 1 x x~-l /t .

x' = x, 1x, -, -- , - -, - . (6)

x lx xl x

The same function is represented by six schemes with different
independent variables

y 1

P

\ y '

P \

^ ' x r }

Change of Exponents. It also follows from the definition that

(7)

4

)

and similarly

^.-j-rf-, ' ", .) - p(+ 8 s ^- >+' A M

\a p y } \OL +6 ft b y +e /

We can thus assign arbitrary values to two exponents at two distinct
singularities, without introducing a new singularity or disturbing the
relation (1). These transformations leave invariant the exponent-
differences (a' a), (jS'jS), (y' y); we may therefore write
P(oLoL,j$ r p,y f --y,x) for the family of functions

, 4

I

Chap. VI, 21 THE HYPERGEOMETBIC EQUATION 85

The Differential Equation. After describing any closed circuit, two
linearly independent branches (y v y 2 ) of the P-function are trans-
formed into branches of the form (#n2/i+ a i22/2> #2 1 2/1+^222/2)* where
(a n a 22 a l2 a 2l ) ^ 0. Hence the determinants

y, Z)?/, D 2 y*

ui Ji yi (10 j

are each multiplied by (a n a 22 a 12 a 21 ), and so any other branch
satisfies an equation of the second order with uniform coefficients

W(y l) y 2 ) -

This has regular singularities at x 0, oo, 1 ; any other singularities
must be apparent, since the solutions are holomorphic. But apparent
singularities are excluded by the condition (1). For consider the
Wronskian of any two independent solutions W(y v y 2 ); in the neigh-
bourhood of x = this may be written as a numerical multiple of
the Wronskian of the principal branches W(P^\ P (a/) ), which is
regular and belongs to the exponent (a+a'l). Hence and by
similar reasoning

W(Vi> 2/2) ^ 0(+'- 1 ) as x -> 0, '

= Ofa-P-P- 1 ) asa;->oo, (12)

^ 0{(lx)Y+y- 1 } as x -> 1.
Accordingly the expression

#(*) = xi--'(l-xY-y-YW( yi ,yz) (13)

is holomorphic for all finite values of x\ and, as x -> oo, we find from
(12) that

<l>(x) = 0(x^-'W-y-y^ l +^) - 0(1), (14)

on account of the relation (1). By Liouville's theorem <f)(x) is a con-
stant; and so W(y lt y 2 ) cannot vanish for any finite value of # other
than zero or unity, and hence there are no apparent singularities, for
the reasons explained in 18.

The differential equation is now found to be uniquely determined
by its singularities and exponents, by the method of 20, (20)-(23),
in the form

86 THE HYPERGEOMETRIC EQUATION Chap. VI, 21

For the general scheme, Papperitz obtained the elegant canonical form

(xa) \(x-a)(x-b)(x-c) * ' 16 '

Reduced Forms. We can reduce one exponent to zero at each of
the singularities x = 0, 1, in accordance with (9), by putting

'a B v \ / oL-4-BA-v \

P 7 x } = x(lxpP\ ^^7 x \ ( 1? )

,a' j8' y' / \a'-a a+j8'+y y'-y /

By interchanging (a, a') or (y, y') we can effect the reduction in four
ways; and since the method is applicable to each of the six schemes
(7), we obtain altogether twenty -four reduced forms in six different
independent variables.

If we introduce the exponent-differences

A = a' a, ju, = j3' j8, v = y'-y, (18)

we find from (1) that

and so the reduced scheme (17) may be written

/O 4(1 A u v) \
P 2V ^ ' A (20)

I \ 1/1 ^k I \ I x '

and on inserting these values in (15) we have the reduced equation

V W<>. (21)

The other reduced forms are found by changing the signs of A or v.

The second reduced form, or invariant form, is found by removing
the middle term of the equation. We must therefore make the sum
of the exponents unity at x = and x = 1, and this can only be done
in one way for a given scheme. The new scheme is

\
i

/

and corresponds to the differential equation

2 ~ - (23)

~ ' ( }

which involves only the squares of the exponent-differences.
The Hypergeometric Equation. While Riemann's equation clearly

Chap. VI, 21 THE HYPERGEOMETRIC EQUATION 87

exhibits the exponents, it is not the most convenient form for
numerical calculations. We identify (21) with the standard hyper-
geometric equation

x(l-x)D*y+[c-(a+b+I)x]Dy-aby = 0, (24)

by writing

X=lc, /* =(a&), v = cab, \

} ( 25 )

The scheme of the hypergeometric equation (24) is therefore

/ a \

P\ x],

\lc b cab I

and we observe that the condition (1) is automatically satisfied.

22. Rummer's Twenty-four Series

Solutions at the Origin. To construct the regular solutions of the
type y = xP ^ c n x n , we use the form

8(8+c-l)y-*(8+a)(8+&)y = (8 = xD), (1)

and obtain the indicial equation

p(p-\-c 1) = 0, (2)

and the recurrence formulae

(n+p+l)(n+p+c)c n+l = (n+p+a)(n+p+b)c n . (3)

If c is not an integer, the principal branches are

p() == F(a,b;c' 9 x), P (a/) x l ~ c F(a c+l,6~c+l;2 c\x), (4)
with the usual notation for the hypergeometric series

From (3) we have lim (c n+1 /c n ) 1, so that the solutions are conver-

n -oo

gent when \x\ < 1. The first series reduces to a polynomial if a or 6,
the second if (a c+1) or (b c+1), is zero or a negative integer.
Prom the four equivalent Riemann schemes

nt a \ /, x fc / a \
P( a;), (lx) c - a - b P( x],

\lc b cab / \lc cb a+bc /

l-c c-b b-a x-

l__c c-a a-6 a;-!

(6)

88 THE HYPERGEOMETRIC EQUATION Chap. VI, 22

we can write down four equivalent expansions of the branch P (a) ,
which is holomorphic at x == 0, namely

F(a,b;c',x), (I-x) c - a ~ b F(c-a,c-b;c',x), \

The expansions in powers of # converge when | x \ < 1 ; those in powers
of xj(x 1) in the half-plane \x\ < \x 1|. In the same way, by a
linear transformation of the Riemann scheme, each of the six prin-
cipal branches can be expanded in four ways, in ascending or descend-
ing powers of x, (lx), or x/(x 1). The domains of convergence of
the various series are the interior or exterior of the circles \x\ = 1 or
\lx\ = 1, or the half-planes bounded by the line \x\ = \xl\.
These series, which were obtained by Kummer, are given in the
following table.

Table of Kummer' s Series

poo
1

F(a, b;c;x)
(l-xf-v-bFfc-a, c-6; c; a)
(l-x)~ a F(a, c-b;c;xl(x~l))
(l-x)~ b F(b, c-a; c; */(* 1))

po

x l - c F(a c+l, b c+1; 2 c;x)
x^l-xy-o-bF^-a, 1-6; 2-c;x)
^-^l-^-^-^a-c+l, l-6;2-c;ar/(a;-l))
x l - c (l~x) c - b - l F(b~c+l, l-a;2-c;x/(x-l))

p(/3)

x~ a F(a, a c+l;a 6 + 1; Ijx)
x-^l-llxf-a-bFil-brC-bia-b + l; I/a)
x- a (l~ I/x)- a F(a, c b; a 6+1; 1/(1 x))
x- a (l-llx) c ~ <l - l F(l-b, a-c+1; o-6+l; l/(l-a;))

pO')

x- b F(h, 6-c+l; 6~o+l; Ijx)
z- 6 (l~l/*) c -- 6 ^(l-a,c-a;6-a+l; I/a?)
x-b(l-l/x)- b F(b, c-a;6-a+l; l/(l-x))
aj-^l-l/^-ft-^l-o, 6-c+l; 6-a+l; !/(!-*))

p(v)

^(a,6;a+6-c + l; l-x)
x l ~ c F(a-c+l, b c+l;o + 6 c+1; 1 x)
x~ a F(a, a c+l;a + 6 c+1; (x l)[x)
x~ b F(b, 6-c+l; a + 6-c+l; (x~l)/x)

p(y')

(l-. x ) c -<*- b F(c-a,cb;c-a-b + l; l-x)
x l - c (lx) c - <l -bF(l~a t l~6;c o 6 + l; lx)
X a - c (lx) c - a ~ b F(l--a, c a;c a 6+1; (x-l)fx)
xb- c (l-x)-bF(l-b, c-6;c-a-6+l; (x-l){x).

23. Group of Riemann's Equation

Invariants. The principal branches of a Riemann function (whose
exponent-differences are not integers) are single-valued in the upper

Chap. VI, 23 THE HYPERGEOMETRIC EQUATION 89

half-plane, and are connected by the relations

otfr PW = a; PW+oy PV>.
The mutual ratios : a l' : i : V (2)

j8 ' <*y <Y

are independent of the choice of the constant multiplier belonging to
each branch, and were determined by Riemann as follows.

Let us consider the effect upon the two solutions (1) of a simple
positive circuit, beginning and ending at a point of the upper half-
plane, and enclosing x = and x = 1. We may regard this as a
sequence of two positive loops, first about x = and then about
x = 1, or else as a negative loop about x oo. The new branches
obtained from (1) can therefore be expressed in the two alternative
forms

We may take the first equation of each pair (1) and (3) and solve for
(P$\ P$'>), and we may do the same with the second equation of each
pair. On eliminating (P ( $\ P^), we have two linear relations between
(P ( y\ PW) which are identically satisfied for all values of the latter;
this will give four relations between the coefficients. Thus the two
expressions for P ( & are

(4)

+oy e^ a '+

and from these and the corresponding forms for P ( P"> we get the
relations

<y e <7rflc sin ?r(cx+^+y') '

+ ay e^

^T? 7 ^' sin TT(OC'
These relations are compatible, because 2<x=l. If we write

4064 N

90 THE HYPERGEOMETRIC EQUATION Chap. VI, 23

(A,/z,v) for the exponent-differences (a'- a, )8' /?,)/ y), we have
sin7r(a:+j8+y) = cos|7r(A+At+^), etc. The relations then become

= cos j7r(A+ju,+v)cosi7r(A /

a y

(6)

The coefficients will be evaluated below in 26.

Exceptional Cases. If we have cos ln(X+fjL+v) 0, we must also
have (ap> = oy) or else (a = = OL' Y ). We can then arrange the
notation and the constant multipliers of the principal branches so
that the relations (1) become

p(a) _. p(0)

The first solution is merely multiplied by a constant after any closed
circuit; and, since it is regular at infinity, its form can only be

P 1 = x (l x)y [a polynomial]. (8)

Similarly, there is a solution expressible in finite terms whenever
cos j7r(A/Ltv) 0. In the hypergeometric form, we find that

A/iv - (l-c)(b-a)(a+b-c) (9)

must be an odd integer. Hence one of the numbers (a,b,ac,bc)
must be an integer. We shall see that these cases are soluble by
elementary methods. When the hypergeometric equation is satisfied
by a polynomial, its group is generated by the transformations

corresponding to circuits about x and x 1 respectively, where
y l is the polynomial and y 2 the other principal branch at # 0.

General Case. If cos %TT(XIJLV) ^ 0, we can reduce the relations
between the solutions at x and x 1 to the form

p (a <) = p(y) + /c p(y') >

where K = = ^--^-v^

Chap. VI, 23 THE HYPERGEOMETRIC EQUATION 91

The generating transformations expressed in terms of the principal
branches at the origin are

Y

' (13)

The latter is obtained by expressing the transformation in terms of
(P(y\ p(/)) and eliminating (JW, P</>) by means of (11). It is evident
that K ^ 0, oo; and it may be verified that K ^ 1, because
sin ?rA sin Try 7^ 0. Thus the relations (13) are well determined.

Associated P-Functions. In the relations (5) the exponents appear
only through the multipliers (e 2/;7Ta ), so that P-functions with the
same multipliers have the same group. Consider three functions

(14)

/V f ~~ I \ 4* ~~l> ~ ' \~ -} > "/7 \ /

Pi 7i

whose respective exponents are congruent modulo 1. If (X t , ^ v t ) are
the exponent-differences (aj a^,^ ft, y^ y^), we have, on account
of the relations T a,- = 1,

(15)

and so A^ A ; -+/^ - />t ; -+^ ^ ~ an even integer. (16)

We may restrict ourselves to the reduced forms

/o KI-A,-^-^) o

where the necessity of the condition (10) is apparent, if the multipliers
at infinity arc equal. We suppose the branches so chosen that the
coefficients in (1) arc the same for i 1,2,3; and we consider the
expression

(18)

This belongs to the lower of the exponents (c^+aj, 0^+0^-) at a; 0,
(&+$} $4~$/) at ^ = oo, (yi+yj>y^+y/) at cc = 1, and these may be

written

a u ~ L a <~^ a j'~r at i~t~ (X ./~~l /l *"~~ / VU

(19)

92 THE HYPERGEOMETRIC EQUATION Chap. VI, 23

Thus x-^(x IJ-w/S^ belongs at infinity to the exponent

and is holomorphic for all finite values of x, so that it is a poly-
nomial of degree

Nti = t[|A < -A / | + l/^-^l + |v < -v y |-2] > (20)

which is an integer because of (16).
Three corresponding branches of the three functions are of the

form

'PW (i=l,2,3), (21)

with the same coefficients (A, A'). These are connected by the relation

Pi &>3+^2 #31 + ^3 S ia = 0. (22)

If (a,y) are the lowest exponents of the triads (oc^) and (y^)
differing among themselves by integers, we can remove a factor
# a (l x)* and find a relation

^1X1+^2X2+^3X3^0, (23)

where xi is a polynomial of degree (<* 23 a+y 23 y-f A^), and so on.

24. Recurrence Formulae. Hyper geometric Polynomials

Contiguous Series. We can illustrate Rieniann's theorems on
associated hypergeometric functions by means of Gauss's relations
between contiguous functions. The six functions

F(al,b' 9 C]x), F(a,bl' 9 cix), F(a 9 b- 9 cl- 9 x)

are said to be contiguous to F == F(a,b\c\x), and are denoted by
F a+ , etc. Each of them can be simply expressed in terms of F and
DF, and on eliminating the latter we get fifteen relations between
F and any two contiguous series, which were given by Gauss. To pass
from F to F a+ , we must multiply the coefficient of x 11 by (a+n) and
remove the factor a\ this is readily effected by the operator (8+a),
where 8 = xD. We thus have at once three of the required formulae

<+ = (&+a)F, (1)

bF b+ = (8+6)JF, (2)

(c-l)lJ. = (8+c-l)JP. (3)
We next write the equation for F a _ in the form

[8(8+c-l)-(8+o-l)(8+6)]^ --= 0,
or

Chap. VI, 24 THE HYPERGEOMETRIC EQUATION 93

on using (1), with (a 1) written for a, we now get

(c-a)F a _ = [(8+c-a)-*(8+6)]J,

i.e. (c-a)F a _ = (I-x)$F+(c-a-bx)F, (4)

and (c-b)F b _ = (I-x)SF+(c-b-ax)F. (5)

Finally we have the equation

[8(8+c)-*(8+a)(8+6)K + - 0,

i.e. [8-jc(8+a+6-c)](8+c)J c+ - (c-a)(c-&)sJF c+>

which gives by means of (3)

(c-a)(c-b)F c+ = c [(lx)DF+(c-a-b)F]. (6)

Our six formulae may now be arranged as follows:

xDF = a(F a+ ~F), (!')

= b(F b+ -F), (2')

(c--l)(^_-n (3')

x(l~x)DF = (c a)jp a _+(a c+6x)^, (4')

c(l a;)!)^ = (c~a)(c-b)F c+ +c(a+b-c)F. (6')

Gauss's fifteen relations now follow by equating two values ofDF.

Associated Series. Gauss showed, by constructing chains of conti-
guous functions, that any three series F(a-{-l,b-\-m',c-{-n: > x), where
/, m, n are integers, are connected by a linear homogeneous relation
with polynomial coefficients. By differentiating the hypergeometric
equation a certain number of times and eliminating intermediate
derivatives, we can show that this is true for any three derivatives of
F(a,b\c\x). Any other associated series can be expressed in terms
of F and DF ~ (ablc)F(a+l,b+l',c+l-x)

by repeated use of the operations (l)-(6), the results being very
similar to a celebrated formula of Jacobi given below.
If A; is a positive integer, we have from (1)

a k F(a+k ) b' J c- ) x) = (S+a)(S+a+l)...(S+a+fc-l) J F

= x l - a D k [x a + k - l F(a, 6; c; x)] 9 (7)

where a k = a(a-\-\),..(a-}-k 1).
Similarly from (3),

(ck) k F(a, b\ cfc; x) = x l -*+ k D k [tf- l F(a 9 6; c; x)]. (8)

94 THE HYPERGEOMETRIC EQUATION Chap. VI, 24

Again, we may write (4) in the form
(c a)x

and hence we have

(c a) k x*- a + k (l x) a + b ~ c - k F(a k,b;c;x)

= (z 2 Z>)*|>'-( 1 - x) a + b ~ c F(a, b ; c ; x)].
But (x*D) k = (x$) k

= x8(8 I)...(S-
= x k +*D k x k -*, (9)

and so we find that
(ca) k F(ak, b\c\x)

^x a - c +\lx) c - a - b+k D k [x^~ a+k -\lx) a+b - c F(a,b\c\x)]. (10)
In the same way we have

(c a) k (c b) k F(a, b\ c-\-k; x)

= c k (l-x) (; - a - b + k D k [(l~x) a + b - c F(a ) b',cix)]. (11)

Jacobi's Formulae.f If M == x c - l (l x) a+b ~ c ) the hypergeometric
equation may be written in the form

D[x(l-x)MDy] = ah My. (12)

Similarly, we may differentiate the hypergeometric equation (& 1)
times and write

D[x k (l-x) k MD k y] = (a+k-l)(b+k~l)x k - 1 (l-x) k - l MD k - l ^/ ) (13)
whence we have the recurrence formula

D k [x k (l-x) k MD k y] - (a+k-I)(b+k-l)D k - l [x k ~ l (l-x) k - l MD k - l y]
- <* k b k My. (14)

This is true for every solution of the hypergeometric equation. If we
put, in particular, y = F(a,b\c\x), and so

we may remove the factors a k b k , provided that F is not a polynomial
of degree less than k, and we then have

D k [x k (lx) k MF(a+k,b+k',c+k;x)] = c k MF(a,b\c\x). (15)
If b = 7i is a negative integer, so that F is a polynomial of degree n,
t C. G. J. Jacobi, Werke, 6, 184r-202.

Chap. VI, 24 THE HYPERGEOMETRIC EQUATION 95

F(a+n, b-{-n\c-\-n\x) is unity, and then we have a finite expression
for the hypergeometric polynomial

c n F(a, n\c\x) = a; 1 - c (la:) n + c - fl J[) w [a*+ w - 1 (l )-]. (16)
Again, if b = n, we have on differentiating n times the hyper-
geometric equation the relation

x(l x)D n +*y+[(c+n) (a+n+l)x]D n+1 y = 0. (17)

For the polynomial (16) we have D n+1 y 0; and for the general
solution &+n(i- x )a-+ij) n +iy = constant, (18)

D n y = C ( x- c -(lx) c - a - 1 dx. (19)

In the general formula (14) we may put k = n and introduce this
expression for D n y\ the second solution is then expressed in a finite
form involving a single quadrature

My AD n \x n (lx) n M f x^-^lx) -^ 1 dx\. (20)

By expanding in descending powers of x, we see that this is the
solution belonging to the exponent a at x oo. We give among the
exercises formulae showing that the hypergeometric equation is
soluble in finite terms, if any of the numbers (a,6,c a, c b) is an
integer.

Another Notation.f To exhibit the polynomials of Legendre and
Tschebyscheff as particular cases of Jacobi's, it is convenient to
place the singularities of the hypergeometric function at x = il, oo,
and to modify the exponents. We write

D"[(x- !)+(*+ 1)^], (21)

and find from (16) after a little manipulation

1, -n; + 1 ; , (22)

the scheme of the differential equation being
'-1 oo 1

P

-n x. (23)

-ft (ot+p+n+1) -OL
Thus P ( %>$(x) is a solution of

= 0, (24)

f G. P61ya and G. Szogo, Aufgaben und Lehrsdtze . . ., 1, 127; 2, 93. R. Courant
and D. Hilbert, Methoden der mathematischen Physik, 1, 66-9, 72-5.

96 THE HYPERGEOMETRIC EQUATION Chap. VI, 24

and a second solution is

xD n \(x l)+(a?+l)0+ J (x-l)-*- n -*(x+l)-P- n ~i dx]. (25)
Generating Function. If </>(z) is holomorphic at z = x and cf>(x) ^ 0,
andif (26)

then an analytic function which is holomorphic at z = x can be ex-
panded by Lagrange's formula

}- (27)

If we differentiate with respect to w and afterwards write /(z) instead
of<f>(z)f'(z), we find the expansion

Let us here write

= 1/^.2 i\ ffv\ = //*._iw/*a_njB ^

(29)

z = x+lw(z*-l).
The root which tends to x as w tends to zero is

z = w~ l [l (l-2xw+w 2 )*], (30)

and (21) and (28) together give

| w(x- l)(

n =

___ [I w(l 2xw+
"~ ^ a +^(l-2a;w;+^ 2 )*

_ (a-l)(o:+l)[l-w+(l-2;EW^
"~ (I~2xw+w 2 )*

(31)
Hence P ( >$(x) is the coefficient of w n in the expansion of

y o,n P ( a ,ft ra: ^ ^ [l-^+(l-2^+^)^[l+^+(l-2^+^)i]-^
n 2 o ^ f n P(x) - .

(32)
If a = j8 = 0, we have the well-known formulae for the Legendre

polynomials

~, .

CO _

n (^-\Y. (33)

Chap. VI, 24 THE HYPERGEOMETRIC EQUATION 97

Orthogonal Relations. If u EE= P ( >&(x) and v == P^^(x), we have
by cross-multiplying the differential equations the relation

= (TI m)(n+m+a+p+l)(I x)*(l+x)&uv. (34)

If the real parts of a,/? are greater than 1 and if m ^ n, we may
integrate between the limits i 1> an d we have

*. = J (l-xni+xP%P>(x)P%>P>(x) dx = (m^ n). (35)

-1

To evaluate the integral when m n, we write

i

.. = J |=^[D"{(i-*) + "(i+a;

J (i )

25. Quadratic and Cubic Transformations

Quadratic Transformation. There arc many known cases of the
transformation of one hypergeometric equation into another by rela-
tionsofthetype tf = ft x ), y' = wy, (1)

and their systematic enumeration was begun by Kummer| and com-
pleted by Goursat.J The only possible transformations of the equa-
tion with three arbitrary exponent-differences are linear; and those
of the equation with two unrestricted exponent -differences are at
most quadratic. The higher transformations are connected with the
theory of the polyhedral functions and apply to equations having
only one free parameter or none.

t E. E. Kummer, J.fur Math. 15 (1836), 39-83, 127-72.

i Soe particularly: E. Goursat, Annales de Vficole Normale, (2) 10 (1881), Suppl.
3-142 ; Acta Soc. Sc. Fennicae, 15 (1884-8), 45-127.

98

THE HYPERGEOMETRIC EQUATION Chap. VI, 26

The quadratic transformation is most clearly expressed by Rie-
mann's scheme

(2)

where (j8+j8'+y+y') = i- It is applicable whenever one exponent-
difference is , or when two are equal. The canonical form of the
equation with two equal exponent-differences is that of the associated
Legendre equation

(lx 2 )D 2 y2xDy-{-\ n(n+l) ^ \y = 0, (3)

whose scheme is

1-1 oo 1

\m Ti+1 l m x W

\m ~-n \m ,

If we put x cos in (4), we have the relations
P(

n

cos 2 <9|

(5)

We thus obtain Olbricht's 72 hypergeometric seriesf in ascending or
descending powers of

(6)

Whipple's Formula.J We can also write (5) in the equivalent form

t E. W. Hobson, Spherical and Ellipsoidal Harmonics (1031), 284-8.
j F. J. W. Whipple, Proc. London Math. Soc. (2), 16 (1917), 301-14; Hobson,
loc. cit. 245.

cos 2 0, sin 2 ^, cot 2 ^,

Chap. VI, 25 THE HYPERGEOMETRIC EQUATION 99

\

cosh a

-I co I

= sinh~*aP< iw+J \+m \n+l cotha). (7)
{ In I \~m \n \

Since the formula is symmetrical in (^+i), we may assume that
the real part of (n+l) is positive. We then obtain two alternative
expressions for the branch belonging to the exponent (n-\-l) at
cosh a oo or coth a = 1. We have by definition, at x co,

X

and so

m -\-n-\- ,.

Again, we have by definition, at x = 1,

and so
Pr-i-\(ooth) = -J- e -(+Wi+OT, i-*;n+|; 1 ~ thtt )

On introducing the expressions (9) and (11) on the left and right of
the relation (7), we obtain Whipple's formula, which was originally
established by transforming contour integrals, namely

.,. (12)

Cubic Transformation. When two exponent-differences are -J-,
or when all three are equal to one another, we can apply the

100 THE HYPERGEOMETRIC EQUATION Chap. VI, 25

transformation

(0 oo

x

(13)

= p { y y y v
y y

where (y+y') = \ and w = exp(2^?r/3). We can also apply the
quadratic transformation to either of the two expressions (13), and
Riemann thus obtains six equivalent P-functions

P( V , V ,v,x 3 ), P(v,\ V ,l,x t ), Pdv.to.faxJ, I

P/l i, i. <v \ P/l lu 1 /v I P/JLw 2 1.. . \ I * '

* \$>* / > a'j'M/j * \a> 2 r ? fj^s/j -Ma^? F> 2 v >^6/> /
where T

2 = 4a? 8 (l ar 8 ) =

(15)

To each of these we can apply the general linear transformation.

Repeated Quadratic Transformation. If one exponent-difference
is J and the other two are equal, we have the equivalent schemes given
by Riemann p( , , p( 9 ,

\ *^2/

If two exponent-differences are |, we have an elementary function,
P (? ' _*Iv X2 ) = P {-1 1 -I (^1)

since

000

26. Continuation of the Hyper geometric Series

Convergence on the Unit Circle. The simple ratio test suffices to
prove the convergence of F(a, 6; c; x) = 2 c ^ n i n ^ ne interior of the
unit circle, but fails on the circumference. Now we have

c a I

= 1
(a+n)(b+n)

Chap. VI, 26 THE HYPERGEOMETRIC EQUATION 101

and

'n+I

(1)

n

hence, by Raabe's test or by direct comparison with ] n~ k , the
series is absolutely convergent on the unit circle if R(cab) > 0.
We observe the significant role of the exponent-difference at x = 1 ;
for if R(c~-ab) > 0, both the principal branches remain bounded
as x -> 1, with any fixed amplitude of (x 1).

If R(c~ a b) ^ 1, the general term does not tend to zero, so that
the series is definitely divergent. A still more delicate test is required
in the doubtful range of values ^ R(cab) > 1. We observe
that in this case |cj -> 0; provided that x ^ 1, we may make the
transformation

aw\ V r VH< / _1_ V (/> n \<*n O\

%) 2, c n x C 0~T 2* \ c n c n-l) x > \*l

and this is expressible in terms of two contiguous series

c(l-x)F = x(b-c)F c+ +c^, (3)

both of which are absolutely convergent. Hence the series converges
(though not absolutely) at all points of the unit circle except x = 1,
when ^ R(cab) > 1.

To consider F(a,b\c\ 1), we use the gamma product

ff r (a+r)(6+r) i _ r( C )r(a+6-c)r mi

H [ (c+ r)(a+b-c+r)\ f(a)f(b) [ ^ \n)\' V ;

which enables us to compare the series with one involving binomial
coefficients

The series of error terms is absolutely convergent, by Raabe's test,
if R(cab) > 1. The sum of the first (n+l) coefficients of the
series (1 x) c ~ a ~ b is the coefficient ofx n in the series (1 a;) c ~ a ~ 6 - 1 , and
this can be verified by elementary methods. This tends to infinity,
and so F(a,b\c\ 1) diverges, when R(c-ab) < 0. If (cab) = 0,
both sides of (4) become infinite. But we can remove the unwanted
factor (a+bc) in the denominator, and so obtain the logarithmic
comparison series 2 % n /n, from which the same result follows.

Gauss's Evaluation of F(a, b\c\ 1). Consider the relation between
contiguous series

(c-a)(c-b)xF c , + -c(c-a-b)xF = c (cl)(lx)(F c _F). (6)
If R(cab) > 0, both F and F c+ converge absolutely in the closed

102 THE HYPERGEOMETRIC EQUATION Chap. VI, 26

interval (0 ^ x ^ 1). The series F c _ need not converge at x 1; but
at any rate its coefficients tend to zero. If we apply to it the trans-
formation (2) and put a; = 1, the sum of the series on the right is
limc n = 0; by Abel's theorem on the continuity of a power-series

n >oo

we have therefore lim[(l x)F c _] 0, and hence

c(cab)F(a,b;c;l) = (c-a)(c-6)^(a,6;c+l; 1), (7)

and accordingly

--

l

F(a,b;c+n;l) (c+r)(c-a-b+r)

If a, 6, c are fixed, the series F(a, 6; c-\-n\ 1) converges uniformly with
respect to n, when n > n say; since each term except the first tends
to zero, we have

limF(a,b',c+n]l) = 1. (9)

?l >oo

The terms grouped in square brackets in the product (8) give an
expression of the type 1 + 01 J ; so that the product converges,

and may be evaluated like (4) by means of gamma functions. We
thus have finally

-">-{Si&

It is convenient, even when the hypergeometric series is divergent,
to write *-,, Nri/ , x

This remains finite unless the argument of one of the gamma functions
is zero or a negative integer.

Kummer's Continuation Formulae. We can now evaluate directly
the coefficients of the continuation formulae in 23; but we must first
remove all ambiguity regarding the branches of the hypergeometric
function. We shall make a cut along the entire real axis from
x = -co to x co, and consider the branches defined in the upper
half-plane by the folio whig conditions.

(asz->0),

(12)

< am(a;) < TT, TT ^ am(l x) < 0.

Chap. VI, 26 THE HYPERGEOMETRIC EQUATION 103

The coefficients of the formulae

will be found by making x -> 1 and x -> 0.

(i) IfR(c a b) > 0, the first expansions in the table of Rummer's
series give, when x -> 1,

P<*> -> $a, 6, c), PM -> <f>(a-c+l, 6-c+ 1, 2-c),

(14)
and hence

(ii) If R(cab) < 0, the second Kummer series of each function
gives, as x -> 1,

p(a') ^ (l_ :r )c-a-6^( 1 _ aj !__^ 2 C), 1 (16)

I

and hence

a , = (f)(c a, c 6, c), ay/ = <(! ^, 1 6, 2 c). (15 b)

(iii) If R(l c) > 0, we have also, as x -> 0,

The relations now give

a^^(a,6,a+6c+l)+ay^(c---a,c--6,c--a--6+l)= 1, | . Jg ,
oL y (f>(a,b,a-}-b c-\-l)-\-oLy<j>(c a,c~ b,c a b+l) = 0. /
(iv) If .#(1 c) < 0, we have, as x -> 0,

' x l ~ c (f>(a c+1,6 c+l,a+6 c+1), } (19)

and these now give

-M-6,c-a-6+l)-0, (18b)

In any particular case, two coefficients are given directly by (15 a) or
(15b), and the others are found from (18 a) or (18b). But it is easy

104 THE HYPERGEOMETRIC EQUATION Chap. VI, 26

to verify, by means of the definition (11) and of the relation

r(z)r(l z) = TrcosecTrz,

that all the eight relations are consistent. The connexion between
the solutions at x = and x = oo are similarly obtained from the
expansions in powers of x/(x 1) and of 1/(1 x). We thus have
always

<t>(a>b, c) <f>(c a) cb,c) \

<f>(a-c+l,b-c+l,2~c) <(i_ a? i_& ? 2-c) /

<f>(a,cb,c) #6,c a,c) \

e^ l - c ty(a-c+l, 1-b, 2-c) e^-^b-c+l, l~~a } 2-c)j

(20)
27. Hyper geometric Integrals f

Euler's Transformation. An effective method of continuing the
hypergeometric series beyond its circle of convergence is to represent
it by a definite integral, where the variable x appears as a parameter.
The oldest method of representation is due to Euler and has been
developed by many writers. Euler's integral was transformed by
WirtingerJ into one involving theta functions, but we shall confine
ourselves to the classical form. Equivalent results were obtained by
Pincherle, Mellin and Barnes||, using integrals with gamma functions;
but we shall omit these, as they are well known and readily accessible
to English students.

A linear differential equation of order n, whose coefficients are
polynomials of degree m, can in general be satisfied by integrals of
the type -

y = j(u-x)f-^(u)du 9 (1)

c

along a suitable path (7, where is a constant to be determined and
where the auxiliary function (f>(u) satisfies a differential equation of
order (m-\-ri). If, however, the coefficient p r (x) of l) n ~ r y is of degree
(nr), the equation for <^(u) is of order n; and in a very special case
it is only of the first order, so that the equation becomes soluble by

t B. Riemann, Werke, 81-3; (Nachtrdye) 69-75; 0. Jordan, Count d'analyse, 3,
251-63; F. Klein, Hypergeometrische Funktion (1933), 88-111 ; L. Srhlesinger, Hand-
buch, 2(1), 405-524.

J W. Wirtinger, Wiener Berichte, 3 (1902), 894-900; A. L. Dixon, Quart. J. of
Math. (Oxford), 1 (1930), 175-8.

|| E. W. Barnes, Proc. London Math. Soc. (2), 6 (1907), 141-77; E. T. Whittaker
and G. N. Watson, Modern Analysis, 280-5.

Chap. VI, 27 THE HYPERGEOMETRIC EQUATION 105

quadratures. Let Q(x) and R(x) be polynomials of degree n and
(n 1) respectively, and consider the equation

2 ( W 7*) r \ x ) Dn - r y~ y 1 1"", ~~ 1 )^ (r) (^)^ n - r - 1 2/ = 0. (2)

r=0 * ' r=0 * '

On introducing the expression (1) for y and removing the constant
factors (n l)(n 2)...(1 ), we find

(n

-fl J (*-*)*--*[ J ^

==

r==0

- J (-*)*-[ 2 (M ^ # r) (*) W) ^ = 0, (3)

G

or, by Taylor's theorem,

J [(n g)(ux)t- n - l Q(u)(ux)t- n Il(u)]<f>(u) du = 0. (4)

The integrand is an exact derivative, if </>(u) is so chosen that

[QMftu)] = R(u)^(u} y (5)

We then complete the integration and determine the path C from the
condition [(-*)f-Q( W )] = 0. (7)

Hypergeometric Integrals. To identify the hypergeometric equation
with the standard form (2), we must have

Q(x) - x(l-x),

(8)

On eliminating the polynomials, we have a quadratic for the para-
meter , namely

tf+a-l)(f+&-l) = 0, (9)

and the root ^ (1 a) gives

E(x) =- (ac+l) (ab + l)x. (10)

From (6) we now get

<f>(u) = Au a - c (l-u) c - b - 1 , (11)

and the path must be chosen so that

[u-*+i(l--u) e -*(u--x)--*] = 0, (12)

4064 p

106 THE HYPERGEOMETRIC EQUATION Chap. VI, 27

with corresponding results for the second root (16). The inte-
grand of the expression

y

== A f u a -(l u) c - b - l (u x)' a du (13)

has in general four singularities u = 0, oo, 1, x.

It may be shown that the six pairs of singularities each yield a
solution corresponding to one of the six principal branches of the
hypergeometric function (any three being of course linearly con-
nected). If the expression (12) vanishes at both the critical points,
an ordinary integral between these limits may be taken; such integrals
were considered by Jacobi and by Goursat. In the most unfavourable
circumstances it is necessary to consider double-loop integrals inter-
lacing each pair of singularities; these were introduced by Jordan
and afterwards rediscovered by Pochhammer. Let P be any con-
venient point in the w -plane, and let a definite initial value be assigned
to the integrand. The three simple loop integrals about the three
finite singularities may be denoted by

(0 + ) (l-f)

L n = J , L, = / , L x = J . (14)

P P P

A positive loop about u = oo is topographically equivalent to a
sequence of negative loops about u = 0, I, x (taken in the proper
order).

If the integrand is holomorpMc at one of the points u 0, 1, x,
the simple loop integral vanishes, but the point can be taken as an
end point of an ordinary line integral or of a simple loop integral
about one of the other singularities. If it is a pole of the integrand,
the simple loop satisfies the condition (12) and furnishes a solution.
In general, the expressions (14) are not themselves solutions; but any
double loop circuit satisfies the condition (12) and the corresponding
integral can be written in terms of the simple loop integrals (14), for
example

(o+,i-f,o-,l-) (o-f) (l-f) (o-) (1-)

f _|_ g2tV(a-c) f \. e 2in(a-b) f I e 2i7r(c-b) f

P P P P

(O-f) (l-f) (0+) (l-f)

= f _J_ c 2iir(a-c) f _ 6 2i7r(c-6) f f

P P P P

. (15)

Chap. VI, 27 THE HYPERGEOMETRIC EQUATION

107

The Forty-eight Eulerian Forms. If we put u = l/t, we obtain
Euler's own forms

(16)

These are respectively valid if (b y c b) or (a, c a) have positive real
parts and if x has any value not lying in the real interval 1 ^ x ^ oo;
when the conditions are not satisfied, an appropriate double or simple
loop contour must be substituted.

Each of the integrals (16) belongs to a set of twenty-four, obtain-
able by linear transformations of the integrand interchanging the
points (t 0, oo, 1, l/x) in all possible ways. For example, the
involutions

xtt'-t-t' + l = 0,

(17)

interchange the points in pairs and give four forms equivalent to the
first of the pair (16), namely

i

J t b -*(l-t) c - b - l (l-xt)~ a dt,

(1 a;)c-a-b J V~b-l(l-t)l>-l(l-xt)- c dt,
o

00

x l - c (l-x) c - a ~ b t-(t-iy- c (xt-l)*>- 1 dt,

OO

x i~ c f t a - c (t I)- a (xt i )'-&-! dt.

(18)

The other transformations lead to Eulerian integrals where x is
replaced by one of the expressions

* x / 1 f\\

, --. (19)

x xl

But it was pointed out by Biemann that there is no elementary
method of transforming the two integrals (16) into one another; he
suggested that both could be derived from the same multiple integral,

* i *

-, 1x, ~,
x lx

108 THE HYPERGEOMETRIC EQUATION Chap. VI, 27

evaluated in different ways, and the required integral was afterwards
found by Wirtinger.

Deformation of Contours. An expression for the hypergeometric
function which is valid in the entire plane, cut along the real axis
from x = I to x = +00, is given by

J t b -\l-t) c -*>- l (l-xt)- a dt

F(a 9 b;c;x) - C - - . ----------- ------ (20)

1

The contour C is in general a double loop interlacing t = and t 1 ;
but in special circumstances it may be replaced by a simple loop,
or a figure of eight, or an ordinary line of integration between the
critical points.

Suppose now that the point x crosses the cut in the plane of x, and
describes a complete circuit about x 1. In the plane of /, the
ningular point t = l/x describes a circuit in the same sense about
t = 1. The new form of the solution (20) is found by deforming the
contour of integration so that it never passes through a singular
point of the integrand. To fix ideas, let the starting-point P lie on
the real axis in the interval (0 < t < 1), and let the initial amplitudes
of t, (I/), and (1 xt) all tend to zero, as t -> in that interval. We
consider the contour integral

(1 + ,0 + ,1-,0-) (l-f) (0 + )

J = [1-e 2 ^] J -[l-e 2 >^>] J . (21)

p p p

After the point t = l/x has described a circuit counter-clockwise

(0 + )

about t = 1, the simple loop integral J is unchanged. But the

deformation of the contour of the other integral is indicated by the
figure

o ? b a

01

Fia. 1
and the new value of the integral is

(l-f ,!/ + , l-Kl/sc-,!-) (1-f)

f = f +e a*r(c-&> . (22)

p p p

Chap. VI, 27 THE HYPERGEOMETRIC EQUATION 109

Hence the integral (21) becomes

(l-i-,0+,1-,0-) (l/a?+,l+,l/a:-,l-)

J + [ e 2M C -&)_ e 2*Vc] J , (23)

P P

and the second double loop integral may be shown to be a multiple

EXAMPLES. VI
1. Verify that

log(l+.T) xF(l, 1;2; x). [GAUSS.]

2. Verify that H (] { r)n _ Iim2j p(~n,6; 26; -x),

b->0

e x --= limF(l,b;I;x/b),
cosh a? = lim F(a, b; J;# 2 /4a&),

a, &~>oo
sirihtf = lim xF(a,b;$;x 2 /4ab). [GAUSS.]

3. Verify that .^u,.^^ J"'

6->oo

o^cja;) = lim F(a,b;c;x/ab). [KuMMEB.]

a.5->oo

4. By means of the equation D^y-^-n^y = 0, show that

( i^ + i i^ + l;i; tan 2 .r),
= cos~ n ~ l x F(^n -{- J, \n -j- 1 ; f ; tan 2 #),

and obtain corresponding forms for cosnx. [GAUSS.]

5. By making n > in the above, show that

= sin # cos # .F( 1 , 1;f;sin 2 o;),

= tan x F(, 1 ; 1 ; tan 2 a;). [G AUSS.]

6. Obtain another solution of the hypergeometric equation satisfied by
F(a, b; 1; x) by evaluating

li- lr

-

7. Show that

= D[x*- l ( l-x) b ~ c + l F(a, b; c; x)]. [T. W. CHAUNDY.]

110 THE HYPERGEOMETRIC EQUATION Chap. VI, Exs.

8. HYPERGEOMETKIC POLYNOMIALS, (i) If a is a positive integer,

(2-c) a _ 1 JP T (l~a,l-6;2-c;a?) =
(ii) If (a c) is a positive integer,

c a _ c F(c-a,c-b;c;x) -- x l
(iii) If (ca) is a positive integer,

9. LEGENDRE POLYNOMIALS, (i) If n is a jjositive integer, the equation

(l x*)D*y 2xDy+n(n+})y ---
is satisfied by P n (a;) = D n (x*~ l)/2"n!.

(ii) By operating with (xD n-\-2) or (arZ)-|-w + 3) on

^P^-tajD + n+lK^-nJP^a?) - 0,
show that

(2n+ 1 )P(^) - DP
(iii) Obtain by integration the relations

Jtn+l)P n+1 (a:) - (2n+l)a;P n (a ; )H-nP fl _ 1 (a ; ) - 0.
Verify that these agree with the recurrence formulae of P|f >0) (
10. (i) If n is an integer, Legendre's equation is satisfied by

(ii) By examining the partial fractions corresponding to the zeros of P n (x),
sliow that

where T^ i _ 1 (a*) is a polynomial expressible in the form

11. If n is a positive integer, show that
P M (.*)= ^(n-fl,-n;l;i

PM- (2n)! X-F( n x " n a n -

Px " X ^~ ~ "~ n

;) + wg n _ 1 (a:) - 0.

12. ASSOCIATED LEGENDBE EQUATION. By means of the scheme of 25, (4),
show that the associated Legendre equation is soluble in finite terms if n or

Chap.VI,Exs. THE HYPERGEOMETRIC EQUATION 111

(mn) is an integer; by the quadratic transformation show that it is also
soluble if (m-f ) is an integer.

13. (i) RODRIGUES'S LEMMA. If (n^m) are positive integers, show that the
associated Legendro equation is satisfied by the equivalent expressions

J ._.

(n-f-w)! (n m}\

(ii) SCHKNDEL'S LEMMA. If n is a positive integer, show that the associated
Legendre equation is satisfied by the equivalent expressions

(r_J.

J/~~

14. TSCHEBYSCHEFF'S POLYNOMIALS. If T w (cos#) ^ cosn^, show that

+ n*T n (x) = 0,

15. JACOBI'S RELATIONS. If C7 /t (cos ^) = - -.,- , show that

sin u

n

(n+D

16. If n is a positive integer and (01 -\-fi-\-n-\-\) --- 0, show that

17. RECURRENCE FORMULAE. Show that

by eliminating DPfr(x) t obtain a recurrence formula connecting three succes-
sive Jacobi polynomials.

112 THE HYPERGEOMETRIC EQUATION Chap. VI, Exs.

18. KUMMER'S QUADRATIC TRANSFORMATIONS. Show that

19. Showthbt

8;a+j8;4(*3-a;)V{(l-^^

20. DIHEDRAL EQUATIONS. If //, ^ 0, show that

/O LL

/O i^t
H - 1

= l/i Vj+V(a;-l)

-

(x = cos 2 ^).

If two exponent-differencos are halves of odd integers, the P-function can be
written in terms of P(J,/i, J,a?) and its derivatives.

Chap.VI,Exs. THE HYPERGEOMETRIC EQUATION 113

21. If p is an integer, Elliot's limiting form of Lamp's equation

d^U

555 + [k z -p(p+l)cosee*e]u -
has the finite solutions

The associated Logendro equation can bo reduced to the above by either of
the transformations

(i) x = icotO, y u, m 2 == k 2 , n p;

[Z. V. Elliot, A eta Mathematica, 2 (1883), 233-60. See also Journal

London Math. Soc. 5 (1930), 189-91.]
22. (i) Show that

satisfies the equation

and obtain other solutions by interchanging (m, 1 m) or (n, \ n).
(ii) Ifu mn is a general solution, verify the recurrence formulae
d

m.l

(iii) If A; 7^ 0, show that

w m> n =
where F(z] is a rational fiuiction of the type

F(Z) - A w _ 1 ^- 1 +^ m _ 2 ^- 2 +...+^ +...+^ 2 _ n 2 2 - w + ^ 1 -n^-^

[G. Darboux, Theorie generale des surfaces, ii. 207-13.]

23. If the exponent -differences at x and x = 1 are halves of odd
integers, the hypergeometric function has two solutions whoso product is a
rational function of x. If in addition the exponent-difference at x = oo is
rational and has the denominator N, there are N solutions whose product is
merely multiplied by a constant after any closed circuit.

[Transform to Darboux 's equation and consider the expressions
^(cot B)F( - cot 6), e lNko F*(cot 9) e -* Nke FX( - cot 0).]

24. If the hypergeometric function has two distinct branches whose product
is merely multiplied by a constant after any closed circuit, the branches become
multiples of themselves or of one another after any circuit. Deduce that
either (i) there are two branches expressible in a finite form, or (ii) two of the
exponent-differences are halves of odd integers.

[A. A. Markoff, Math. Ann. 28 (1887), 586-93; 29 (1887), 247-58;
40(1892), 313-16. Comptes rendus, 114 (1892), 54-5. E. B. van Vleck,
American J. of Math. 21 (1899), 126-67. E. G. C. Poole, Quart. J. of
Math. (Oxford), 1 (1930), 108-15; 2 (1931), 90-6.]

4064 Q

114 THE HYPERGEOMETRIC EQUATION Chap. VI, Exs.

26. (i) Show, by means of the relation 2 = 1, that Riemann's equation
cannot have two solutions expressible in finite terms if the exponent -differences
are not integers.

(ii) If a, 6 are integers of opposite sign and c is not an integer, verify that
the finite expressions F(a,b;c;x) and x l ~ c (l x) c - a - b F(l~a, 1 -b;2-c;x) are
branches belonging to the same exponent at infinity.

(iii) If a, 6, c are integers and > a > c > 6, the complete primitive is a
polynomial. Show that any solution vanishing at x and x -~ 1 must
vanish identically. Examine the equation

#(1 x)D*y (3-lx)Dy 12y = 0.

26. LOGARITHMIC SINGULARITIES. Show by the methods of 26 that

27. GROUP OF LEGENDRE'S EQUATION, (i) When n is not an integer, two
independent solutions of Legendre's equation are

Vl = P n (x) =

(ii) lim

(iii) The transformations corresponding to positive circuits about x = b 1 are

1 ^-- 2/i \ Y i ^ 2/i + (2iin7r/i)2/ 2 ,

2 = (2isin7rn)2/ 1 + 2/ 2 , / ^ 2 == 2/ 2 -

Y 2

(iv) The transformation corresponding to a negative circuit about x oo,
or to positive circuits about x = 1 and a; = 1 in turn, is

(v) The characteristic equation is

1 A, 2ishi7m _

2i sin Tin, 1 4sin 2 7m A
and the multipliers A = exp( = t2i7rn) are unequal unless n is half an odd
integer; in that case the invariant factors are [(A-j-l) 2 j 1] and the singularity
x oo is logarithmic.

(vi) From the formulae of Ex. 10 above, write down the corresponding
transformations of two linearly independent solutions, when n is an integer.

28. SCHLAFLI'S INTEGRALS. Sol ve Legendre's equation by means of Euler's
transformation, and hence show that, when 1 1 x\ < 2, P n (x) can be expressed
in either of the forms

U+.a+)

/

Chap.VI,Exs. THE HYPERGEOMETRIC EQUATION 115

where P is a point t = t > 1 on the real axis and initially |am( - x)\ < TT.

Show that the integrals can be evaluated as residues when n is an integer,
and deduce Rodrigues's formula.

29. If R(n-\-l) > Oand \x\ > 1, show that

/n+1 n + 2 1

30. If P n (x) is defined by the first integral in Ex. 28, show that, after x

<l + .as + )

has described a positive circuit about x = 1 , the integral J becomes

p

(l + .aj + .-l + .as + .-l-.z-)

J ; and show that the latter gives the value

p

o
in agreement with Ex. 27.

3 1 . ASSOCIATED LEGENDRE EQUATION. Show that branches of the associated
Lcgondre function are represented by the Eulerian integrals along suitable

contours (*- !)*/ J (_ \)(t-xr*--* dt,

(x-\-l) m l 2 (x~l)- m l 2 J (t- l} n + m (t+ \} n - m (t-x)- n ~ l dt.

c

Show that alternative forms are obtained by interchanging (n+1, n) or
(m, m). When can these be evaluated as residues in finite terms ?

[Cf. E. W. Hobson, Spherical and Ellipsoidal Harmonics (1931), ch. v.]

32. COMPLETE ELLIPTIC INTEGRALS. If A/ 2 = (1 k 2 ), show that the com-
plete elliptic integrals of the first kind

wa

K --- J ( 1 - & a sin 2 0)-* dO - i7r^(i, i; 1 ; k 2 ),
o

K' - jV-A^sin^-'etf- \irF(^ i; l;Aj /a ),
o

are both branches of PI ![ & 2 1. Show that this is equivalent to a

Logendre function of order J. By putting t 2 ~ sin 2 0-f A;~ 2 cos 2 ^, show that

ilk
K' = J (-l)-*(l

116 THE HYPERGEOMETRIC EQUATION Chap. VI, Exs.

33. The complete elliptic integrals of the second kind are given by

77/2

E = J (l-Fsin 2 0)* dO
o

W2

E' = f (l-fc /2 sin0)* dS =

By means of the relations between contiguous functions, show that

E = kk'* ^ -f fc' 2 K, #' = k'k* ~
ak dh

By means of Abel's relation, show that the expression
EK'+E'K-KK' =

where x = k 2 , is a numerical constant. By examining the asymptotic form
of K, K? as x -> 1, show that

Hence obtain Logendre's relation

EK'+E'K-KK' = ITT.

34. (i) Two independent solutions of the equation of the periods in elliptic
functions , x(l-x)D*y+(l-2x)Dy-~y =

are y l =f(x) = K and y 2 = if (I x) = iK' t

i

where K = $ J H( 1 - )-*(! -a*)-* *.

o

17*

i

(ii) Show, both by the method of power-series and by the method of de-
formation of contours, that the transformations corresponding to positive
circuits about x and x I are

(iii) If z =z: 1/2/i/i, show that after any closed circuit z undergoes a trans-
formation of the modular group

z^ +b

cz-\-d
where a, d are odd and 6, c even integers, and (ad be) = 1.

[J. Tannery and J. Molk, Fonctions elliptiques, iii. 188-214; E. Picard,

Traite d'analyse, iii. 358-64.]

35. LOGABITHMIC SINGULARITIES, (i) Any hypergeometric function with
a logarithmic singularity at x is expressible in terms of the derivatives of

x\

o b 1-0-6 x r

Chap.VI,Exs. THE HYPERGEOMETRIC EQUATION 117

(ii) The branch F(a, 6; 1 ; x) is a multiple of the Eulerian integrals
(0+,aH-)

p

(0\.x+)

or J u b -*(l u)~ a (u x)~ b du.

(oi-.iuo-.i-)
(iii) The integrals f or their degenerate forms (when they vanish

p
identically) give the branches

x- a F(a,a-,a 6; l/x) and x~ b F(b,b ',b-a; l/x).

[Cf. W. L. Ferrar, Proc. Edinburgh Math. Soc. 43 (1925), 39-47. The
notation has been altered to conform with the text.]

VII
CONFORMAL REPRESENTATION

28. Schwarz's Problemf

P-Functions whose Group is Finite. THE cases where the hyper-
geometric function is algebraic, or has only a finite number of
branches, were enumerated by Schwarz in 1873. His methods and
some of his results had been anticipated by Riemann in his lectures
in 1858-9, but these remained unpublished until 1902.

If only a particular branch is algebraic, the complete primitive
being transcendental, that branch must be merely multiplied by a
constant after any closed circuit; for otherwise we should have two
independent algebraic branches and the complete primitive would
be algebraic. The particular branch is accordingly the product of
# a (l X)Y and a hypergeometric polynomial, and the equation can
be solved completely in finite terms by means of one quadrature, in
accordance with Jacobi's formula.

If every branch is algebraic, the exponents must be rational. For

after m circuits about x = 0, say, any branch

transformed into {Ae 2i7Tmoi PM+A'e zi7TmOL 'PM}', and, if (a, ex') were

irrational, the number of distinct determinations would be infinite.

We can easily dispose of cases where one of the exponent-differences
is an integer. For we know how to determine whether the correspond-
ing singularity does or does not involve logarithms. If it does, the
solution is certainly transcendental; if it does not, we can make a
transformation of the Riemann equation which reduces the singu-
larity at x = 1 (say) to an apparent singularity, where every branch
is holomorphic. The two principal branches at x either resume
their initial values, or are multiplied by constants, after any closed
circuit (supposing that (a' a) is not also an integer). Accordingly
each of them is the product of a power of x and a hypergeometric
polynomial. If two exponent-differences are integers and the solution
is free from logarithms, we can arrange that x = and x 1 shall
both be apparent singularities, and the complete primitive will then
be a rational integral function.

t H. A. Schwarz, Gesammelte mathematische Abhandlungen, ii. 211-59 ( J. fur
Math. 75 (1873), 292-335) ; B. Riemann, Oesammelte mathematische Werke (Nachtrage),
Leipzig (1902), 67-93.

Chap. VII, 28 CONFORMAL REPRESENTATION 119

Reduced Sets of Exponents. Let A, p, v be any real rational numbers,
but not integers; and let Z, m, n be any integers whose sum is even.
Then we saw in 23 that all the Riemann functions

have the same group. To pick out the simplest function of the group,
we first choose exponents satisfying the conditions

A! A, ui = a, v^~v (mod 2), ) ...

-KA^, !/!<!; / (

and, because P(iA 1} i^i, i"ij#) a ^ have the same group, we may
arrange without loss of generality that

0<A 1 ,/, 1 ,v 1 <l, (2)

integer values being excluded. We now consider the four associated
functions p,^ ^

P(X,.l-u,.I-v v x),

and we can readily verify that at least one satisfies the conditions

0< A ,/x ,v < 1, j ^

A set of exponent-differences satisfying these more stringent condi-
tions is called a reduced set.

We can reduce all the associated P-functions to hypergeometric
functions in such a manner that the branches whose exponents are
zero at x and at x == 1 shall correspond to one another. If
F(a, b\ c; x) is the hypergeometric function with the reduced exponent-
differences, all the others can be expressed linearly in terms of
branches of F(a, b \c\x) and DF = (ab/c)F(a+ 1, 6+ l;c+l; x), multi-
plied by rational functions of x.

Quotients of Solutions. Let us now consider the function defined
by the quotient of two branches of a given P-function, together with
its inverse function

After the description of any closed circuit, z is transformed into

(ad -bc*o), (6)

120 CONFORMAL REPRESENTATION Chap. VII, 28

where (a, 6, c, d) are certain constants, while the inverse function has
the automorphic property

for the same sets of constants. The relations expressing the periodicity
of the circular and elliptic functions are particular cases of such
automorphisms .

If(Z,z) are connected by a linear relation with constant coefficients
of the form (6), we have

(adbc)Dz

___ 2cDz
DZ "" 7te~~"

, .
and hence

rin f , /0 ,

The expression {2;, } = _ - -11 (8)

was called by Cay ley the Schwarzian derivative. As Schwarz pointed
out, it had been used by Lagrange and Kummer, and it appears also
in Riemann's lectures.

The property (7) may be written

=' -, (9)

CZ~\~CL j

and we have in particular

(10)

Any function z = <f>(x), which undergoes a linear transformation
with constant coefficients when x describes any closed circuit,
satisfies an equation of the form {z, x} F(x), whose right-hand side
is a uniform function of x. Moreover, if we put (ad be) 1, we

can write the relations between Z --~ and z in Riemann's form

Z(DZ)-i = - ( '

Chap. VII, 28 CONFORMAL REPRESENTATION 121

these show that (Dz)~* and z(Dz)~* undergo a homogeneous linear
transformation after any circuit, and therefore satisfy an equation of
the second order with uniform coefficients.

The Schwarzian Equation. Before considering the quotient of two
solutions of D 2 y-\-p l Dy-\-p 2 y 0, it is convenient to reduce the
equation to the invariant form D 2 y-\-ly = (/ ^ p^^P \~\P\)
by putting y yexp[ J J Pi(x) dx\. We shall suppose this to have
been done. This implies that Riemann's scheme must be taken in
the form H(l-A) -1(1+ M ) 1(1-,) \

\i(l+A) -Kl-ii.) i(l+v) /'
with the equation

Now if we have

z - ?i, IK/i+Ii/i = (t = 1, 2), (14)

we find Dz (y^y\y\Dy^)\y\, (15)

which reduces, by Abel's relation, to

Dz = Cy^ 2 . (16)

Hence (Dz)~* is a solution of (14), and so we have

or {z, x] -27, (18)

as is easily verified.

We can solve completely the equation D 2 y-\-Iy = 0, if we know
any solution of (18). The case 7 = being trivial, the solution z can-
not be a constant; hence, using (9), we see that z~ l is a distinct
solution, and so we have two solutions of (14) given by

which are evidently linearly independent.

Change of Variables. If we put in D 2 y+Iy the forms

we get the equation

^+/* = 0, (21)

where 7* =

4064

122 CONFORMAL REPRESENTATION Chap. VII, 28

Since z ~ y^y^ = u^u^ we have

{z,x} = 21, {z,t} = 21*, (23)

and so, from (22) and (23), we have the relation

(24)

This identity was given explicitly by Cayley, but was used implicitly
in Kummer's investigation of the transformation of the hyper-
geometric equation. By putting t ^ z in (24), we have

(25)

By means of (24), we can prove that, if z is a particular solution of
{z, x} 27, then the most general solution is

Z = (az+b)l(cz+d) (ad-bc ^ 0),
the solution suggested by (9). For we have the relations

(26)

( {Z,*} = 2/ = { Z ,*}; j

hence {Z, z} = 0,

or ' = 0; (27)

dz*\dz) v '

hence _ = (cz+d)~ 2 , (28)

dz

and so Z = a ~ (ad-bc = 1). (29)

CZ~ j (Ji

From these relations and (19) we see that the solution of the equations
D^y+Iy = and {z, x} = 21 are equivalent problems.

Every equation of the second order is formally equivalent to
d z w/dz 2 = 0; for if {(/>i(x), <f> 2 (x)} are known independent solutions, we
have only to put y = ^(xjw and z = ^ 2 ( a? )/^i( a? )-

29. The Reduced Curvilinear Triangle

Conformal Representation. We shall now examine the mapping
of the upper half-plane of # by the quotient of two distinct branches

(i) If the exponents are subject only to the general condition
== 1, we can prove that the relation between z and x is locally

Chap. VII, 29 CONFORMAL REPRESENTATION 123

one-to-one (schlicht), except at the points (x 0, oo, 1). For the
branches [y^x), yz( x )] are holomorphic and have at most a simple
zero at any ordinary point, and they cannot vanish together. Thus
either z or l/z is holomorphic; without loss of generality, let us
suppose that y 2 ^ 0, so that z is holomorphic. Then, by the Abel-
Liouville formula, we have

Hence, at an ordinary point (x = , z = ) we have, by the implicit
function theorem,

*--Ci(z-)+c 2 (z-0 2 +... (2/2^0), (2 a)

or else, when z has a pole,

S-f =C 1 /3 + C 2 /3+... (2/ 2 = 0). (2b)

(ii) We next restrict the exponents to real values, and consider
more particularly the points corresponding to real values of x other
than the singular points. In the typical interval (0 < x < 1) the
principal branches are expressions of the type Ax^l x)yF, where
F is a real hypergeometric series, and we can select two branches
(2/i > 2/2) which remain real in the entire interval. Thus z is also real,
and the formula (1) shows that it is monotonic (say increasing). If
the denominator vanishes at any point, z passes discontinuously
from oo to oo and begins to increase again as x increases, and it
must pass through the value z before it can again become infinite,
so that the zeros of (y l9 y 2 ) must separate one another.

This particular quotient z therefore travels steadily along the real
axis of the z-plane, but it may pass over the same point more than
once; the general form Z = (az-\-b)/(cz-{-d) gives a point which travels
steadily around a circle, but the arc corresponding to the interval
(0 < x < 1) may overlap itself.

(iii) At the typical singularity x 0, wo have in particular

= Mx*[l+c 1 x+c 2 x*+...]. (3)

If (A,jz,v) are real, and x is real and sufficiently small, the series in
brackets converges and takes real values. Hence as x describes the
segments oo < x < and < x < 1 of the boundary of the
upper half-plane of x, the quotient z {} describes segments of two

124 CONFORMAL REPRESENTATION Chap. VII, 29

straight lines enclosing an interior angle TT\. The most general form
of quotient z = y^y^ thus maps the upper half -plane of x upon a
domain bounded by three arcs of circles (which may overlap them-
selves) enclosing interior angles (TT\ TT/Z, TIT) at the points correspond-
ing to x = 0,oo, 1.

Reduced Exponent-Differences. When we have
0<A,/*,v<l, \

< p+v, v+A, A+/I < 1,1 (4)

< A+^+r < f, J

the parameters of the hypergeometric series

a = Kl-A-^-v), b = i(l-A+/x-v), r, = (l-A) (5)
are found to satisfy the corresponding conditions

-K<i 0<b,c-b<l. (6)

Following Wirtinger,f we choose as denominator the branch

i

F(a,b;c;x) = r(b ^-b) J ^-^-'(l-a*)- dt, (7)

'

and we show that this does not vanish in the interval (0 < x < 1),
nor in the plane of x, cut from x I to x -\-co. For the integral
converges when the value of x is not real and greater than unity. If
we determine the amplitude of (1 xt)~ a in the cut plane by assigning
the value zero at x 0, we have, in the upper half -plane either

< am(l xt)~ a < |TT (0 < a < i),
or > a,m(lxi)- a ^ \n (0 ^ a > J).

In either case the real and imaginary parts of the integrand (7) do
not change sign, and so y 2 ~ F(a,b\c\ x) ^ 0. Hence the arcs corre-
sponding to the segments ( oo < x < 0) and (0 < x < 1) do not
overlap themselves; and, by considering another special case, we can
show that the arc corresponding to (1 < x < oo) does not overlap
itself either. Thus as x describes the real axis from oo to oo, with
indentations above the singularities x = Q,l,z describes once counter-
clockwise the boundary of a curvilinear triangle with interior angles
(TrA, TTyz, TTV ), whose sides do not overlap. We can now prove, by a
classical argument, that there is one-to-one correspondence of the

f Summarized by O. Haupt in his new edition of F. Klein, Hypergeomelrische
Funktion (1933), 326-30.

Chap. VII, 29 CONFORMAL REPRESENTATION 125

two domains. For consider the increment of

(27r)-iam(3-) = (2ir)-iam{/(a)-}, (9)

as the points z and x each describe once counter-clockwise the boun-
daries of their respective domains.

The expression on the left increases by unity or zero, according as
does or does not lie within the triangle. The function f(x) = y^\y^
has no poles in the upper half -plane, on account of the way in which
the denominator has been chosen; hence the increment of the expres-
sion on the right is equal to the number of zeros of {/(#) } in the
upper half-plane. Accordingly it has exactly one zero when lies
within the triangle, and none when it lies outside.

The Rectilinear Triangle. If (A+ju,+y) = 1, we have a 0, and
then Riemann's equation is soluble by quadratures,

= A ( x*- l (l-xy~ l dx+B. (10)

J

The branch ?/ 2 ~ 1 fulfils the condition of not vanishing in the upper
half-plane of x\ and the quotient

~ l (lx) v - 1 dx (11)

is the ordinary Schwarz-Christoffel formula giving the conformal
representation of the upper half-plane of x upon a triangle (vrA, 777*, TTV}.

30. Symmetrical Continuationf

Schwarz's Principle of Symmetry. If an analytic function z = f(x)
is holomorphic in a domain intersected by the real axis of x and real
along the segment of the axis, it takes conjugate complex values
z, z at conjugate points x, x in the domain. If we make linear trans-
formations of both planes

az+6 _a'x+b' m

& r~7J -^ -- / T~J/> V 1 /

cz-\-d c x-\-d

we find that iff(x) is holomorphic in a domain intersected by a circle,
and if points on the circumference lying in the domain correspond to
points lying on a circle in the z-plane, then inverse points with respect
to the circle in the a:-plane correspond to inverse points with respect
to the circle in the z-plane. This is obvious if we remember that
inverse points are common points of a family of circles orthogonal

t H. A. Schwarz, Ge#. math. Abhandlungen, ii. 65-83; or J. Jur Math. 70 (1869),
105-20. See, for example, E. C. Titchmarsh, Theory of Functions (1932), 155.

126

CONFORMAL REPRESENTATION Chap. VII, 30

to a given circle, and that the transformations (1) convert circles or
straight lines into circles or straight lines.

Now the function z /(#), which maps the upper half -plane of x
on a triangle of circular arcs ABC, is holomorphic except at the
points x = 0, oo, 1 corresponding to the corners. Hence continuation
into the lower half-plane across the segment ( oo < x < 0) gives a
new triangle of circular arcs ABC', by inversion in AB\ and a return
to the upper half-plane across (0 < x < 1), which completes a posi-
tive circuit about x 0, gives a fresh triangle AB"C' by inversion
in AC'. The two successive inversions give a linear transformation
Z = (az+b)/(cz+d).

1
a-plane

z -piano
FIG. 2. Dihedral configuration (n 3).

Similarly, if we had left the original domain across (0 < x < 1) or
(1 < x < oo), we should have found different representations AB'C
or A'BC of the lower half-plane, by inversion with respect to CA or
EC respectively. Continuing this process, we obtain a pattern of
(say) black and white triangles, corresponding respectively to the
upper and lower half-planes of x. The pattern may or may not over-
lap; it may cover the whole or only part of the z-plane; and in certain
cases the number of triangles may be finite.

Types of Reduced Triangles. The circles EC, CA, AB have a
unique radical centre (which may lie at infinity if the centres are
collinear). If O is exterior to the three circles, they have a real
common orthogonal circle with as centre. The reduced triangle
ABC is found to lie wholly on one side of this orthogonal circle. We
may invert AB, AC into straight lines, and then the arc EC will be
convex to A, and the sum of the angles is less than two right angles,

Chap. VII, 30 CONFORMAL REPRESENTATION 127

that is to say (X+p+v) < 1. We now find that all the successive
images are in the interior of the fixed orthogonal circle, which can
never be filled by any finite number of repetitions. Since the number
of values of z corresponding to any given x is infinite, the relation
z = f(x) cannot be algebraic.

Similarly, if (A+/^+^) = 1> the triangle can be inverted into a
rectilinear one, having an infinite number of distinct repetitions.

A necessary condition for a finite number of repetitions is accord-
ingly that (A+/x+v) > 1. In this case the radical centre O is interior
to the circles. The pattern can be more easily visualized by an
inversion in three dimensions converting BC, CA, AB into great
circles on a sphere. The centre of inversion V must lie on the normal
at to the plane of the figure, and OF 2 must be the power of the
radical centre with respect to each circle, so that OF is the geometric
mean of the segments of any chord through 0. It can then be shown
by elementary geometry that the circles EG, CA, AB are inverted
into great circles on a sphere passing through F. Inverse points with
respect to the circle BC, in the plane figure, are common points of
a family of circles orthogonal to BC. They therefore become the
common points of a family of small circles cutting orthogonally the
great circle BC, in the corresponding spherical figure. The operation
of inversion in the plane thus corresponds to reflection in the plane of
the great circle of the sphere.

Steiner's Problem. We must now find all spherical triangles having
only a finite number of distinct repetitions on the sphere. Every side
lies in a plane of symmetry, and the number of such planes must
be finite, because two different planes of symmetry give different
reflections of a figure. These planes cut out a finite number of spherical
triangles. Let PQR be a triangle of minimum area. If the angle
at P is 770, and 6 is irrational, the number of planes of symmetry
passing through the diameter at P is infinite. If is a fraction in its
lowest terms 6 = p'/p, there are p such planes passing through the
diameter at P; and if p' > 1 we can choose one of them cutting off
a spherical triangle of smaller area than PQR. For a triangle of
minimum area with a finite set of repetitions the angles must be
aliquot parts of TT, say (nip, Tr/q, 7r/r), where (p,q,r) are integers
greater than unity, and

+-

128

CONFORMAL REPRESENTATION

Chap. VII, 30

The number of solutions is limited, and each solution gives the planes
of symmetry of a regular solid. Arranging the numbers in ascending
order, we find the following sets.

Bipyramid:

Tetrahedron:

Cube and Octahedron:

Dodecahedron and Icosahedron:

(2, 2, r arbitrary)

(2, 3, 3)

(2, 3, 4)

(2, 3, 5)

(3)

By picking out all the reduced triangles cut out by the planes of
each configuration, Schwarz enumerated fifteen cases.

Table of Schwarz 's Reduced Triangles

Spherical Excess

No.

A, fa, v

7T

Configuration

1

1

I

1 A -
" 2 'n

n

Regular Bipyramid

jl

A l 1

A 4

III

2 3 3

i. i. *

A

J-24

Tetrahedron

IV
V

*. i> i

M-t

TV = *
i = 2B

Cube and Octahedron

VI

i j i

2 3 6

aV - C'

VII

*. i. i

A = 2C

VIII

f . i t

Jy-26'

IX

*.f*

TV = 3C>

X

-I, i, 1

A- - 4C

XI

I, !, !

t=8C

Dodecahedron and Icosahedron

XII

l.t.*

J- = 6C

XIII

*. 1. 1

t == 66'

XIV

i. !. i

a ? tf - 70

XV

i J. 4

i - JOO

Uniform Schwarzian Functions. In general, when z is the quotient
of two branches of P(\,IJL,V,X), the pattern of triangles covers the
2-domain with an overlapping Biemann surface of many sheets.
A given complex number z corresponds to points differently situated
in different overlapping triangles; and so the inverse function
x (f)(z) is many- valued. If, however, A is the reciprocal of an integer,
equal numbers of black and white triangles fit exactly around the
points corresponding to corners TrA, and a unique value of x corre-
sponds to I/A values of z near such a point, so that x = <f>(z) is locally
uniform. If (A, ^, v) are all reciprocals of integers, x = (f>(z) will be
uniform everywhere. There are a limited number of cases where
(A+/i+v) ^ 1, giving the polyhedral triangles and space-filling recti-

Chap. VII, 30 CONFORMAL REPRESENTATION 129

linear triangles, covering the entire z-plane. There are an infinite
number of uniform Schwarzian functions where (A+^+v) < 1, the
example (J, J, J) being illustrated in Schwarz's memoir. In these
cases the sides of every triangle are orthogonal to a fixed circle or
line, forming a natural boundary beyond which continuation is
impossible.

31. Some Special Cases

The Dihedral Equation. If the reduced exponent-differences are
reciprocals of integers, and their sum is also greater than unity, the
function x = </)(z) is both uniform and algebraic, and therefore is a
rational function. These are the cases I, II, IV, VI of Schwarz's
table.

The first case Pi -,-,-, x] is soluble by elementary methods; for
\2 2 n )

the quadratic transformation

oo

= P-

" -1

1

X

-

I

= P-

2n

to

^~2n

oo 1

1 1 1

2n 2n ']

1 1

271 2iifi

oo

. o v*

1
1_

271

2n

(1)

gives an equation with only two regular singularities, which is equi-
valent to Euler's homogeneous equation. The general solution being

i, ( 2 )

we can choose the quotient

giving the inverse function

The 2;-sphere is divided by the equator and n complete meridians
into 2n pairs of triangles, whose corners form a bipyramid. In

130 CONFORMAL REPRESENTATION Chap. VII, 31

general, 2n distinct values of z correspond to each value of x\ but
these become united in pairs at the corners \TT, along the equator, as
x -> or x -> oo ; and as x -> 1 they are united in two sets of n at
the poles.

The Octahedral Equation. A cube and a regular octahedron are
inscribed in a unit sphere, so that the edges of the cube are parallel
to the diagonals of the octahedron. If the pole of coordinates is at
a vertex of the octahedron, the point z = e^ cot |0 is the stereo-
graphic projection on the plane of the complex variable z of the
point (9, </>) on the unit sphere. The corners of the octahedron are
then given by 2 = 0,oo,l,t. (5)

At the corners of the cube cos# = :tl/V3, and their stereographic
projections are found to satisfy the equation

z 8 +14z 4 +l - 0. (6)

The middle points of the edges correspond to the affixes

(z 4 +l)(z 4 -tan 4 7T/8)(z 4 -COt 4 7r/8) =

or (z 4 +l)(z 4 -17+12V2)(z 4 -17-12V2) = 0,

or (z 12 -33z 8 -33z 4 +l) = 0. (7)

The three fundamental polynomials satisfy the identity

108z 4 (z 4 -l) 4 -(z 8 +142 4 -|-l) 3 +(z 12 -33z 8 -332; 4 +l) 2 ^ 0. (8)

Consider the function x = <f>(z) defined by the hypergeometric
equation with exponent-differences (-J-, |, J). The fundamental spheri-
cal triangle with angles (^TT, ITT, |TT) is one-sixth of an octant, or one
forty -eighth of a sphere, and is bounded by planes of symmetry of
the octahedral configuration. Since each half-plane ofx is represented
on twenty -four different triangles, the equation x <j>(z) is of degree
24 in z. When x 0, the points on the sphere are united in threes at
the corners of the cube; when x = oo, they are united in pairs at the
middle points of the edges; when x = 1, they are united in fours at
the corners of the octahedron. Hence we must have

In order that this may have the root z = when x = 1, we have to
put .4 = 1, and then the identity (8) shows that we can write

(z 8 +14z 4 +l) 3 (s ia -333 8 -332 4 +l) a 108z 4 (z 4 -l) 4
- _ - _

_ - _ ___

Chap. VII, 31 CONFORMAL REPRESENTATION

131

The Tetrahedral Equation. Two adjacent octahedral triangles
(^77, |-77, 77) joined along the sides opposite to the angles 77, make
up one tetrahedral triangle (77, 77, ^77). There are twenty-four such
triangles on the sphere, each of which gives a complete representation
of the #-plane in the octahedral relation (10), but only half of the
plane of x l corresponding to the tetrahedral relation with exponent-
differences (J, \ 9 \). Two adjacent tetrahedral triangles contain four
complete octahedral triangles, and give a single representation of the
plane of x l9 and a double representation of the plane of x. Thus to
every value of x there correspond two values of x l9 and it is found that
we can pass to the tetrahedral equation by putting

*/(*-!) = 4^(1-^). (11)

The transformation of Biemann's equation is as follows

(0 oo 1 \ (1 oo 1

?., x, \ = P| f,

2^-1!

-i *

f

00

1

= P

h

u

~~~24

i

'0

00

i

= p<

&

I*

Hi

\

(2^-1)

4^(1-.^)

(12)

From (10), we have algebraically

(13)1= (2X 1 1)' 1 = -

2 1Z 332 8 332 4 +l'

and on taking the upper sign, we get the tetrahedral relation
(24_|_2^V3z 2 +l) 3 (z 4 2&V32 2 +1) 3 12W3s 2 (2 4 I) 2

x l x 1 l 1

We have the identity

(14)

(15)

which corresponds to the division of the vertices of the cube into
those of two desmic regular tetrahedra, whose edges are the diagonals
of the faces of the cube. The corners of one tetrahedron correspond
to the centres of the faces of the other, and the middle points of the
edges of either lie at the corners of the octahedron. When x = 0,
the twelve roots of the tetrahedral equation x l = <f>i(z) are united in

132 CONFORMAL REPRESENTATION Chap. VII, 31

threes at the corners of one tetrahedron; when x l = 1, they are
united in threes at the corners of the other; when x 1 = oo, they are
united in pairs at the middle points of the edges. The algebraic
identity corresponding to (10) is

(24 +2 tV3z 2 +l) 3 Hz 4 -2W3z 2 +l) 3 12tV3z 2 (z 4 -l) 2 == 0. (16)

The Icosahedral Equation. The triangle (TT, JTT, ITT) corresponds to
the sixth of a face of the regular icosahedron, and to the tenth of a
face of the regular dodecahedron. As there are 120 such triangles on
the sphere, the equation x = </>(z) is of degree 60 in z, with quintuple
roots at the corners of the icosahedron (say x oo), triple roots at
those of the dodecahedron (say x = 0), and double roots at the mid-
points of the edges. With a suitable orientation of the figure, the
equation may be written

1728z 5 (z 10 +llz 5 -l) 5 (Z 20 -228z 15 +494z 10 -f228z 5 +l) 3

1 x

(z 30 + 522z 25 - 10005z 20 - 10005z 10 - 522z 5 + 1 ) 2

(17)

l-x

For full details, reference should be made to Schwarz's memoir or
the writings of Klein, f

Composite and Associated Triangles. The remaining cases of re-
duced triangles giving algebraic functions can be solved by the
method of adjunction of domains, used above to obtain the tetra-
hedral function from the octahedral. The method was used in Rie-
mann's lectures (posthumously published in 1902), but the funda-
mental theorem was more explicitly established by Burns! de; J it has
been applied in numerous problems by Hodgkinson.|| When the
reduced equation is solved, it is possible to solve the associated non-
reduced equations having the same group.

The Modular Function. The limiting case P(0, 0, 0, x) gives a
uniform transcendental Schwarzian function x = <(z), which is
obtained by inverting the quotient of the elliptic integrals z iK'/K,
where x = k 2 . The initial triangle is bounded by two parallel lines

f F. Klein, Vorlesungen uber das Ikosaeder (1884); Vorlesungen uber die hyper-
geometrische Funktion (reprint 1933, edited by O. Haupt). See also A. R. Forsyth,
Theory of Functions, ch. xx; Theory of Differential Equations, iv. 174-90.

| W. Burnside, Proc. London Math. Soc. (1) 24 (1893), 187-206.

|| J. Hodgkinson, ibid. (2) 15 (1916), 166-81; 17 (1918), 17-24; 18 (1920), 268-73;
24 (1926), 71-82.

Chap. VII, 31 CONFORMAL REPRESENTATION 133

perpendicular to the real axis of z and a semicircle touching them on
the axis. The generating transformations of the modular group

correspond to circuits about x and x = 1. The function </>(z) has
an automorphic property analogous to those of the simply and doubly
periodic functions,

for all linear transformations with integer coefficients such that
(ad be) = 1 and a^=d~l,b^EC~Q (mod 2). The function is
the subject of a large literature. f

EXAMPLES. VII

1. CIRCLES ON A SPHERE, (i) If two circles on a sphere cut orthogonally,
their planes are conj ugate with respect to the sphere.

(ii) Families of planes through two fixed lines in space, which are conjugate
with respect to a sphere, cut the sphere in two families of orthogonal circles,
and conversely.

(iii) If two points of a sphere are collinoar with the pole of a circle on the
sphere, their stereographic projections are mutually inverse with respect to
the storeographic projection of the circle.

2. ROTATION. If the stereographic projection of points on a sphere is given
by the relation z = e^cot^O, show that the linear transformation

corresponds to a rotation of the sphere.

3. KLEIN'S PARAMETERS, (i) The above rotation may be written

, _ (d+ic)z (6 id)

where

(a:b:c:d) = (sinj^sinflcos^: sin0sin0sin<: sinj^cosfl: cosje/r).

(ii) The rotation (a, 6, c, d) followed by the rotation (a', &', c', d') is equivalent
to the rotation (A, B, <?,>), where

A ad'-\-a'd'bc f -\-l>'c>
B = bd'+b'd ca'+c'a,
C = cd'+c'd ab'+a'b,
D = aa' W cc'+dd'.

t See, for instance, W. Burnside, Theory of Groups (1911), 372-427; H. Weber,
Algebra (1908), iii; F. Klein and R. Fricke, Elliptische Modulfunktionen (1890-2), i-ii.

134 CONFORMAL REPRESENTATION Chap. VII, Exs.

(iii) Verify that the resultant rotation is given by the quaternion product

Ai+ Bj+Ck+D = (a'i+b'j+c'k+d')(ai+bj+ck+d),
where ij ~ k ji, jk i = kj, ki j ik.

4. RECTILINEAR SPACE -FILLING TRIANGLES. The only rectilinear triangles
whose symmetric repetitions fill the plane without overlapping are those with
angles (^,^,0), (fafafr), (!*,}*, JTT), (fafalir).

The triangle with angles (TT, i?r, JTT) gives a double covering of the plane.

[SCHWARZ.]

5. (i) The relation z arcsinrr xF( t ;f ;x 2 ) maps each quadrant of the
07-plane conformally on a semi-infinite rectangular strip of breadth JTT.

(ii) By the method of symmetric continuation, show that x = sinz is a
uniform analytic function, and obtain geometrically the relations
sin(z-fTr) = sin 2, sin( z) = sinz.

6. The conformal representation of an isosceles right-angled triangle on a
half -plane is effected by the relation

X

z = f[4x(l-xfi~*dx.

This is equivalent to the relation

4x(l-x) = l/p 2 (z),

where $p(z) is the Weierstrassian elliptic function satisfying the equation
'

The relations [p()-e]* = G *~ 0} *. ( a = i, 2, 3)

show that x is a uniform function of z.

[A. E. H. Love, American J. of Mathematics, 11 (1889), 158-71.]
7. The triangle (|7r, JTT, JTT) is conformally represented on a half-plane by
the relation x

This is satisfied by putting

x ^ 1/[1 P 3 (z)], where p /2 (z) = 4p 3 (z)-4. [LovE.]

8. Obtain the representation of the equilateral triangle from the above by
putting x x\, or x = 2i/$o'(z). Obtain the representation of the isosceles
triangle with a vertical angle 2?r/3 by putting x 1 x\, and verify that
x 2 2p*(z)/p x (z) is not a uniform function. Obtain also the Sch warz- Chris -
toffel formulae expressing z in terms of x l and x 2 . [LovE.]

9. REGULAR POLYGON, (i) The triangle whose angles are {?r/2, 7r/n,
7r(n 2)/2n} is conformally represented on a half-plane by the relation

z= cJar*({c-l)< 1 -")/ n dte.

(ii) The representation of the same triangle on a semicircle is given by
putting x = (l-}-t)*/(l t) 2 , or

- !)-*/ eft.

= C' f

Chap, VII, Exs. CONFORMAL REPRESENTATION 135

(iii) The representation of the interior of the regular polygon with n sides on
a circle is given by putting t = w n , or

z s= C" J (w n -l)-*l n dw.

[H. A. Schwarz, Gesammelte tnathematische Abhandlungen, ii. 65-83

(=; J.filr Math. 70 (1869), 105-20).]

10. THE ICOSAHEDRAL TRIANGLE. In tho equilateral spherical triangle with
angles 2;r/5, show that

tana = 2, tan# = 3-V5, sin2r = f, tanp = J(V5-j-l),

3 V6.

The angles a and 2r arc those subtended at the centre by edges of the regular
icosahedron and dodecahedron.

11. Verify that tana --- 2cos2?r/5 = (V5- 1)/2. Hence show that tho
corners of the regular icosahedron can bo stereographically projected into the
points

s = o, oo, e"(6 + e 4 ), e^eH e 3 ) (v = 0, 1, 2, 3, 4),
where exp(2wr/5). Verify that these values (except 200) satisfy the
Cation Z ( z "+n s -l)--0.

12. Verify by spherical trigonometry the expressions given for the stereo -
graphic projections of the corners of the regular dodecahedron and for the
middle points of the edges.

VIII
LAPLACE'S TRANSFORMATION

32. Laplace's Linear Equationf

Form and Singularities. THE equation

[P(D)+xQ(D)]y = I (a r +b r x)D-ry = (1)

r =

is a generalization of the equation with constant coefficients, and is
completely soluble by an artifice of Laplace. If 6 0, we may take
unity as the coefficient of D n y, and the only singular point is seen to
be x = oo, which is irregular. If 6 ^ 0, we shift the origin and write

xD*t/+(a l +b 1 x)D*-*y+... + (a n +b n x)y - 0. (2)

The equation now has one regular singularity at x = and one
irregular one at x = oo. The exponents at x = are

(0, l,2,...,w 2,?& 1 fl^).

If a is not an integer, we find (nl) holomorphic solutions forming
one Hamburger set, and one regular solution belonging to the
exponent (n 1 a ) forming another. If % is an integer, all the
exponents belong to the same set. It is easily verified that there are
always at least (nl) solutions which are free from logarithms and
holomorphic. Since there is only one winding point in the finite
part of the plane, and since (nl) independent solutions are uniform,
the group properties of the equation are trivial. The feature of
interest is the behaviour of the solutions at the irregular singu-
larity x oo.

Laplace's equation furnishes a simple illustration of an interesting,
but difficult, theorem due to Perron. If p (x) is a polynomial of
degree s < n, and {p r (x)} are integral functions of x, the equation

Po(x)Dy+p I (x)D-*y+...+p n (x)y = (3)

has at least (n s) linearly independent solutions, which are integral
functions of x.

Solution by Definite Integrals. Laplace uses the transformation

y = je*f(t)dt, (4)

c

t C. Jordan, Cours ^analyse, iii. 251-65; E. Picard, Traitt & analyse, iii. 394-402.
% O. Perron, Math. Ann. 70 (1911), 1-32.

Chap. VIII, 32 LAPLACE'S TRANSFORMATION 137

which may be regarded as a limiting form of the Eulerian trans-
formation of 27 as -> oo. The equation

je*[P(t)+xQ(t)]f(t)dt = (5)

c
can be integrated exactly if f(t) is chosen so that

or

We must now find n contours giving linearly independent solutions
and satisfying the condition

Suppose first that 6 7^ 0, so that we can use the form (2), where
Q(t) is of degree n and P(t) of degree (n 1). We have in general, if
the roots (J3 r ) are unequal,

(9)

Q

W)

Let / denote a point at infinity, taken in such a direction that xt is
real and negative. Then the condition (8) is satisfied by the simple
loop integrals ( +)

f(t)dt (r=l,2,...,n). (10)

If a r is a positive integer, Z> r EEE 0; but the condition is then satisfied
by the definite integral ^

L* = fef(t)dt. (11)

/

If oi r is zero or a negative integer, am[/()] returns to its initial value
and the contour may be shrunk to a small circle about t = /? r . The
integral can be evaluated as a Cauchy residue. If we expand f(t) and
e xt _ e xp re x(t-p r ) m ascending powers of (tj3 r ) and evaluate, we
obtain an integral function which is the product of e x $ r and a poly-
nomial.

If x is very large and if the contour consists of a small circle about
t = j8 r , together with a straight line to infinity described twice, it
may be shown that the dominant term in the integral is given by the

138 LAPLACE'S TRANSFORMATION Chap. VTII, 32

portion of the contour near t f! r . Thus we find L r ~

and since (fi r ) are supposed unequal, the n solutions are linearly

independent.

If a r , a. 8 are not integers, consider the double-loop integral

L r9 = e**f(t)dt

i

= (l e 2iw ^)L r (1 e 2 *^)^ (12)

Since the double circuit restores the initial amplitude of /(), we may
contract this path to a finite contour interlacing t = j8 r , $,, and clear
of all singularities of the integrand. Since f(t) is bounded and the
contour is of finite length, we can expand e xi and evaluate term
by term; the resulting expression is a power-series converging for
all finite values of a?, i.e. an integral function. Putting r = 1,
8 = 2, 3,...,n, we have (n1) independent integral functions.

If a r or oi s is a positive integer, L rs EE 0; but we can then satisfy the
conditions by a simple loop from one singularity about the other, or
by an ordinary definite integral between t = f$ r and t = j$ s .

For the last solution, we take an infinite simple loop O enclosing
all the singularities t = j$ r . If we enlarge the loop sufficiently, we can
write along the contour

/(O = **-" 2 ^U-*, (13)

r-O

since 2 r = a v We now integrate term by term, using Hankel's
integral (0+)

and find the solution

dt = 2ni f ^ r o: n + r - ct i- 1 /r(w+r-a 1 ). (15)

f

If a x is not an integer, this is regular and belongs to the exponent
(n^l) at the origin.

The expression (15) cannot give a solution belonging to a negative
integral exponent, nor a solution of logarithmic type. In fact the
method fails when 04 is an integer, because the set of solutions (12)
and (15) are not linearly independent. But we may retain one
simple loop integral (10), or one definite integral (11), and the logarith-
mic property can be exhibited by deformation of the path as am(o;)

Chap. VIII, 32 LAPLACE'S TRANSFORMATION 139

increases by 2?r. It appears that the integral in question is augmented
by a multiple of the solution (15).

Exceptional Cases. The appropriate paths when Q(t) is of lower
degree than P(t), or when it has multiple zeros, have been indicated
by Jordan. If Q(t) is of degree (nX), we have

\ 71-A

(16)

The space at t = oo is divided into (2A+2) sectors where f(t) is
alternately very large and very small. We can make a festoon of
(A+l) paths, beginning and ending in sectors where f(t) is small and
crossing the intervening sectors in the finite part of the plane. These
and the (n A 1) double loop circuits interlacing (/3 r ) make up the
n required paths.

Similarly, if t = j8 is a zero of multiplicity A, we have

B(t-)*-\ as t -> 8.

The space about t = ft is now divided into (2A 2) sectors where
f(t) is alternately large and small. We can construct small loops
entering and leaving t = ]8 in sectors where f(t) is small, and crossing
at a finite distance the intervening ones where it is large. This set of
(A 1) loops just compensates for the loss of (A 1) simple zeros of

GW-

33. The Confluent Hypergeometric Equationf

Canonical Forms. Just as Biemann's equation can be transformed
in twenty -four ways into a hypergeometric one, so the equation

ieDy+(A +A 1 x)xDy+(B +B 1 x+B t a*)y = (1)

can be transformed in four ways into Rummer's first confluent hyper-
geometric equation xD , y+ (c _ x)Dy _ ay = . (2)

We write y = x?eP x y', x' == (' /?)#, (3)

where p is either root of the indicial equation

0, (4)

t E. T. Whittaker and G. N. Watson, Modern Analysis, ch. xvi; G. N. Watson,
Bessd Functions (1922), 100-5, 188-93; H. Bateman, Differential Equations (1918),
75-9, 110-15.

140 LAPLACE'S TRANSFORMATION Chap. VIII, 33

and (/?,/$') are the roots of

If jg = j8' 9 we obtain similarly Rummer's second equation

xD*y+cDy-y = 0, (6)

or an elementary one of Euler's homogeneous type. The typical
solutions of (2) and (6) are respectively

q(q+l) 2 ,
2!c(c+l)

~2

Again, the equation

i> a y+(-4 +^i)^+(5 +5 1 ar+^ a * a )y - (8)

can be reduced by putting

y = ePx+^y, x' = &Z+Z, (9)

to one of the forms

D*y+(2n+l-x*)y - 0, (10)

, D*y-9xy - 0, (11)

or to an elementary equation with constant coefficients. The equa-

tion (10) was found by Weber on transforming the equation of the

potential V 2 F ~ to parabolic cylindrical coordinates

It is reduced to Hermite's equation

D 2 y2xDy+2ny (12)

by putting y = e~^ xZ y f ; and then to an equation of Rummer's first

by the change of variable z x 2 .

The equation (11) is satisfied by a definite integral considered by
Airy, and on changing the variable to z x 3 it is reduced to Rummer's
second type 9 ,

Finally, Rummer's second equation is transformed by the change of
variable z = 4Vr and the substitution y e^ z y' into one of his
first type ,

z + (2c-l-z)-~(c-\)y = 0. (15)

Chap. VIII, 33 LAPLACE'S TRANSFORMATION 141

The equations (2), (6), (11), (12) are soluble as they stand by
Laplace's method, and their solutions can also be expressed in terms
of the standard power-series (7). But it is convenient to regard (2)
as the ultimate canonical form. The analogue of Schwarz's form of
the hypergeometric equation is obtained by removing the middle
term. We thus find Whittaker' 's equation

exhibiting the exponents (im) at the origin. The solutions of (16) are
written M ktm (x), where

M kfM (x) = x*+ m e-* x L F 1 [%+m--k\ 2m+l;x). (17)

If fc 0, the equation is invariant when x is replaced by x and is
reducible to Rummer's second form.

Solutions in Power-Series.f Rummer's first equation (2) is trans-
formed into another of the same type by putting

y = e?y f , x = x', a = c'a', c = c f ; \

or y e^-'V, x = x', a = 1 a', c = 2 c'. J

From the standard solution (7) we can write down two alternative
expressions of each principal branch, when c is not an integer. We
thus have Rummer's linear transformation

These four expressions correspond to Rummer's twenty-four hyper-
geometric series; and the relation is the limiting form of Euler's
identity

F(a,b;cix) == (lx) c - a - b F(ca,cb;2c;x), (20)

when x is replaced by x/b, and b -> oo.
The principal solutions of (6) are

when c is not an integer. Rummer's transformation of (6) into (15) is
the analogue of Riemann's quadratic transformation and gives the
identities

; -2), (22)

where p = (c J).

f Cf. Ex. V, 4-7.

142 LAPLACE'S TRANSFORMATION Chap. VIII, 33

Bessel's Equation. If A = B v the equation (1) is invariant
when x is replaced by x. It may then be reduced to Bessel's
equation x*D*y+xDy+(x*-k*)y = 0, (23)

or to an elementary equation of Euler's homogeneous type. This
property is analogous to the symmetry of the associated Legendre
equation

p] [ i-l- j ' = ' (24)

with the scheme

oo 1

Ti+1 \m fi (25)

n |ra
If we write /z. = m/ix and let ra -> oo, we get an equation

x*^+[x*-n(n+l)]w = 0, (26)

d/x

having a regular singularity at x with exponents (n+l,n)
and an irregular singularity at infinity. If we now put w = xty, the
equation is reduced to Bessel's equation of order k = (n-{- J), with
the exponents k at the regular singularity x = 0.

When k is not an integer, the solutions of Bessel's equation are
J k (x) and J- k (x), where

(27)

Let n be positive and let m tend to infinity by real positive values.
We have with Hobson's notation

_ e^ r(n+ro+l)r(j)

-

As m -> oo, we have by Stirling's formula F(7i+m+l) rw m n+1 P(m),

where S= Jim , ; + } ; - (30)

m->oo \ Z J W/

The series being convergent uniformly with respect to m for a fixed

Chap. VIII, 33 LAPLACE'S TRANSFORMATION 143

value of x, we may evaluate the limit term by term and so get

Hence, with k = (ft+i), we have finally

Urn [e--e(|)/r ( m)] = .(^*W). (31)

Finite Solutions. From Rummer's relations (19), we see that one
of the four series is finite, if a or (a c) is an integer. As a typical
case, let a n. As in the proof of Jacobi's formula in 24, we
find for any solution of (2) the relation

D k \tf+ k - l e~ x D k y\ = a k xr- l e- x y, (32)

where a k ~ a(a+ 1 )...(+ k I).

In particular, if y ^ ^(ajcjrr), we get

Z> fc [2^-VVl(0+*; c+&; x)] = c k ^- l e-\F^(a\ c; a?). (33)

If now n is a positive integer, and a n, k n, we get the
expression

c n iFi(n\c\x) = x*- c e x D[x^ n - l e- x ]. (34)

The standard polynomials of this class are known as Sonine poly-
nomials.

34. Integral Representations of Rummer's Series

Standard Forms. To solve Rummer's first equation, we write in
32,(4)-(8),

P(t) = ct-a, Q(t) = t*-t,

and obtain the expression

y = J c ^-i(l<) c ~ a " 1 M, ( 2 )

c

where the contour C must satisfy the condition

ty-] c =0. (3)

Independent solutions are in general given by the simple loop

(o+) (i-f)

integrals L = f and L J , where / is the point xt -co. From

144 LAPLACE'S TRANSFORMATION Chap. VIII, 34

these we construct the combinations

(o-Ki+,0-,1-)

J = J

4> I

f <V + )

J - J - L Q +e****L l9

n I

(4)

which are linearly independent, provided that e 2tVc ^ 1 and that
I/o, L do not vanish identically.

The contour <D may be deformed into a fixed double loop inter-
lacing t = 0,1, and lying entirely in the finite part of the plane. On
expanding e xt under the integral sign and evaluating term by term,
we get

/* r

gxtia-in iY~ a ~ l dt = -,FJ((t\CyX) I t a ~H\ t) c ~ a ~ 1 dt y (5)

J J

<t> *

the integral on the right being a generalized beta function.

The contour D may be enlarged so that \t\ > 1 everywhere; on
expanding f(t) in descending powers of t, and evaluating term by
term by means of Hankel's integral for the gamma function, we

have

(o-f)

X J 6 U J 1 J du

J \ ui

""

(0 + )

-*) ?l J c tt tt c - a - w

-c;x). (6)

&

If a or (c a) is an integer, the contours must be modified. If a is a
positive integer, L Q EEE 0; but t may now be taken as a limit of
integration. The holomorphic solution is obtained as in (5), except
that the double loop is replaced by a simple one from t about
t 1. The other solution is

J e xt t a - l (t-l) c - a ' 1 dt - JDa- 1 !"^ J e*'-^-!)'-"-! dt I,
n '- n J

r (0 + ) -j

^a-l La^a-c J e ^c-a-l ^ 1
L f J

9^

Chap. VIII, 34 LAPLACE'S TRANSFORMATION 145

= rr~^ x^F(a-c+ l;2-c;x),

L (4C)
= p^^f l - CeXF ( l - a > 2 - C ^ X ^ ( 7 )

as may be verified by elementary methods.

If a is zero or a negative integer, L Q can be evaluated as a finite
series of Cauchy residues, and the contour <D gives a numerical
multiple of L Q .

(0 + ) -a (0 + )

J^j A rv>n r
J-4 72,1 I
n-0 ' ^

Similar considerations apply to j^ when (c a) is an integer, and so
we obtain integral representations of the Sonine polynomials. When
a, (ca) are integers of the same sign, x = is an apparent singu-
larity or reducible thereto, and there are two solutions expressible
in finite terms.

If c is any integer and a not one of the integers lying between zero
and c, x is a logarithmic singularity, and the solutions (4) are not
independent, because e 2i7TC = 1. A simple circuit enclosing t ~ and
t 1, and lying in the finite part of the plane, gives the holomor-
phic solution y l = (Jv -[-e 2l7m -i). The solution y 2 ~ is logarithmic.

FIG. 3.

For as am(x) increases from zero to 2?r, am(Z) at / decreases from
TT to TT. As the path of integration swings round and bends to
avoid the singularities, we get the new branch

7 2 =

K-l)y v (9)
whose form shows the presence of a logarithmic term.

146 LAPLACE'S TRANSFORMATION Chap. VIII, 34

Alternative Forms. According as we operate on Kummer's equa-
tion as it stands or on one of the equivalent forms, we have the four
Laplace integrals

* (A)

e* J e-**F- a - l (l-t) a - 1 dt, (B)

(10)
(C)

e x x l-c I e ,-a*t-a(l t) a - c dt, (D)

which correspond to the four Eulerian hypergeometric integrals

1

f ( 1 _

! Hx

(11)

x l ~ c (l-x) c - a ~ b \ (l-xt) b ~ l t- a (l-t) a - c dt. (D)

I/as

The latter all converge and give multiples of F(a,b\c\x), if
1 > c > a > 0. In each set, the pairs (AB) and (CD) belong to the
same sets. Kummer's linear relations 33 (19) are given by putting
t' (lt) in (10); and Euler's relation 33 (20) is similarly given by
putting t' = (lt)l(lxt) in (11).

There is in general a curious reciprocity of paths between the two
pairs. The branch y l is given by the double loop O in (AB) and by
the infinite loop Q, in (CD), and the paths are interchanged for the
branch y%. There is an exception, however, in the logarithmic case,
where we saw that O and D both reduce to the same branch. In that
case only, the two types can be transformed into one another by
elementary methods, so that the same path gives the same branch
for each. Without loss of generality, we suppose that c is a positive
integer. We take the representation (A) and integrate by parts (c 1)
times, the integrated parts vanishing around any admissible contour,

f

f

O

Chap. VIII, 34 LAPLACE'S TRANSFORMATION 147

Now in Jacob! ' s identity

(13)
let us write a = (ac),/3 = a,n (c1); then we have

and so
Hence

f ex i t a-i(i__ t )c-a-i dt ^ ! (-a;)!-* f
la c+1 J

l(a c+1)

Both sides vanish around any closed contour if

(a_l)(a_2)...(a-c+l) = 0, (16)

these values of a giving an apparent singularity when c is a positive
integer.

Kummer's Second Equation. The equation of 33 (6), which is
satisfied by ^(c; x), also admits two alternative solutions of Laplace's

(17)

But these can be transformed into one another by putting t' = l/xt.
The most interesting feature of this solution is the choice of contours.
One solution is given by a loop from xt = -co about t 0, and
another by a small heart-shaped Jordan loop entering and leaving
the origin on the side where E(t) is negative, so that the condition

[ c rf+i//F] c = (18)

is satisfied. The infinite contour and the small loop are interchanged
when we pass from one representation to the other.

35. Bessel's Equationf

Integrals of Poisson's Type. Bessel's equation

SVK* 2 ~*% = o (i)

is reduced to an auxiliary equation of Laplace's type

xD*w+(2k+l)Dw+xw = (2)

t G.N.Watson, Bessel Functions (1922).

148 LAPLACE'S TRANSFORMATION Chap. VIII, 35

by the substitution y = x k w. On solving by the usual rule, we find

(l-* 2 )*-* dt, (3)

where the contour C must satisfy the condition

[e<*(l-* a )*+*] =0. (4)

There is an alternative solution where k is everywhere replaced by
k. If k is half an odd integer, the solutions can be evaluated in
an elementary form; in the one case they are given as residues by
small circuits about t 1, and in the other as elementary definite
integrals between limits t 1, ixt = oo.

In general, an infinite loop from ixt oo enclosing t = ^1 and
a figure of eight interlacing these points give independent solutions
J k (x) and J-k(x) [cf. 33 (27)]. But when k is an integer,

*(*) = (- )*(*).

Except when 2k is an odd positive integer, independent solutions
are always given by Hankel's simple loop integrals

i+

f

J

I

(-1-)

J

(5)

the phases of these integrands being so adjusted that

J k (x) = #HP(X)+HP(X)} = -|^p J eW(<- 1)*-* ctt.

7 (6)

If we choose /<; with the real part positive, these expressions may be
replaced by convergent definite integrals. For example, if x is real

and positive, we can show by putting t ( 1 J that

/ 2 \ig-<(3-lfcw-j7r) f / 97/\fc-l

= s) V+ir J -" M ( 1 -) dM -

(7)

Chap. VIII, 35 LAPLACE'S TRANSFORMATION 149

Asymptotic Expansions. From Hankel's integrals we obtain
approximate values of the Bessel functions, when x is large with a
given amplitude. We shall take x and k both real and positive.
Then, using Cauchy's form of the remainder in Taylor's series, we
write in (7)

(8)
Tx - r(*-r+ f)H \2x p ' ( '

where

Ifp>k,

ivti k ~v-

< 1, since u9/2x is real and positive. Hence

2x

the remainder is numerically smaller than the first term neglected.
On integrating term by term, we have the asymptotic representations

/ 2 \W-***V r(t+r+i)/,-y

(1)V*

\irx)

The series continued to infinity would be divergent, the ratio of two
successive terms being T / - \ > which tends to infinity.

But when x is large the terms at the beginning of the series decrease
very rapidly, and the error committed by keeping p terms is numeri-
cally smaller than the (p+ l)th. A very general theory of asymptotic
solutions was developed by Poincare.f

BessePs Integral. Another way of using Laplace's transformation
is to write z = x 2 in (2) and so obtain the auxiliary equation

. CO

This is satisfied by / / l \

w= explzt~\t k - l dt, (12)

c
along a contour chosen as for Rummer's second equation so that

=0. (13)

t H. PoincarS, American J. of Math. 7 (1885), 203-58; Acta Math. 8 (1886),
295-344.

150 LAPLACE'S TRANSFORMATION Chap. VIII, 35

If we restore y = x n w, z = x 2 and put xt = \u y we have the forms

= Jexp |L--|L^- 1 ^,

which can be obtained from one another by putting u f = 1/u.
Loops from infinity give distinct solutions, except when k is an
integer; we can then use a path from u = to u = oo, if we approach
the limits in such directions that the integrand tends to zero.
For integer values of fc, we have

J k (x) = (~) k J k (-x) = ^ J ^P-"'*- 1 du >
and so we find the generating function

On putting u = e ie in (15), we have

27T

1 j k (x) = JL f exp(ia;8infl-i*tf) dO, (17)

27T J

and on putting 6' = (27T6) and adding, we have Bessel's integral

27T

J k (x) = f coa(xBin6k0) dO. (18)

2?r J
o

EXAMPLES. VIII
1. RECURRENCE FORMULAE. If F == ^(ajcja;), show that

2. Show that Bessel's equation may be written in either of the forms

Hence prove that

Chap. VIII, Ers. LAPLACE'S TRANSFORMATION 151

2k

3. If A; is a positive integer, show that

4. Show that

/ 2 \* / 2 \*

= W ** nx> J -* (X) = W COSa; '

5. Show that, if k is a positive integer

-1

I, j / 1^,\ Jfc i '/*
^ V 2 *" / I

6. Deduce Mehler's expression J Q (x) Km PJ cos -I from Laplace's

n _^oo \ n/
integral w

P n (cos^) = - f(cos^-f ismdcos</>) n d<l>.

o
7. If R(k+ 1) > 0, show that

= pxJ k (ocx)J' k (px)--ctxJ k ((XK)J k (px) f
o

2 jxJ k (ccx) dx = (^ 2 - ~) J|(
o a

If a ^j3 and J k (a.)lotJ'(ct) - J k (P)lpJ'dp), show that

o
8. If A; is a positive integer, prove that Poisson's integral

152 LAPLACE'S TRANSFORMATION Chap. VIII, Exs.

can be transformed into Bessel's integral

277

J k (x) = - f exp(ixaind-ik6) dd
o

by integrating by parts and using Jacobi's formula, Ex. VI, 15. [JACOBI.]
9. If a?, a, (a c -f- 1) are all positive, show that

.a?
f

j

du

V = 11 ;S.

1 (c)

oo

Show that the alternative representations of iF-^a; c; x) as integrals of Laplace's
type are obtained by evaluating the double integral in two ways.

10. SCHEBK'S EQUATION. Show that the equation

D n y-xy =

CO

n r r t n ^ i

by y =^C r \ expL/otf -- dt,

r=o J * n-f-lJ

is satisfied

r=o

o

where o> = exp[2^7r/(M + l)] and 2^=0-

r-=0

[H.F. Scherk, J./wr 3fa^. 10 (1832), 92-7; C. G. J. Jacobi, Werke, iv. 33-4.]
11. HEEMITE'P EQUATION. Show that the equation

D*y 2xDy -\-2ny =
is satisfied by

; i; -a; 2 ),

Obtain integral representations, along appropriate contours, of the types

J 6 2-i r n-l ^ J e^^-l-Jnjj.^^J+Jn rf5> x J e *"--( !-*)* rf 5 .

[C. Hermite, (Euvres, ii. 293-308; E. T. Whittaker and G. N. Watson,
Modern Analysis, 341-5; R. Courant and D. Hilbert, Methoden der
mathematischen Physik, 76-7, 261, 294.]

12. HEBMITE'S POLYNOMIALS, (i) Prove by Jaeobi's method that the equa-
tion is satisfied, when n is a positive integer, by the polynomial

and by e xt H_ n _ 1 (ix), when n is a negative integer.
(ii) Show that

(iii) Prove the orthogonal relations

? Bl / = (m ^ n),

J \ 2 n nl^7r (m n).

Chap. VIII, Exs. LAPLACE'S TRANSFORMATION 153

oo

(iv) Show that </>(x,t) = e 2 *'-*' = ^ .H n (x),

and verify tho above formulae from the relations

13. LAGTJERRE'S POLYNOMIALS, (i) Find tho integral representations of the
polynomials

^ = n , ^.^ 1;!r)-
(ii) Show that

(iii) By means of Lagrange's series, show that

and hence or otherwise prove that

i n+ i(a;)~(2n+l-.a;)L fl (a;) + n 2 Zr fl-1 (a;) - 0.
(iv) Show that

D m L m+n (x) =

n!

[Courant-Hilbert, loc. cit., 77-9; see also E. Schrodinger, Abhandlungen

zur Wellenmechanik (1928), 131-6.]

14. SONINE POLYNOMIALS. A set of polynomials (</> n (x)} are denned by the
generating function r _ ,

(l-t;)-i-*exp U- T = 2 *#(*).

Li VJ WC= Q

Show that

(i) (k+nty^x) = n^ n (rc)-o;0;(a;);
(ii)

(iii)

;:::;:

[N. J. Sonino, Ma^. ^4nw. 16 (1880), 1-80; H. Bateman, Partial
Differential Equations of Mathematical Physics (1932), 451-9; G. P61ya
and G. Szego, Aufgaben und Lehrsatze, ii. 94, 293-4.]

IX
LAMP'S EQUATION

36. Lam6 Functionsf

Ellipsoidal Harmonics. A SYSTEM of confocal quadrics being given by

where ei+e 2 +e 3 = (e t > e 2 > e 3 ),

the analogy of spherical harmonics makes us look for polynomials
in the Cartesian coordinates satisfying Laplace's equation V 2 F = 0,
whose nodal surfaces belong to the confocal system. Since even and
odd terms must satisfy V 2 F separately, we expect to find eight
types of polynomials

*l #2*3 }

!^..^ (2)

where 0^ is an expression of the form (1) with i written for 6.

For any assigned point (o^, x 2 , x 3 ), the equation (1) in 9 has three
real roots (A, jz, v) lying in the intervals

e 1 > A > e 2 > n > e 3 > v. (3)

These correspond respectively to the hyperboloid of two sheets, the
hyperboloid of one sheet, and the ellipsoid of the system passing
through the point, and are called its confocal coordinates. From the

identity 3

a V *i T _ (fl

U - Z e.-O ~ ( ei

we have by partial fractions

--,etc. (5)

On substituting the expressions (4) and (5) for the several factors of
V in (2), the expression V breaks up into a product of three similar
factors y _ C E(\)E(n)E(v), (6)

f G. H. Halphen, Fonctions elliptiques, ii. 457-531 ; E. T. Whittaker and G. N.
Watson, Modern Analysis, 536-78; E. W. Hobson, Spherical and Ellipsoidal
Harmonics, 45496; P. Humbert, Memorial des sciences mathe^matiques, x (1926);
M. J. O. Strutt, Ergebnisse der Mathematik, I, 3 (1932).

Chap. IX, 36

where

LAMP'S EQUATION

155

P(e) = (e _ gi )(e-e t )...(e-9j.

We now make the well-known transformation of Laplace's equa-
tion to confocal coordinates,

2^= <*L + j , *!, (8)

y

4(A-c 1 )(A-e a )(A-c 3 j

This implies that we can find (A, B) to satisfy the relations

Lf (WA) = AX+B!

where /(ff) = ^-eJ^-Cj)^-^) = 4S*- gi O-g a . (11)

On dividing by F, we get from V 2 F = the equation

(13)

The last two equations show that (^4,-B) cannot depend on A; and
similarly they cannot depend on JJL, v, so that they are numerical
constants. Thus E(Q) satisfies a linear differential equation of the
second order

f(0)E' r (0)+tf'(e)E'(0)-(A6+B)E(0) = 0, (14)

which is called Lame's equation.

156 LAMP'S EQUATION Chap. IX, 36

This is an equation of the Fuchsian class having four regular
singularities. At 9 e t the exponents are (0, |), while at 6 oo the
indicial equation is

4p(p+l)-6p-4-0. (15)

But, if n is the degree in the Cartesian coordinates of the expression
V in (2), the corresponding solution (7) belongs to the exponent

\n at infinity; and since this must satisfy (15), we must have

A=n(n+l), (16)

where n is a positive integer.

If n is even, E(9) must be a function of the first or third types,
having (K^ all zero or two of them equal to i; if n is odd, E(B) must
be of the second or fourth types, having one or three of (/c^) equal to f .
In each case it is convenient (following Crawford)f to expand the
polynomial P(6) in descending powers of (0 e 2 ).

Solutions of the First Type. The solution belonging to the exponent

00

\n at 6 oo can always be written E(Q) = 2 c r (Q~ e zfi n ~ r \ but

r^O

this will not jbe a polynomial unless n is an even integer and B is
properly chosen. We write the equation in the form

+ 12e 2 (6-e 2 ) 2 E"+l2e 2 (d-e 2 )E'[B+n(n+l)e 2 ]E+

+f(e 2 )[(0-e 2 )E''+$E'] = 0, (17)

and obtain the recurrence formulae

2(2n-l)c l +[B+n(l-2n)e 2 ]c () = 0, ^

(2r+2)(2n-2r-l)c r+1 +[B+n(l-2n)e 2 + (18)

+ I2r(n-r)e 2 ]c r -lf'(e 2 )(n-2r+l)(n-2r+2)c r ^ - 0. J

The necessary and sufficient condition for E(6) to be a polynomial
of degree m = \n is c m+1 0. For this automatically gives c m+z 0,
c m+3 = 0, etc. Now if c = 1, c r is a polynomial of degree r in B;
and so B must satisfy an algebraic equation of degree (m+ 1). It was
proved by Lam6, and then more simply by Liouville, that the values
of B are real and unequal. For the relations (18), where f'(e 2 ) < 0,
show that (c r ) is a Sturm sequence of polynomials in B, and that
c r _j and c r+1 take opposite signs when c r = 0. No changes of sign
are lost in the sequence as B varies, except when B passes through

t L. Crawford, Quart. J. of Math. 27 (1895), 93-8, 29 (1898), 196-201.

Chap. IX, 36 LAMP'S EQUATION 157

a zero of the last polynomial c m+1 of the set considered. But since
the highest term of c r is a positive numerical multiple of ( ) r B r ,
we see by putting B = i o that c m+l must change sign for (ra+1)
real values of B.

Other Types of Solutions. If an expression of the form (7) is sub-
stituted for E(0) in Lame's equation, the equation satisfied by P(6)
takes the form

4(6-e 2 )*P"+(4n-8m+6)(6-e 2 )*P r -2m(2n-2m+l)(e-e 2 )P+
+ 12e2(0-e 2 ) 2 P''+[(8rc-16m+12^^

+f(e*)[(0-e 2 )P'+Q+2K 2 )P'] = 0, (19)
where m (^nK l K 2 /c 3 ) and

B* = B+n(n+l)e 2 +(2K 2 +2K 3 +8K 2 K 3 )(e 2 -e l ) +

e 3 ). (20)

If we expand in a descending series P(0) = ]T c r (9 e 2 ) m ~ r , we get

r=0

the recurrence formulae

(21)

As before, c m+l is the necessary and sufficient condition for
P(B) to reduce to a polynomial, and this is satisfied for (ra+1)
distinct real values of B.

The Lame functions of any given order and type, corresponding
to distinct parameters (B r ), are linearly independent. For if we had
2 A r E r (0) = 0> we should obtain also

[3 -i

jS*+ra(4ra 8n)e 2 8m % /^eJc = 0,

(2r+2)(2n~2r-l)c r+l +[B*+(m-r)(m-4r--8n)e 2 -
-8(m-r) 2 / c.

for all positive integers Tc\ and this would imply A r = 0.

If n is even, we have %(n+2) independent solutions of the first
type and \n of the third; if n is odd, we have |(n+l) solutions of
the second type and \(n 1) of the fourth. This gives in either
case (2r&+l) ellipsoidal harmonics, which is the same as the number
of independent spherical harmonics of order n. These ellipsoidal

158 LAMP'S EQUATION Chap. IX, 36

harmonics are themselves linearly independent. For if we had an
identity ]T C r V r ~ 0, we could write it in the form

2 C r E r (X)E r (n)E r (v) EE 0;

and, if /i, v are fixed, this is an identity between Lame functions
2 A r E r (X), which has been proved to be impossible.

We note that the same value of B cannot give two Lame functions
of distinct types and of the same order n. For they would have to
be independent principal branches, belonging to distinct exponents
(/c t -) and (J K t ) at each of the points 6 = e t \ and in the one case the
positive integer n would be even, and in the other odd.

Zeros of Lam6 Functions.! The polynomial P(9) cannot have a
double zero; for otherwise the differential equation would show that
it must vanish identically. Accordingly, if P(6 r ) 0, we have
P'(6 r ) ^ 0; and so (19) can be written

If P(6) SB (d-e r )Q r (6) y we have

1 P'(0 r ) _ 2Q' r (6 r ) v 2 .

'~ ~

and so (23) becomes

2 7

2

If any of the roots (6 r ) are complex, let 9 1 be the one with the
numerically greatest imaginary part (say positive). Then every term
of the relation (25) corresponding to r = 1 would have an imaginary
part with the same sign (say negative); and, since this is impossible,
none of the roots can be imaginary.

Again, suppose any of the roots are greater than e x ; if 8 now
denotes the greatest root, every term of the relation corresponding
to r 1 would be positive; and, since this is impossible, no root can
be greater than e v and similarly none can be less than e 3 .

We shall show that, for each value k = 0, l,2,...,m, there is one
polynomial P(6) whose roots are distributed in the following manner

e 1 >6 l >6 2 > ... >6 k >e 2 > 6 k+l > ... >6 m > e 3 . (26)

t E. Heine, Kugelfunktionen (1878), 382; F. Klein, Math. Ann. 18 (1881), 237-46;
T. J. Stieltjes, Acta Math. 6 (1885), 321-6; G. Polya and G. Szego, Aufgaben und
Lehrsbtze (1926), ii. 57-9, 243-5.

Chap. IX, 36 LAMP'S EQUATION 159

For as (O r ) vary in the intervals (26), the expression

is bounded and continuoiis. It therefore has a maximum, at which
the conditions (25) are satisfied. The polynomial

(28)

then vanishes at every zero of P(0), and so must be of the form
(A6-{-B)P(6), and we get a Lame function.

Stieltjes interprets (25) as the conditions of equilibrium of a system
of collinear particles. Three of these have masses (i+/c t -) and fixed
coordinates (e t -), and m have unit mass and variable coordinates
(9 r ). If they repel one another directly as their masses and inversely
as their distances apart, there will clearly be a position of equili-
brium where k of the movable particles lie in one interval and (m k)
in the other.

37. Introduction of Elliptic Functions

Uniformization. We may now drop the physical interpretation,
and consider as a purely analytical problem the solution of Lam6's
equation

(1)

f-^l ' 2 3 i J

Further progress depends on the introduction of elliptic functions,
and we use the Weierstrassian form defined by

00

f dx

x = #(), u = J /f _ 7 _ rJ7 __^__ v (2)

Suppose first that (e t ) are real (e 1 > e 2 > e 3 ). Then (2) is the Schwarz-
Christoffel formula giving the conformal representation of the upper
half-plane of x upon a rectangle in the plane of u. We construct a
Biemann surface covering the ^-domain, with winding points at
x = e v e 2 , e 3 , oo. When x describes a circuit about one of these points,
the analytical continuation of u = u(x) is given by the principle of
symmetry. We thus obtain a pattern of rectangles covering the
-w-plane without overlapping. The path in the w-plane corresponding

160

LAME'S EQUATION

Chap. IX, 37

to any given closed circuit in the #-plane can be determined without
ambiguity; and, conversely, each value of u gives a unique value of
x, so that x = @(u) is a uniform analytic function. We make the

I

FIG. 4.

domains correspond as in the figure, w l and eo 3 being the positive real
and positive imaginary semi-periods of the elliptic functions, and
(w 1 +a> 2 +a> 3 ) = 0. We have >(c^) = e it and from (2) we have

u ~ or* (x -> oo), u Wi = O^xe^] (x -> e t ). (3)
Thus u = is a double pole of fr>(u), and u = t^ is a double zero of
{$p( u )~ e i}- We can express the radicals (x e^ as uniform functions
of u by the well-known formulae

"\ w / v \ w l/

These relations still hold when (e^ are complex.f Each period
parallelogram of sides (2a) v 2o> 3 ) contains two complete pictures of

Fia. 5.

the a>plane. The triangle (0, 2co ly 2co 3 ) contains one complete repre-
sentation of the plane bounded by three curvilinear cuts from
x = 6i to x = oo. If we take any value of x and an initial value of
u within this triangle, then when x describes a circuit about e l9 c 2 ,
or e 3 , u is transformed into (2a} u), ( 2co 2 -i*), or (2^ 3 u), the
reflections of the initial point in the mid-points of the sides of the
triangle.

t For a detailed study see C. Jordan, Coura d? analyse, ii. 413-29.

Chap.IX,37 LAMP'S EQUATION 161

Lame's equation now takes the form

= 0. (5)

The singularities x = e t , with exponent-difference |, have become
ordinary points, by virtue of the relations (4). The point x = oo
corresponds to singularities at the lattice -points u = (modulis
2o) l5 2o> 3 ). At u = 0, we have )(u) ~ u~ 2 i the indicial equation is
now (p-}-n)(p n 1) = 0, the exponents being doubled. If n is an
integer, the exponent-difference (2n-\-l) is also an integer. But since
(5) is invariant when we change the sign of u, one solution contains
only odd and the other only even powers of u. Thus the complete
primitive of (5) is free from logarithms everywhere and is a uniform
analytic function of u.

Special Cases. Suppose first that B has a special value giving the
Lame function

3 ifYi __ x ,. __

I I e -J 7 ^* *" W ^ I! cr^-i-^jcr^ WJ I ,^v

I 1 cr(w)cr(a> v ) J I I I V 2 (u)cr 2 (u r )
i_-i L \ / \ t/ J r = 1 L v y v r; j

where A:^ = or J

This is a uniform function of u having w-tuple poles at the lattice-
points u ^ and n zeros in each period -parallelogram. The eight
types are distinguished by whether they do or do not vanish at
u o>f, or by whether (2a^.) are full periods or semi-periods. In all
cases (f) 2 (u) is an even doubly -periodic function with periods (2co l5 2o> 3 ).
The second solution is now given by the usual formula

y = tf)(u) \ [^(^)]~ 2 du. (7)

To effect the integration, we must resolve the integrand into its
principal parts at the poles. Let u = a be any zero of <f>(u), so that
u = a is also a zero. We must have </>'(a) = 0, or (f>(u) would
vanish identically; and the differential equation (5) gives <f>"(a) = 0.
From this it readily follows that the residue of [<f>(u)]- 2 is zero; and
since [^(0)]~ 2 we get the expression

(8)

162 LAMP'S EQUATION Chap. IX, 37

summed over the n poles in a period-parallelogram. We now obtain
by integration

There are m pairs of identical terms arising from poles other than

U = O)p

As in the case of Legendre's equation, we know in general the
coefficients of the Lame function, but not its factors. But the second
solution can be found by the method of undetermined coefficients,
without knowing the roots of the first. For the expression (9) is of
the form

V =

}, (10)

where P is a known polynomial of degree m, and Q an unknown one
of degree (n m 1), and where L, M are constants. If we substitute
the expression (10) in Lame's equation and remove the factors
IT {$ ) ( u )~ e i}*~ Kt > we have a polynomial of degree (nm) which must
vanish identically. This gives (n m+ 1) conditions determining the
ratios of L, M and the coefficients of Q.

Halphen's Transformation. Every Lame function is an elliptic
function of u, admitting (2a>i, 2o> 3 ) either as full periods or as semi-
periods. If we put u = 20, it is an elliptic function of v with periods
(2^, 2cu 3 ) and is rationally expressible in terms of $)(v) and $>'(v).
In fact the irrationals (x e s )* are removed by means of the identities
of the type

[p(2t;)- Cl ] = [p a ()-2e 1 p()-e?-c ]l e,]/p'(). (11)

The Lame function y = (j>(u) = <(20) is either an even or an odd
function of v, according as n is even or odd. It has four poles of
order n in each period-parallelogram, at the lattice-points v == 0,
coj, a) 2 , o> 3 . If we multiply it by [$>'(v)] n , three of the poles are can-
celled, and we have in all cases an even function of v with poles of
order n at the lattice-points 0^0. The resulting function must
therefore be a polynomial of degree 2n in p(v).

Accordingly Halphen puts in (4)

u - 20, y = [p'(v)]-2, (12)

Chap. IX, 37 LAMP'S EQUATION 163

and obtains for z the equation

= 4[n(n+l)p(2)+B>. (13)
But we know that

and so (13) becomes

S-* $ +<[<*"- W">-*]' = o. (is)

If now we put = p(), we have (with our previous notation)

This is an equation of the Fuchsian class with four singularities; the
exponents are (0,%+J) at each of the points f; e l9 e 2 ,e 3 , and
( 2n,^ n) at oo. Following Crawford's procedure, we may
expand the solution belonging to the exponent 2n at = oo in
the form

and obtain the recurrence formulae

B* CO = o, \

(18)

rKw+J-r^^O (r=l,2,...),
where 5* = +^(1 2n)e 2 .

The necessary and sufficient condition that z should be a poly-
nomial is c 2n+l = 0, which entails c r = (r > 2n). If c = 1, c r is a
polynomial in B of degree r, so that there are (2n-{-l) special values
of B corresponding to the Lame functions of different types. If
< r < 2w+l, and c r for some value of B, then c r _j and
c r+1 have opposite signs, except when r n, when c /l _ 1 and c n+1 have
the same sign. A Sturm sequence of polynomials is, however, formed
by the set

+ C 0> 4~ c l> + C 2>'"> + c >i> c +l> + C w+2> c n+3'"> ( ) ?l+ C 2n-fl (^^)

and we can prove that the values of B are real and distinct.

Brioschi's Solutions. If n is half an odd integer, the exponent-
difference (n-\-\) is an integer and the solution is in general logarith-
mic. But for certain special values of B we have algebraic solutions;

164 LAMP'S EQUATION Chap. IX, 37

these were discovered by Brioschi, but are most simply exhibited
by the analysis of Halphen and Crawford. Putting n = (ra+ 1) in
(18), we have the recurrence formulae

(r+I)(m-r)c r+1 +[B*+3r(2m+l-r)e 2 ]c r - (20)

where B* B w(2m+l)e 2 .
The critical equation of the set is

[B*+3m(m+l)e 2 ]c m -tf'(e 2 )(m+2)c m _ l = 0, (21)

where c m+l does not appear. If the solution is free from logarithms,
the first (m+1) equations must be compatible, so that B satisfies
an equation of order (m+ 1). When this is the case, c and c m+l may
be arbitrarily assigned, and it turns out (owing to the vanishing of
a certain determinant) that c r (r > 2m+l), so that the solu-
tion is a polynomial. The explanation of this is the fact that, if
<f>(u) is a solution of (4), then <j>(u-{-2w> 1 ) is another solution. In terms
of v, these solutions are respectively

(23)

If we put c r = (ez-e^^-e^c^^ (r < 2m+ 1),
c r (r>2m+l),

in (20), it is found that the formulae for (c' r ) are of the same form
as those for (c r ). Thus the one condition satisfied by B gives two finite
and compatible sets of equations for (c ,c 1 ,...,c 2m+ i), and then

38. Oscillation and Comparison Theorems

A General Orthogonal Property. If V v V 2 are distinct solid ellip-
soidal harmonics, we have by Green's theorem

-*

Chap.IX,38 LAMP'S EQUATION 165

the integral being over an ellipsoid of the confocal system and the
derivatives along the outward normal. In confocal coordinates
we have therefore

and on expressing the ellipsoidal harmonics as products of Lame
functions, we get (on the ellipsoid v constant)

J J(

00 - = o. (3)

If we now put (A,/z,,v) = ($)(u), fi(v), $)(w)} y we have w = constant
on the ellipsoid. The factor depending on w does not vanish identi-
cally unless E^v) ^ E 2 (v), which is excluded. Hence we have the
relation

/ J [#()-#()]&()&()&()&() dvdv = 0, (4)

fc>! a 3

between any two distinct Lame functions. On a fixed ellipsoid we
have e > @(u) > e 2 > (p(v) > e 3 , and @(u), @(v) will be real if we
assign to u a fixed real part w l and to v a fixed imaginary part a> 3 .
The range of integration covers the entire surface twice.

Functions of the Same Order. If the functions are of the same
order n, but belong to unequal parameters B v B 2 , we have

and hence by cross -multiplication

b
a

Since Lame* functions of all types admit the periods (4^, 4o> 3 ), we
I (j>i(u)<j) 2 (u) du ^^ (BI T 2 ^ B 2 ). (7)

a

The eight types of Lame functions are characterized by three
boundary conditions of the form

166 LAMP'S EQUATION Chap. IX, 38

according as they do or do not vanish at the lattice -points u *= o^.
For two functions of the same order and the same type, we have
from (6) the relation

J ^(u)^(u) du = Q (i,j =1,2, 3). (9)

Wi

From this we infer that the critical values of B are real (as has
been otherwise proved already). For if the two functions corre-
sponded to conjugate imaginary values of J5, their product would
not change sign on the segment parallel to the real axis joining w 3
and (^3+wj); and the relation (9) would be impossible.

Oscillation Theorems.*)* Upper and lower bounds for the critical
values of B, and further information about the zeros of the Lame
functions can be obtained very simply and directly by the methods
of Sturm and Liouville. The parameters fa) being real and 2 e t = 0,
let </>(u) be any Lam6 function and B the corresponding critical value
of the parameter. We can choose a coefficient c so that Y c</>(u)
shall be real in the interval {e 3 < @(u) < e 2 }, where the imaginary
part of u reinains fixed and equal to a half-period, say I(u o> 3 ) = 0;
we put X = (u o> 3 ) and consider the graph of a point whose
Cartesian coordinates are (X, Y). If d 2 Y/dX 2 and Y have the same
sign, the curve is convex towards the axis of X\ in any interval
where [n(n-\-l)@(u)+B] remains positive, the curve cannot cross
the axis more than once, after which Y, dY/dX, d 2 Y/dX 2 all retain
the same sign and the curve moves steadily away from the axis.
Such a curve cannot represent a real periodic function with the period
4:0) v and so the minimum values of the expression in the interval
must be negative, i.e. [n(n-\-l)e 3 +B] < 0. But similar reasoning
applied to the real curve traced by the point [i(u o^), c'<j>(u)\ in the
interval {e 2 < (p(u) < ej shows that [n(%+ 1)^+1?] > 0. Hence all
the critical values lie in the interval

w(w+l)(e a +e s ) < B < n(n+l)(e 1 +e 2 ). (10)

Now let B i > Bj be two of the (2^+1) critical values, and
[<f>i(u),</>j(u)} the corresponding Lam6 functions. By means of (6)
we can prove that the zeros of <f>j(u) separate those of ^(u) in the
interval {e 2 > fi)(u) > e 3 }, the situation being reversed in the interval

t M. Bocher, Lea M&hodes de Sturm (1917) ; Die Reihenentwickelungen der Potential-
theorie (1894); F. Klein, Oesammelte matheniatische Abhandlungen, ii. 512-600.

Chap. IX, 38 LAMP'S EQUATION 167

(e l > @(u) > e 2 }. For let u = a, b be adjacent zeros of <f>i(u) in the
range

< a w 3 < ba) 3 < co l9 e 3 < g)(a) < $(b) < e 2 . (11)

We can make <f>i(u) real and positive on this segment, and let us
suppose (if possible) that <t>j(u) does not change sign in the interval,
say (f>j(u) > 0. Then the relation

(u)(u) du =

gives a contradiction. For the left-hand side is positive; but at
the end-points $(a) > 0, $(&) < 0, <^.(a) > 0, <f>j(b) > 0, so that
the right-hand side is negative. Thus <j>j(u) must have a zero in the
interval, and a similar argument applies to the segment where
{e 2 < 0(co) < ej.

By considering the two segments, we can arrange in order the
(2n-\-l) Lame functions corresponding to the sequence

(B, >B Z > ... > B 2n+1 ).

For example, if n 2, we find two functions of the first type
[p(^)ifc^] } where 6 is the positive root of (120 2 g 2 ) = 0, and three
of the third type \{$(u} ej{p(w) fy}]*, and the sequence is

(13)

The zeros of solutions of Lame's equation for other than critical
values of B have been considered by other methods by Hodgkinson.f

39. The General Equation of Integral OrderJ

Multiplicative Solutions. We saw in 37 that, if n is a positive
integer but B is unrestricted, every solution of

jg_[n(+l)f)(tt)+B]y = (1)

is a uniform analytic function of u. When u increases by the period
2a> v the equation resumes its initial form; hence if/(^) is a solution,
then /(w+2o> 1 ),/(^+4a> 1 ),... are also solutions. If they are not all
multiples of the same solution, any three of them must be linearly

t J. Hodgkinson, J. of London Math. Soc. 5 (1930), 296-306.

j C. Hermite, (Euvres, iii. 118-22, 266-83, 374-9, 475-8; iv. 8-18; F. Brioschi,
Comptes rendus t 86 (1878), 313-15.

168 LAMP'S EQUATION Chap. IX, 39

connected; and so we have an identity

f(u+4a> 1 )+2f(u+2a> l )+Pf(u) = 0. (2)

Now we have j8 = 1; for when u is changed to (u+2a)^, the Wron-
skian W[f(u+2a> 1 ),f(u)] becomes

TT[/(tt+4ci 1 ),/(tt+2ci 1 )] = pW]J(u+2uj^f(u)}. (3)

But we know that its value is constant, and so the multiplier ft = 1 .
(i) If a 2 9^ 1, the equation (A 2 +2aA+l) = has unequal roots
(^,/Lt" 1 ); and (2) can be written

or

Thus [f(u-\-2a) l )ijr 1 f(u)] and [/(^+2o> 1 ) pf(u)\ are linearly inde-
pendent solutions admitting the multipliers O^,/*" 1 ) respectively for
the period 2^. If one is called <f)(u), the other must be a multiple of
(f>( u); for if c/)(u-{-2a) l ) p^>(u) t then <f)(u-2a) l ) = ^-^(u).
Now consider the solution ^(w+2o> 3 ); we must be able to express

this in the form

</>(u+2a> z ) = a<t>(u)+b<t>(-u), (5)

and we no^ have two alternative expressions for

Since this is a single-valued function and /* 2 ^ 1, we must have
6 = 0; hence ^(^+2ct> 3 ) = v(f>(u) say, and so <f>(u) admits the multi-
pliers (/it, v), and $( u) the multipliers (/*"~ 1 ,v~ 1 ) for the periods
(2a> 1? 2o* 3 ).

(ii) If a 2 = 1, we can find at least owe solution such that
= </>(u). If we repeat the argument with ^(u),^(^+2o> 3 ),
), we obtain either two distinct solutions with unequal multi-
pliers (v,v~ l ) for the period 2o> 3 and the same multiplier for the
period 2ca 1? or at least one solution with multipliers (l 5 dzl) for
the two periods. This is a doubly -periodic function with periods
(40)!, 4co 3 ), in the least favourable case, and so is a Lame function,
which we know how to construct.

There cannot be two independent doubly-periodic solutions. For
then every solution would admit (2o) x , 2o> 3 ) as full periods or semi-
periods. In particular, the solution belonging to the exponent (n-\~ 1)
at u = would be everywhere bounded, and therefore constant, by
Liouville's theorem; and this is absurd.

Chap. IX, 39 LAMP'S EQUATION 169

In general <f>(u) is a doubly-periodic function of the second kind,
and a method for constructing it was published by Hermite. An
easier method, which we here follow, was given in Hermite 's lectures
and independently published by Brioschi.

Product of Solutions. If (y^y^) are any solutions of (1), their
product Y = y\y<i * s found to be a solution of an equation of the
third order

)Y = 0, (7)

or, if x = $)(u), f(x) == 4# 3 g 2 x g, 3 ,

v/ .dY

4[n(n+l)x+B]^-2n(n+l)Y = 0. (8)

ttX

This has four regular singularities, three with exponents (0, 1, 1) at
x = e t and one with exponents (n-{- 1, J, n) at x = co. Now suppose
in particular that Y is the solution <f>(u)<f>(u)\ this is an even doubly-
periodic function having poles of order 2n at the lattice-points
u == 0, and so it is a polynomial of degree n in x $(u), and must
be of the form v

Y = i i c r (x-e^-r. (9)

r=0

We now rewrite (8) in the form
4(x-e 2 )*Y'"+lS(x-e 2 )*Y"+[12-n(n+l)](x-e 2 )Y'-^

+ I2e 2 (x e 2 ) 2 Y"'+36e 2 (x e 2 )Y" 4[B+n(n+l)e 2 ]Y'+
+ f'(e 2 )[(x-e 2 )Y'"+%Y"]^0 ) (10)
and find the recurrence formulae

2 n 3)e 2 B]c Q = 0, \

l +^r-n)[(2n 2 -n-3)e 2 + (11)

' (e 2 )(r-n)(r-n- ^(r-n-l)^ = 0. J
These evidently give c n+l = c n+2 ... 0, and we have the required
polynomial; on resolving it into factors we have

The solution <f>(u) of (1) cannot have a double zero without vanishing

170 LAMP'S EQUATION Chap. IX, 39

identically; and <f>(u) and <f>(u) cannot have a common zero unless
they are multiples of one another; each would then have the multi-
pliers (1, 1) and would be a Lame function, a case which is here
excluded as it has been more simply discussed already.

The Invariant. To complete the solution we require the Wronskian,
which is a numerical constant

W{t(-u),<l>(u)} == <f>(-u}<j>'(u)+<}>(uW(-u) = C. (13)
We observe that

V(u) = i(-uW(u)-<l>(u)<l> f (-u), (14)

and if we choose the notation in (12) so that </>(-{- u r ) = 0,<f>(u r ) ^ 0,
we find <t>'( +Ur ) = -V(-u r ) = <t>(-u r W(u r ) = C. (15)

The 2n zeros of O(it) lying in a period-parallelogram can thus be
divided into two sets, according as they give Q>'(u) = (7; and the
zeros of the one set belong to (f>(u) and those of the other to </>( u).
We now have VW J>'(-u) _C_

' ( )

The expression on the right is an even elliptic function with 2n poles
in each parallelogram and zeros of order 2n at the lattice -points
u ~ 0.

From (15) we see that the residues are ( + !,!) at (u r , u r )
respectively, and so we must have

We now integrate and fix the constant of integration by putting
u = 0, which gives

and on combining this with (12) we have the required solution

= (-)-/Wi ft \ Q( ^\ e^l
1^1 [a(u)a(u r ) J

(19)

40. Equations of Picard's Typef

Commutative Linear Transformations. We now consider more
generally an equation

t E. Picard, J.fiir Math.9Q (1880), 281-302 ; Trailed' analyse, iii. 437-53 ; G. Floquet,

Chap. IX, 40 LAMP'S EQUATION 171

whose coefficients are uniform doubly -periodic functions. Suppose
that each singularity in a fundamental parallelogram has been
examined, and that every solution has been found to be uniform.
If {</>i(u)} is any fundamental system of solutions, then

is another, (p,q) being any integers and (26o l5 2o> 3 ) a pair of funda-
mental periods. In terms of the original system we shall have

(S): h(u+toi) = ay^(tt) (i = 1,2,...,*),
(T):

and because {</>i(u)} are single-valued, the alternative forms of
{^(w+2co 1 +2co 3 )} given by ST and T$ are identical.

Hence the product of the matrices may be written in either of the
forms

c ij = 2 * A* = 2 b ik a kj (i,j - 1, 2,..., w). (3)

A: - 1 & - 1

n

If we choose a different fundamental system ^ =

matrices corresponding to (a^) and (6^) will be (d^a^d^)- 1 and
(dij)(b i j)(d i j)-' L , which are also commutative. If (^) are the roots of
the characteristic equation A(A) = |^j"A8^.| 0, we may suppose
$ reduced to the Jacobian semi-diagonal form

the notation (a i;f ) being retained schematically for the new coefficients.
If A /z x is a root of multiplicity m, we can arrange the reduction
so that

.,n). (5)

We shall now show that the m solutions belonging to the charac-
teristic root jitji of 8 are permuted only among themselves by the
other transformation T.

Consider first the expression ^ 1 (^+2oj 3 ) = ] bytyu), and let b l8

Annales de rticole Norrnale (3) 1 (1884), 181-238; C. Jordan, Cours d" analyse, iii.
287-301.

172 LAMP'S EQUATION Chap. IX, 40

be the last coefficient which does not vanish, so that b ls ^ 0,
b^ = (j > a). Then we shall find from (3) that

and this would give a contradiction if s > m, /x x ^ /x,,. Hence
^ 1 (^+2o> 3 ) is expressible in terms of the first m solutions. Suppose
that this is also true of 2 (^+2a> 3 ),...,^ A .(w+2co 3 ); and that in the
expression for <^ +1 (^+2a> 3 ) the last coefficient which does not vanish
is b( k+1 ) 8 = 0, b (k+1 )j = ( j > s). Then we shall have

and again we find a contradiction if s > m, /z s ^ /^ fl .

We can therefore resolve the fundamental system, first into sets
belonging to each distinct characteristic root of S, and then further
into sub-sets belonging to each distinct characteristic root of T.
The solutions of a sub-set belonging to a particular pair of multipliers
(JJL,V) will then be permuted only among themselves in all trans-
formations of the group.

Construction of a Multiplicative Solution. It is not in general
necessary to construct ab initio a complete system of n distinct
solutions; for if f(u) is any solution, other solutions are given by
/(w+^coj), f(u-\-4:0} l ),..., and we naturally choose as many of these
as are linearly independent as part of our fundamental system. Let
y i ^2/{u-\-2(i 1)0^} (i = 1,2, ...,&) be linearly independent, then if
we have

we begin by reducing to the canonical form the transformation,

(9)

We can form at least one combination g(u) of these solutions, such
that g(u j r2a) 1 ) = i^ig(u); by applying the same argument to
g(u-{-2a) 3 ), </(^+4o> 3 ),..., we can then always find one combination
<j>(u), such that <f>(u-{-2uj l ) = ^^(u) and also </>(u-\-2a) 3 ) = vi</>(u).

It will not be necessary to construct any entirely new solution
until all solutions of the type f(u-\- 2pct>i +2^o> 3 ) have been accounted
for and shown to involve less than n independent solutions.

Canonical Sub-sets. It was shown by Jordan that a sub -set
belonging to the multipliers (JLI, v) can be so chosen that the matrices

Chap. IX, 40 LAMP'S EQUATION

are both of Jacob! 's type

Q

00
v

^ v

173

(10)

where or = TCT.

The theorem is obvious for a sub-set consisting of only one solution.
We assume it true for a set of (k 1), and prove it for one of k solu-
tions. By operating within the sub-set as we did for the system as a
whole, we can find at least one multiplicative solution <f>i(u), such
that (j> l (u+2aj 1 ) = ^(u) and <f> 1 (u+2a) 3 ) = v<f>(u). We choose ^(u)
as the first solution of the canonical sub -set, and retain (kl) of
the others to make up k independent ones altogether. We can now
rewrite the transformations in terms of these solutions. If we ignore
the first row and column of the matrix, we can reduce the trans-
formation of the last (kl) to the canonical form, because the
theorem was assumed true for (kl). When we add these relations
(with the terms in ^(u) on the right) to those expressing (f> 1 (u+2a> l )
and <f>i(u-\~2w 3 ), the &-rowed matrices will be in the required canonical
form.

Analytical Form of the Solution. The two multipliers (ju, v) being
never zero, we can reduce both to unity by writing fa(u) ==
where x( u ) * s an auxiliary function of the form

For the relation cr(u-\-2<jt> i ) = e 2(u+att>v i i cf(u) gives

X(u+2w i ) = x(u)e*"*-#<> (i = 1, 2, 3); (12)

and because 2(77^3 173 o^) = ^7^0, we can choose (a, b) to
make the multipliers corresponding to (2o> l5 2o> 3 ) have any assigned
values (fji, v). If we write

*l*(u) EH {^( w +2o> 1 ) $(u)} and A*^(w) = {0(w+2o> 3 ) ift(u)} 9
the two transformations become
A^(tt) = 0, A Vi(w) = 0,

where the matrices (Ay) and (By) are commutative.

(13)

174 LAMP'S EQUATION Chap. IX, 40

It is evident that ^i(u) is a uniform doubly -periodic function of u\
we shall prove by induction that $ k (u) is a polynomial of degree
(& 1) in {u y (^)}, whose coefficients are uniform and doubly -periodic.
For this purpose, we take the periods in the order given by
~ +^> an d introduce the expressions

ITT ITT

which have the properties

&ot(u) = 1, Afi(u) 0,

A*a(w) = 0, A*/?(M) = 1.
Any polynomial of the type in question can then be written in the
ionn f/*.\ __ \p v* TT D / i ^ i.\ (16)

(17)

Suppose expressions of the required form have been found for
1 (^),...,j^ A; _ 1 (^); at the next stage we are given the expressions

where

__ ^^...(cx-A^+l) Q _j3QS I)...(j8t+l)
ot A . _ , ^- , p k ^

The condition (A^)(B^) = (B^A^) is equivalent to

A*A^ A () = AA*^();
and the relations (18) will be compatible if, and only if,

n$ +1) = njtu-

Accordingly we now write

W)= (( |.g ) n fc , yai ft, (20)

where H M = n|}L 1M or n i>w = !!$_. (21)

All these are given doubly -periodic functions, with the exception of
IT^ 00 , which is indeterminate. But the relations (18) now reduce
to (AII fc 00 = A*II A . f00 ), so that this is also a doubly -periodic
function. The theorem thus holds for the kth function if it holds
for the (k l)th, and the induction is complete.

Chap. IX, 40 LAMP'S EQUATION 175

For the actual construction of the solutions, reference should be
made to works on elliptic functions.

EXAMPLES. IX

1. BBIOSCHI'S IDENTITY. If Y == yy 2 is the product of two solutions of
Lamp's equation, show that the equation of 39 (7) has a first integral

= K -

Show also that

2. Show that, in Jacobian elliptic functions, Lame's equation may be
written in either of the forms

By changing the periods, reduce to Lame's form the equation

3. WHITTAKEB'S INTEGRAL EQUATION. Show that the Lame functions of
order n which are rational in snu satisfy the integral equation

4K

y(u) -- X ( P n (ksnusnv)y(v) dv.

o
[E. T. Whittaker, Proc. London Math. Hoc. (2) 14 (1915), 260-8.J

4. LAME-WANGEKIN FUNCTIONS, (i) Show that Laplace's equation V 2 F
admits solutions of the type V = -rodeos w^W(w,?j), where (-ar,z,(f>) are cylin-
drical and (u,v,(/)) curvilinear coordinates, connected by the relation

and tlrU - 8 * W

and that

(ii) Show that this has normal solutions W = G(u)H(v) if F(w) - snu?,
, or clnz/;; and that O(u] and //(?') then satisfy Lame's equation of order
(w-J).

(iii) In the most general transformation giving normal solutions, show that

[A. Wangerin, Berliner Monatsber. (1878), 152-66; E. Haentzschel,

Reduktion dcr Potentialglelchung (1893).]

5. PENTASPHERICAL COORDINATES. If (S k ) are the powers of any point with
respect to five mutually orthogonal spheres of radii (-R^), show that

176 LAMP'S EQUATION Chap. IX, Exs.

* I (iX) -

X.V.Z

5

fc-1
If V is homogeneous of degree \L in (S k ), show that

[G. Darboux, Le$ons sur les systkmes orthogonaux (1910), 287-93;
Principes de geometrie analytique (1917), 379-404, 462-7.]

6. CONFOCAL CYCLIDES. If x k = S k /R k and if (e k ) are unequal, three surfaces
(6 A,/x, v) of the system

pass through any point.

6 5

If P === 2 e k xl t f(0) --. n (^ ^), prove that

(A-/*)(i>-A)dA 8

16 Z, /(A)

7. Show that V 2 F - is satisfied by V = P-W
all solutions of the equation

cfc-^-Bto - 0.

[A. Wangerin, J. fur Math. 82 (1876), 145-57; G. Darboux, loc. cit.
or Comptes rendus, 83 (1876), 1037-40, 1099-1102; M. Bocher, Die
Reihenentwickelungen der Potent ialtheorie (1894).]

8. THE FLAT RING. Cyclides of revolution of the family

(O^: W--Z) a _ (q2 + m 24. Z 2)2 __

(9-1 e-(l/k z ) ~

Chap. IX, Exs. LAMP'S EQUATION 177

are given by 6 sn 2 w, 9 = Bu 2 iv, where

(m+iz) (a/k')[dn(u-\-w) kcn(u+iv)].
If V 2 F =- 0, show that

and that there are solutions V -ar~*cosm<f>F(u)G(iv), where F(w) 9 G(w)
satisfy ^ ^ 2

[E. G. C. Poole, Proc. London Math. Soc. (2) 29 (1929), 342-54;

30 (1930), 174-86.]

A a

X

MATHIEU'S EQUATION

41. Nature and Group of Mathieu's Equationf

Introduction. JUST as Bessel's equation is a limiting form of
Legendre's, so Mathieu's equation (which is now to be discussed) is
easily derived from Lame's. Let the latter be written in the Jacobian
form j% nl

= 0, (1)

and let n(n-}-l)k 2 = A be kept fixed as Jc -> and n -> oo. In the
elliptic functions 2K > IT and 2iK' -> oo; the singularities, which are
situated at the points u = [2pK+(2q-{~l)iK'} recede to infinity, and
we are left with the equation

?j t -(AsuAt+B)y=0, (2)

which is Mathieu's equation.

In celestial mechanics there occurs the celebrated equation of Hill

+ [0 +2e i cos20+20 2 cos40+...> - 0, (3)

of which Mathieu's is a very special case. This equation was the
occasion for the introduction into analysis of infinite determinants
by Hill, and their subsequent justification by Poincare. In this
problem, the coefficients of the equation are given and we have to
determine the character of the solution. We here confine our atten-
tion to the special case of Mathieu's equation, where the difficulties
regarding convergence are trivial and where the determinants are
of the special type associated with continued fractions.

The equation arose in a different manner in Mathieu's problem of
the vibrations of an elliptical membrane, and in analogous problems
regarding the potentials of elliptic cylinders. We transform the two-
dimensional wave equation

_

dx* dy* c 2 dt 2 v '

t E. Mathieu, Liouville, J. de Math. (2) 13 (1868), 137-203 ; G. W. Hill, Acta Math.
8 (1886), 1-36; H. Poincar6, Lea mtthodes nouvelles de la mecanique celeste (1893),
ii. 228-80; E. T. Whittaker and G. N. Watson, Modern Analysis, ch. xix; P.
Humbert, Memorial des sciences mathernatiques, x (1926); M. J. O. Strutt, Ergebnisse
der Mathematik, i, 3 (1932).

Chap. X, 41 MATHIEU'S EQUATION 179

to confocal coordinates (x+iy) = acosh(g-\-irj), and look for normal
solutions of the form F == e ikci F()G(r)). We then have

( }

and both sides of the equation must be equal to a numerical constant
p, whose value is unknown. F(i6) and G(6) will then both satisfy
Mathieu's equation

^+(p-*Vcos*%==0. (6)

In most physical questions, F must be a single-valued function of
position, and so^> must be chosen so that #(7?) shall admit the period
277. These periodic solutions are called Mathieu functions, and the
attention of writers on mathematical physics has largely been con-
centrated upon them for evident practical reasons.

Group of the Equation. The equation (6) has no singularities
except 6 oo, so that every solution is an integral function of 0. At
the ordinary point = 0, two independent solutions are determined
by the conditions (y ^ Q,y' 0) and (y = Q,y' ^ 0). The equation
being invariant when we change the sign of 0, one solution will
involve only even and the other only odd powers of 6, say

/<,W=/o(-0)> AW = -A(-0). (?)

Now the coefficients of (6) admit the period TT and the solutions are
uniform functions of 9; in accordance with Floquet's theory, we
express the solutions / (^+7r),/ 1 (^+7r) in the form

Since the Wronskian W{f Q (9) ) f l (0)} must be a constant, we find on
changing 6 into (O+n) that

cxS-^y = 1. (9)

We now combine (7) and (8), and obtain the transformation

On repeating this transformation, we must have the identical one

f-ffy. 8-)x /i oy

y (-8), p-fr) \0 l) '

180 MATHIEU'S EQUATION Chap. X, 41

(i) If possible, let these relations be satisfied with (a 8) = 0.
Then /? = y and a = 8 = i 1 ; the relations (10) now reduce to

either / (ir-0) - / (0), /i(*r-0) - A(0), 1

/ y /J\ / //1\ / / n\ f / /\\ I * '

In the one case, every solution is even, in the other odd in 0' ES (0 Jrr).
But this is impossible; for \TT is an ordinary point of (6), and on
taking this as origin we find (by symmetry) that there is always both
an odd and an even solution in 0'. In all cases, therefore, we have

a = 8. (13)

The relations (a 2 j8y) = 1 = (8 2 j3y) now follow automatically
from (9) and (13).

(ii) The multipliers of the transformation (8) are given by

A 2 -2aA+l = 0, (14)

and are unequal provided that a 2 ^ 1. We write them (e iirh ,e~ iirh ) 9
where a = COSTT&. The corresponding multiplicative solutions
{Af Q (0)+B^ l (0)} are given by the condition

B~ \-cc~ Y
since a = 8. We may therefore write them as

f(-e) = yyoW-W^).

Their product ^(0) =/(0)/( 0) is an even function admitting the
period TT, whose construction is the crucial step in Lindemann's
method of solution, based on the Hermite-Brioschi solution of
Lamp's equation. We can easily verify that

F(0) == y/J(fl)-j8/!(fl) (17)

is the only independent expression of the type

such that ,F(0) = F(0) = F(*B).

In Hill's method of solution we use the property that, if the
multiplier e ih>n belongs to /(0), then <f>(0) EEE e~ ihe f(0) is an integral
function of with the period TT. It is therefore a single-valued
analytic function of z ^ e 2iff t whose only singularities are z = 0,oo;

Chap. X, 41 MATHIEU'S EQUATION 181

oo

and so it is expressible as a Laurent series 2 c n zn - W G therefore

n=-oo

have oo

/()= 2 c n tf*+*, (18)

n co

where h is at first unknown. But Hill obtained and solved a trans-
cendental equation for /, from the condition that the series (18)
should converge.

Periodic Solutions. If the coefficients of Mathieu's equation are real,
those of the linear transformation (8) are also real, and the character
of the solution depends on the magnitude of a 2 .

(i) If a 2 > 1, the multipliers (A, A" 1 ) are real and unequal; the
expression /(#+2ra7r) = A 2m /(0) then tends to infinity or zero for
large integers m. Such solutions are called unstable.

(ii) If a 2 < 1, the multipliers are conjugate imaginaries of modulus
unity, and then |/(0+2w7r)| |/(0)| for all integers m; such solutions
are called stable. The values of h satisfying cos rrh a are real, but
not integers (since a 2 < 1).

If h has a rational value h r/s, where the fraction is in its lowest
terms, the solutions both admit STT as a semi-period or full period,
according as r is odd or even. For example, if a 0, h = 1, we have

/o(0+7r) = j8A(0), A(0+7r) = y/ (0) (0y - -1); (19)
and then we have, on repeating the operation,

Ue+tor) = -/ (0), / 1 (fl+2r) = -/!(*), (20)

so that 2?r is a semi-period of every solution.

(iii) If a 2 1, the multipliers are given by (Ail) 2 = 0. From
(9) and (13) we get f$y = ; hence one at least of the solutions admits
7T as a semi -period or as a full period. We find four types of periodic
solutions

\
/

The four types are distinguished by their parity as even or odd

functions of 6 and of (0 ITT).

Ince's Theorem. f It was proved by Ince that there cannot be two
independent solutions of period TT or %TT. Suppose there are two

t E. L. Ince, Proc. Cambridge Phil. Soc. 21 (1922), 117-20.

182 MATHIEU'S EQUATION Chap. X, 41

solutions, in sines and cosines of even multiples of 9 say. Then the
recurrence formulae will be (2 kW-4p)a Q +(kW)a 2 = 0, "

r

b-d I (22)

If we eliminate the middle coefficient we get

271-2 2n ^ 2n+2 > ^3)

Wnere ^2n = ~ 2n 2n+2 2n 2n+2' '

But since both series are to represent integral functions of 6, we
must have, as n -> oo,

2n -> 0, b 2n -> 0. (24)

Hence A 2/l = for- all values of n\ and in particular A = a Q b 2 = 0.
This relation and the relations (23) can only be satisfied if one of the
assumed solutions vanishes identically. A similar proof applies to
series of odd multiples of 9.

Since /?, y cannot vanish together, the invariant factors of the
characteristic matrix are [(A^l) 2 , 1]. The fundamental solutions can
then be reduced to the canonical forms

(25)

where /(#), g(0) are integral functions of opposite parity, with the
same minimum period TT or 2?r.

42. The Methods of Lindstedt and Hill

Continued Fractions. In considering Lame's equation, it was
mathematically convenient to solve first the cases of greatest
physical interest, where there is a periodic Lame function expressible
in finite terms. But when we pass to Mathieu's equation (a confluent
Lame equation of infinite order) there is no simple expression for the
interesting solutions of period TT or 2?r; in the special cases, the
periodic solution is about as intractable as the solution in the general
case, and the associated non-periodic second solution much more
so. We shall therefore begin with the problem as it arises in celestial
mechanics, the coefficients of the equation

g + (00+20! cos 2% = (1)

uu

being known, but the solution not necessarily periodic.

Chap. X, 42 MATHIEU'S EQUATION 183

We write a multiplicative solution in the form

V = e ihe I c^e*"*, (2)

n oo

and obtain the recurrence formulae

eiC 2 n-2 + [0-(*+2) a >2+lC a n + = ( = 0, 1, 2,...). (3)

Following Lindstedt, we use a method applied by Laplace in the
theory of the tides on a rotating globe, and greatly developed by
Kelvin, Darwin, and Hough. f Let us put for brevity

u '2n+2 ~ ' LJ 2n ~f\ V*^

C 2n ^1

then the recurrence formulae may be written

1

Let us choose any large value of n; then if u 2n is not small, u 2n+ %
will be large and the subsequent ratios will tend to infinity.

But we require a solution (2) which shall converge for all finite
values of 0-, and this is only possible if, for all large values of n, we
have approximately

U 2n ~ T~~ "^ ( n ~* )- ( 6 )

^2n

But we can now rewrite (5) in the form

! = r^~ . (7)

^2n ^2n+2

and obtain an expression as an infinite continued fraction

M2n= i L L .... (8)

We now apply the same argument to the terms of (2) with negative
indices; we now find

and since u_ 2n must now become large for convergence, we get the
expansion , ,

Of. H. Lamb, Hydrodynamics (5th ed.), 309-35.

184

MATHIEU'S EQUATION

Chap. X, 42

Finally, we substitute the expressions (8), (10) for u 2 ,u_ 2 respec-
tively in the relation

and so obtain the transcendental equation for h

r \ l l l loJ l l 1 1
LQ== \T~ ~T T "' + r" 7~ 7~~ - r
L/> 2 At AJ J |>-2 ^-4^-6 J

Hill's Determinant. The relations (3) can in general be written

and we obtain an expression for the mth convergent of the continued
fraction u 2n c 27? /c 2rt _ 2 , by using m successive equations (13) and
putting in the last of them c 2n+2m = 0; on solving the modified
system of linear homogeneous equations in (c 2s ) we get

(m) __ a n jK"(w+l,W+W 1)

where

(14)

(15)

If 7i remains fixed and ri -> oo, we have an infinite determinant of
von Koch's type.f We have

2 71

K(n,n-\-m 1)

1

Pn

OW

1

/U

1

Pn'-l

OL n ,

1

K(n,n') = K(n,n f -l)-oi n ^ n ^ 1 K(n ) n f -2), (16)

and this shows that every term in K(n,ri) occurs also in the product

n'-l

Y[ (1 ^r+iPr)) so ^ na ^ ^ ne determinant converges absolutely if

r ~n

2 lWi&l converges. This condition is satisfied, provided that there
is no finite value of n for which L 2n = 0; for, as n -> oo, we have
ot n = j3 n = 0(?^~ 2 ). We may therefore write (8) and the analogous
formula (10) in the form

c an _g

K(n,co) c_ 2n+2 K( oo, i

t H. von Koch, Comptes rendus, 120 (1895), 144-7.

-, (17)

Chap. X, 42 MATHIEU'S EQUATION 185

and we have in general

(-K, -1).
The condition (11) may be written

= 0, (19)

which is the expansion in terms of the elements of the middle row
of the doubly infinite determinant

A(A) == #(-oo, oo) = 0. (20)

This is a special case of Hill's determinant

n<n-Hfc) = a n(n-k) =

Should one or both of the numbers (&o) be an even integer, the
corresponding L 2n 0, and the relation (3) becomes

(c 2 n-2 + C 2 tt+2) = 0.

In one row of the infinite determinant (or in two at the most) the
triad of non-zero elements (a n , I,j8 w ) must be replaced by (1,0,1).
But we can still expand the modified determinant as a finite com-
bination of convergent infinite determinants of von Koch's type.

Evaluation of A(A). Let (m^.), (n k ) be any two sequences of positive
integers tending monotonically to infinity, and consider the sequence
of rational analytic functions of the complex variable h defined as
the determinants

f k (h) = K(-m ky n k ) (k = 1,2,...). (22)

The poles of these functions are the points

A=0J2 (n = 0,l,2,...); (23)

if these are all excluded by small circles of given radius p, we can
show that the sequence {f k (h)} converges uniformly in the rest of the
plane, and so its limit A(^) is an analytic function of h, whose only
singularities are poles at the points (23). Corresponding to {f k (h)}
we construct as non-analytic functions of h the products of positive
terms

= ft {l+KI+l&D- (24)

186 MATHIEU'S EQUATION Chap. X, 42

Now we have always

IAWI

, 2 .

L/VW-AWI

for in each case the modulus of every term on the left occurs among
the terms on the right, which are all positive. Accordingly the uni-

00

form convergence of 2 {KI+IA-Q in any domain, which implies

r= oo

that of the sequence {P k (h)}> also ensures that of the sequence of
analytic functions {f k (h)}, whose limit A(^) will (by Weierstrass's
theorem) be a uniform analytic function of h. Now in this instance
we have ,

..... (26)

0J| V '

Now among the terms |&+2r+0J| ( r = 0, !>)> there is at most
one numerically smaller than unity, which is itself not smaller than
p outside the system of circles. Accordingly we have

uniformly in the said region; and so

A(A) = lim/^A) (28)

is a single-valued analytic function, whose only singularities are
at the points (23). By renumbering or rearranging the rows and
columns of the determinant, we now find

A(-A) = A(A), A(A+2) = A(A). (29)

Thus A(^) is an even periodic function, with the period 2; its values
everywhere are deducible from those in the strip ^ E(h) ^ 1.
But in this strip the expression Z = cos nh assumes once every value
in the plane of Z, cut from oo to 1 and from 1 to oo. Hence
A(A) is a uniform analytic function of cos irk, which is also an even
analytic function of h with the period 2. The points (23) are those
where cosrrh = cos7r0J; and as h approaches one of these points,
one or at most two factors of the product (24) tend to infinity, but

Chap. X, 42 MATHIEU'S EQUATION 187

in such a manner that [COSTT& cosTrOj]^^) remains bounded. A
fortiori, the expression [COSTT& cos 7r0J]A(/t.) remains bounded also;
and so A(^), regarded as a function of COSTT&, has a pole of the first
order only. If K is the residue there, the function

#A) = A(*)-_~ -*-- -v,,., (30)

[cos rrh cos 7T0]

remains bounded for all finite values of h. But if h -> foo, A(7i) -> 1
and the second term in (30) tends to zero; and since <f>(h-}-2) = </>(h),
we see that <j>(h) is everywhere bounded, and therefore is a constant,
by Liouville's theorem.
Hence we have finally

(31)

p - ~ - -^

[COS TTll COS 770 JJ

The coefficient K is determined by substituting any convenient value
of h, such as h 0, which gives

K = [A(0)-l][l-cos7T0*]. (32)

The equation A(&) = is thus equivalent to

sm 2 7Th = A(0)sin 2 i7r0*. (33)

This equation may be read in two ways. In the astronomical problem,
where the coefficients of the differential equation are given, it serves
to determine the multipliers e i7rh of the solutions. On the other hand,
if is not given, it may be read as a transcendental equation to
determine , when the solution has assigned multipliers, for example,
when it admits the period 4-77- .

43. Mathieu Functionsf

Mathieu's Method. We must now examine more particularly the
solutions of period TT or 2?r, and we begin by remarking that, when
0! = 0, the critical numbers become = n 2 and give the solutions
cos n0, sin n9. By the methods of Sturm and Liouville we know also
that, for any given real 1} the critical numbers giving the even and
odd Mathieu functions with n zeros in an interval of length TT must
satisfy the inequality

|0 -n| < 2I0J. (1)

f See (in addition to previous references) E. Heine, Kugelfunktionen (1878), i.
401-15; E. L. Ince, Proc. Roy. Soc. Edinburgh, 46 (1925-6), 20-9, 316-22; 47 (1927),
294-301 ; S. Goldstein, Trans. Cambridge Phil. Soc. 23 (1927), 303-36; Proc. Cambridge
Phil Soc. 24 (1928), 223-30.

188 MATHIEU'S EQUATION Chap. X, 43

Accordingly Mathieu assumes that the solution and its corresponding
critical number can be expanded as power-series in X , and gives
rules for evaluating those series term by term. We write in Mathieu 's
equation ( 42 (1)) a solution of the type

ce n (0) -

(2)

n 11 1 2 > I /g\

Q = W 2_^ j g i i _|_^ 2 02_p ? j

where ri is zero or a positive integer. We now equate to zero the
coefficients of each power of v which gives for the even Mathieu
functions the set of conditions

Ul+n*U l +(A l +2cos20)cosnO = 0,
U" 2 +n*U 2 +(A l +2cos2d)U l +A 2 cosnd = 0,

0,

At each stage we first determine A k by the condition that U k (0) shall
be periodic; and then we determine U k (B) uniquely by the condition
that cos nd, sinnO must appear only in the leading term. Now a
particular integral of the first equation (4) is

TT , m _ cos(n+2)6 cos(n2)0 A l 9sinnB . n
U W- 4 (n+l) 4fa=Ij 2rT ( n > ^

or, = JcosSfl - 1(1+ A^O sinO (n = 1), ' ^

or, = J cos 20 \A^e* (n = 0),

which is not periodic unless

A l = (n^l), or^ 1 = 1 (n = 1). (6)

The complementary function Ccosn(d 6 Q ) need not be added, in
accordance with our rule, because all such multiples of 6 are to appear
only in the first term. We have therefore definitely

(n= 1).
At the next stage we have in general (if n ^ 2)

(tt 4)0

Chap. X, 43 MATHIEU'S EQUATION 189

and the condition for a periodic solution gives

A * = a&=T) (n>2} ' A * = I* (w==2); (9)
and we then have a unique expression

The second term is dropped if n 2; the modified forms correspond-
ing to n = 0, 1 are left to the reader. It is evident that the formal
process can be continued indefinitely, with proper precautions from
the Tith stage when terms in cos( n6) first appear. The same method
is applicable to the odd Mathieu functions. It was pointed out by
Mathieu that the series giving for ce n (0) and se n (0) begin alike,
but differ after n terms. The coefficients of the trigonometrical
polynomials {U k (0)} and {V k (B)} begin to differ as soon as negative
multiples of 6 make their appearance. Various improvements in the
method have been made by Whittaker, and it has been shown by
Watson that the series converge if 0j is sufficiently small.

As X increases, however, the method breaks down from one of two
causes. The true connexion between and 0^8 found by putting
h = or h 1 in Hill's determinantal equation 42 (33). This is
a transcendental equation for , whose roots can be expanded as
convergent power-series in 1? if and only if X is sufficiently small.
Again, in constructing the nth even Mathieu function, we divided
by the coefficient of cos nO. But, for certain values of 1} the coefficient
in question vanishes; and, if we try to equate it to unity, the other
coefficients become infinite. These limitations of Mathieu 's method
have been more or less clearly realized by mathematicians since
Heine, who applied the method of continued fractions; this method
was employed by Lindstedt in the associated astronomical problem,
as we have already seen, and was also employed in connexion with
the solutions of period 2s7r (s > 1) by the writer. The most recent
and complete results relating to the Mathieu functions of period
TT or 277- have been based on the use of infinite determinants or
infinite continued fractions. The former were used by Ince to calcu-
late the critical values of , and the latter by Goldstein for expand-
ing the Mathieu functions and also the associated second solutions.

Continued Fractions. To construct the functions of period TT, we
put h = in the analysis of 42. It is easily verified that, by

190 MATHIEU'S EQUATION

symmetry, we have

C -2r?-2

C -2

and this gives either

or c = 0, c an = -c_ 2n .
For the even solution we get

L c Q +2c 2 = 0,

and so 42 (12) is replaced by the condition

^L =
or 42 (20) by the simply -infinite determinant

Chap. X, 43

(11)

(12)

(13)

(14)

1, 2 ao , 0, 0, ...
i, 1, ft, 0, ...

0, 2 , 1, ft, ...

0.

For the odd solution, we have in the same way
2 c 2 +c 4 = 0,

+c 4 = 0, |

, 2n+2 =Q ( n > 2); J

which give

1 1

or

1, !, 0, 0, ...

2> !> ft >

> a 3> 1> ft>

= 0.

(15)

(16)
(17)

(18)

For the solutions of period 2w we put h = 1 and adopt the more
convenient notation

V= I e***

(19)

Chap. X, 43 MATHIEU'S EQUATION 191

We then have, by symmetry

C 2n+l __. c -2n-l /2Q)

C 2n-l C -2n+l

and so in particular

Ci C,

For the even solution we get now

(1-^)^+03=0,

C 2n-l L 2n+l C 2n+l+ C 2n+3 = (^ ^ 1),

giving the condition j ,

1^-JL J- .... (23)

Ai A>
For the odd solution we get

= 0,

(24)

giving the condition

0= l + j^-.-L J_ .... (25)

The critical values are calculated by the method of successive
approximations, following Hough's procedure in the theory of the
tides. By the methods of Sturm and Liouville, we know that one of
the roots of (14), for example, is given to a first approximation by
L 2n = 0. To evaluate this root more exactly, we write (14) in the
equivalent form

J_. J_ ...l

2n+4~ -^2+8~ J

(26)

The value of given by L 2n is now introduced in the two expres-
sions on the right, and a second approximation is then obtained, and
so on.

The Second Solution. When one solution is a known Mathieu
function, the other can be expressed by a quadrature; but a more
convenient construction based on the method of infinite continued
fractions has been given by Goldstein. As a typical case, suppose
we know the Mathieu function

(27)

192 MATHIEU'S EQUATION Chap. X, 43

and the appropriate value of . According to 41 (25), the second
solution will be of the form

in 28+1 (0) = 0ce 2s+1 (0)+ f oi n+1 sin(2n+l)fl, (28)

n-O

and the recurrence formulae are

(29)

We now introduce an auxiliary sequence (A 2n+1 ), defined by the
relations -(l +Ll )A 1+ A 3 = 0, (3Q)

This must diverge to infinity; for otherwise we should have both
an odd and an even Mat
in 41 (23), we find that

an odd and an even Mathieu function for the same value of . As

' = ....... (31)

= 2a 1 A l ^ 0.

Following Goldstein, we write in (29)

2 + i = I <4SSi u ( = 0, 1, 2,...), (32)

r-0

where the (r+l)th set of terms is defined by the relations

) = (n * r) (33)

= 2(2r+l)a 2r+1 /0 1 (w = r).

To satisfy these conditions by a sequence of values tending rapidly to
zero, we put

n 2r+l) _ \ A n in <? r\

a 2n + I A 2r+l^ 2n+l a 2r+l \ n ^ r )

(34)
where \2r+i * s determined by the condition

(35)
By using (30) and then (31), this is reduced to

r+1 ; (36)

Chap. X, 43 MATHIEU'S EQUATION 193

hence we have

ST+i 1) = -(2r+l)XA 2n+1 al +l (n < r), |

= -(2r+l)A^ 2r+1 a 2r+1 a 2/l4 . 1 (n > r), /
where A = (A 1 a 1 S 1 )~ l . The expression (32) now becomes

rH~ 1 T p QO

(38)
Now for large values of n we have

//. /4 .

- co, (39)

lim 2+32* + 8 _ lim (2n+3)Z, 8yy+1 = x

n^oo(2TO+l)^4 2n+1 a 2n+1 n^co(2^+l)L 2A?+3

For any fixed positive a, we can choose M so that

+i; (41)

and so the first series in (38) has the upper bound

2 " +1 - (42)

On the other hand, the second series in (38) converges with extreme
rapidity and (except for those values of ^, limited in number, for
which a 2n+l = 0) its sum is nearly equal to the first term, for n > n Q
say. Hence

= 0[(2n+l)\A 2n+l al n+1 \]

^\a 2n+l \]. (43)

From (38), (42), and (43) we now have

in + i = #I>2n+i!(l+) 2?l+1 ] (n > n ); (44)

and since ( 2n+1 ) are the coefficients of an integral function, the series
2 a 2n+i sm ( 2ri +l)0 i s a ^ so convergent for all finite values of 6.

44. The Methods of Lindemann and Stieltjes

Product of Solutions. We conclude our account of Mathieu's
equation with the solution of Lindemann and Stieltjes, modelled on

4064 c c

194 MATHIEU'S EQUATION Chap. X, 44

the Hermite-Brioschi treatment of Lame's equation. It is con-
venient to revert to the notation of 41 (6), namely

+ (p-iVcos% = 0. (1)

av 6

The product of any two solutions, Y = y^y^ satisfies the equation

= 0; ( 2 )

du

and on multiplying this by 2Y and integrating, we have the
Brioschi identity

'A Y\ 2 ,72 Y

W/ ~ dQ* - 4 ^-*

If we put Y = y l y% and use (1), the left-hand side reduces to the
square of the Wronskian; when Y is known, we can calculate (7 2
from (3) and we then obtain the numerical value of the Wronskian

which will enable us to complete the solution by a quadrature.

Now whether the equation ( 1 ) has or has not a solution of period
TT or 277, it is easily verified that (2) has always one and only one solu-
tion which is an even function of 6 with the period 77, This solution
is either the square of a Mathieu function, or else the product
F(6) =/(#)/( 0) of the two principal multiplicative solutions of (1).
We could expand F(6) either in the form J A 2n cos 2nd, or as an even
power- series in cos or sin 9. The latter forms are more convenient
in connexion with the relation (3). For to calculate C we insert
some particular value 6 or \-n\ and in the one case Y and its
derivatives are expressed as infinite series, and in the other in a
finite form. We therefore put z = cos 2 and obtain

z(l-z)Y m +^(l~2z)Y f/ +(p-l~k^z)Y f ~^k 2 a^Y = 0, (2*)
z(l-z)(Y'*-2YY")-(I-2z)YY'-(p-kWz)Y* = i<7 2 , (3*)

where primes denote differentiation with respect to z. Now the
equation (2*) has an irregular singularity at z = oo, and two regular
singularities free from logarithms, with exponents (0,^,1), at
z = 0,1. We can express each of the principal branches at z
in terms of the three principal branches at z = 1 ; and we must
construct a combination of the two branches holomorphic at z == 0,

Chap. X, 44 MATHIEU'S EQUATION 195

which shall not involve the branch belonging to the exponent

\ at z = 1. This solution will be holomorphic for all finite values of

z, and so it will be the required integral function. Now every solution

Y = 2 @n zn > which is holomorphic at z = 0, satisfies the recurrence

formulae

n(n+J)(n+l)C7 w+1 = n(n-ii)C' ll +*W(n-i)C7 B . 1 (n=l,2,...), (5)

which may be written

,

/ _
\ n+1

C n

In general, if u n is not small for some fairly large value of n, the
sequence (u n ) tends to the limit unity. If the ratio u is arbitrarily
chosen, the solution converges when |z| < 1, but has a singularity
at z = 1. But, for a certain properly chosen value ofu l9 the sequence
tends to zero and we have approximately

-- o

*
We write (5) in the form

(8)

which gives in general

= _^_^_n^l.... (9)

This infinite continued fraction is equivalent to the quotient of two
infinite determinants

** = -*^L ) . <

where &, _ x

Pn + l

(11)

This converges by von Koch's rule, because 2 ^n+i^n * s absolutely
convergent. The coefficients of the solution may now be written

C n = (-) n C' cx ia2 ... an ^(^+l,oo)/^(l,oo), (12)

and the series 2 ^n^ 71 converges for all finite values of z.

4064 c c 2

196 MATHIEU'S EQUATION Chap. X, 44

If p is the square of an integer (p = m 2 ), we have exceptionally,
for that one value of the suffix,

= _

M '

The expressions (12) as they stand are indeterminate of the type
oo/oo ; but the correct values are found by omitting ot m , whenever it
occurs as a factor of the numerator, and at the same time replacing

,1 i / -, n \ i /* ^ m(m+i)(m+l\ ,, -, ,

the elements K n ,l,/3 m ) by fl,Q ? -^^ 2 ^^ j m the deter-

minants K (n, oo), where n < m.

The Invariant. The series Y = ^ C n z n being now known, we
substitute it in the identity (3*) and put z = 0. This gives the value
of the constant Z __ (14)

If there is a Mathieu function of period TT or 27r, the constant C
vanishes and/(0) = f(0)', the solution is then more conveniently
constructed by other methods. We accordingly suppose C 2 ^ 0; it
is immaterial which sign is given to C, as this merely changes the
notation by /interchanging f(6) and/( 6). We now have

(15)
f'(0)J'(-6)

~

--

both sides reducing to unity as - 0, since /(O) ^ when there is
no solution of period TT or 2?r. If we put 6 TT in (16) and use the
multipliers, we have Stieltjes's formula

Finally, we obtain the solutions

e i
1 [CdO]

2) F(8)\

n J

(18)

Chap. X, Exs. MATHIEU'S EQUATION 197

EXAMPLES. X

1. The four types of Mathiou functions are the solutions of the equation

=

determined by the boundary conditions y ~ or -~ = 0, at the two points

du

6 = and 6 = \TT.

2. Mathieu functions belonging to different critical values p l9 p 2 satisfy the
orthogonal relation Q7r

{ /i(0)/(0) d6 = 0.

o

If Pi > p 2 and k 2 a 2 is real, the zeros of/^0) separate those of / 2 (0) along the
axis of real values of 0.

3. If w is a solution of the auxiliary equation

show that

If n? = p, n\ = pk 2 a z (k z a 2 > 0), the real zeros of /(#) separate those of
cosn 2 (0 ) and are themselves separated by those of cosn 1 (0~0 ).

[The zeros in the complex plane are examined by E. Hille, Proc.
London Math. Soc. (2) 23 (1924), 185-237.]

4. INTEGRAL EQUATIONS, (i) If x -- cos 6, show that Mathieu 's equation
may bo written in the form

(lx 2 )D*y-xDy-}-(pkWx 2 )y == 0.
(ii) Show that this can be solved by Laplace's transformation

y = | e lkaxt (j>(t) dt,

c

provided that (lt*)(f>"(t)-t(f>'(t) + (p k 2 a 2 t 2 )(/>(t) = 0,

and [e^ axt {ikax(l-t 2 }<)(t}+t(>(t)-(l-t 2 }\t)}] = 0.

(iii) By making <^(cos#) a Mathieu function, obtain Whittaker's equations
of the form

ce ln (0) = A J cos(A:acos0cos0 / )ce 2n (0 / ) d6' 9
o

7T

Ce 2n+i(^) A J sin(fcacos0cos0 / )ce 2n+1 (0') dO'.
o

(iv) By taking sin0 as variable, obtain similarly Whittaker's equations

77

ce 2n (0) = A J cosh(A;asin0sin0 / )ce 2n (0 / ) dB',

198 MATHIEU'S EQUATION Chap. X, Exs.

9T

se 2w +i(0) = A J sinh(kasm6smd')se 2n+l (e') d6'.

o
[E. T. Whittaker and G. N. Watson, Modem Analysis, 400-2.]

5. By expanding as a series of Mathieu functions 2 ^n-^nd)^n(^) tne solu '
tions of the equation of wave -motion

e ikx f e ik Vf

obtain Whittaker 's equations and the following:

rr

= ^ f sin0sin0'cos(fcacos0cos0 / )se an+1 (0'

1 (0) = A J
o

7T

n (^) = A J cos^cos^smh(;tasin^sin^)se 2n (^) dd'.

o

6. If ce WH .j(^), se OT4 ,^(0) are tho even and odd solutions of period 4TT, whose
limiting forms are cos(m-f %)0, sin(m-f %)Q, show that

.' 2,

cWfl) - A j K(e,6')ce m+ i(6') dO f ,
o

se m+J ((9) - A J X^+TT.^-fTrJse,^^') dff.

2C08i0C08}0'

where X(ft^) = d**<**o*** J e~ ikatt dt.

o

[Transform to parabolic cylindrical coordinates
(x+a + iy) = ia(^+^) 2

^ 2 F ^ 2 F
the equation + r-^- ^ 2 F = 0, and expand in a series of products of

Mathieu functions of period 4?r the particular solution

{ ?
V = d3cn(?-rt [ e-* kat *dt.

E. G. C. Poole, Proc. London Math. Soc. (2) 20 (1922), 374-88.]

7. By means of Whittaker *s integral equation, show that

co 2w (0) = a 2r cos2r0 = ATT ] (-) r a 2r J^acos^).

r= oo /= oo

[Heine; Whittaker; Goldstein, loc. cit.]

8. The solutions ^(0), t (^) of Mathieu's equation with the parameter p^ are
determined by the conditions

Ci (0) - 1, cj(0) - 0, *i(0) - 0, f(0) = 1.

Chap. X, Exs. MATHIEU'S EQUATION 199

Show that

[Z. Markovi6, Proc. London Math. Soc. (2) 31 (1930), 417-38.]
9. KECUEBBNCE FOBMULAE. (i) If (x+iy) = acosh(f -f ^), show that

8V 1 f . , f. dV , , . &V\

xz- -~ _ i gmn ^ cos f] -r cosn sin ft }

c)x tt(cosh^ cosr?)L ^c ci^l

(ii) If V 53 ce w (*)ce n (77), show by expanding dV/dx as a series of Mathiou
functions that

//-^ f cos-nce rt (-n)ce rt _ 1
i(tf)J - c ^i~ 2

== -4 n . 1 ce B-1 (^) J cel_i(i?)dty.
ir

fE. T. Whittaker, J. of London Math. Soc. 4 (1929), 88-96.]

SHORT BIBLIOGRAPHY!

A. GENERAL TREATISES IN ENGLISH

H. BATEMAN, Differential Equations (Longmans, 1918).

A. R. FORSYTH, A Treatise on Differential Equations (6th ed., Macmillan, 1929).
A. R. FORSYTH, Theory of Differential Equations, Part III, Volume 4
(Cambridge, 1902).

E. L. INCE, Ordinary Differential Equations (Longmans, 1927).

F. R. MOULTON, Differential Equations (Macmillan, New York, 1930).
H. T. H. PIAGGIO, Differential Equations (Bell, 1920).

E. T. WHITTAKER and G. N. WATSON, A Course of Modern Analysis (4th ed.,
Cambridge, 1927).

B. FOREIGN GENERAL TREATISES

E. GOURSAT, Cours d' analyse mathematique (5th ed., Paris, 1927).

J. HADAMARD, Cours d 1 analyse prof ess6 a Vficole Polytechnique (Paris, 1927).

L. HEFFTER, Einleitung in die Theorie der linear en Differentialgleichungen

(Leipzig, 1894).

C. JORDAN, Cours d'analyse de Vficole Polytechnique (3rd ed., Paris, 1909).
E. KAMKE, Differentialgleichungen reeller Funktionen (Leipzig, 1930).
E. PICARD, Trpite $ analyse (3rd od., Paris, 1922).
L. SCHLESINGER, Handbuch der Theorie der linearen Differentialgleichungen

(Leipzig, 1895).
L. SCHLESINGER, Vorlesungen uber lineare Differentialgleichungen (Leipzig,

1908).
CH. J. DE LA VALLEE POUSSIN, Cours d y analyse infinitesimale (4th ed., Louvain

and Paris, 1921).

C. WORKS OF REFERENCE

E. HILB, 'Lineare Differentialgleichungen im komplexen Gebiet' (1916),

Encyklopadie der math. Wiss. II, B (5), 471-562.
P. HUMBERT, 'Fonctions de Lam6 et fonctions de Mathieu' (1926), Memorial

des sc. math. x.
L. SCHLESINGER, 'Bericht uber die Entwickelung der Theorie der linearen

Diflerentialgleichungon seit 1865" (1909). Jahresbericht der deutschen

Mathematikervereinigung, XVIII.
M. J. O. STRUTT, 'Lam6-sche, Mathieu-sche und verwandte Funktionen in

Physik und Technik' (1932), Ergebnisse der Mathematik, I (in).
E. VESSIOT, 'Gewohnliche Differentialgleichungen: elementare Iiitegrations-

methoden' (1900), Encyklopadie der math. Wiss. II, A (4 b), 230-93

(especially 260-74).
A. WANG ERIN, 'Theorie der Kugelfunktionen und der verwandten Funktionen'

(1904), Encyklopadie der math. Wiss. II, A (10), 695-759.

f Further references to memoirs and treatises of a more special character are
given in the footnotes.

INDEX OF NAMES

Abel, 12, 13, 102.
Airy, 140.
Aitken, 27, 60.
d'Alombert, 16, 70.
Appell, 44.

Baker, 2.

Barnes, 104.

Bateman, 139, 153.

Berg, 18.

Bessel, 79, 142, 147-51.

Birkhoff, 57.

Bocher, 27, 36, 60, 166, 176.

Boole, 18.

Borel, 32.

Brassine, 34.

Brioschi, 163, 167, 169, 175, 180, 194.

Burnside, 132, 133.

Caque, 2.
Carson, 18.

Cauchy, 2, 4, 17, 18, 32.
Cayley, 120, 122.
Chaundy, 109.
Christoffol, 134, 159.
Courant, 95, 152, 153.
Crawford, 156, 163, 164.

Darboux, 36, 42, 43, 113, 176.
Darwin, 183.
Dixon, 104.

Elliot, 113.

Euler, 26, 53, 104, 107.

Ferrar, 117.
Floquet, 170.
Forsyth, 132.
Fricke, 133.

Frobenius, 42, 63, 70-4.
Fuchs, 4, 6, 16, 55, 74-9.

Gauss, 92, 101, 109.

Goldstein, 187, 189, 191, 192, 198.

Goursat, 97, 106.

Green, 164.

Haentzschel, 175.
Halphen, 154, 162, 164.
Hamburger, 53-5, 65.
Hankel, 79, 138, 148, 149.
Haupt, 124, 132.
Heaviside, 18-22, 30-2.

Heffter, 63, 70.

Heine, 158, 187, 189, 198.

Hermite, 32, 140, 152, 167, 169, 180,

194.

Hilbert, 95, 152, 153.
Hill, 178, 180, 182-7.
Hille, 197.
Hilton, 60.

Hobson, 98, 115, 154.
Hodgkinson, 132, 167.
Hough, 183, 191.
Humbert, 18, 154, 178.

Irice, 181, 187, 189.

Jacobi, 7, 43, 51, 59, 94, 106, 111, 152,

171, 173.
Jeffreys, 18.
Jordan, 27, 51, 104, 106, 136, 139,

160, 171, 172.

Kelvin, 183.

Klein, 104, 124, 132, 133, 158, 166.
von Koch, 184.
Kronecker, 8.

Kummer, 80, 87-8, 97, 102-4, 109,
112, 120, 139, 140, 143-7.

Lagrango, 37, 120, 153.

Laguerre, 153.

Lamb, 82, 183.

Lame, 154-77.

Laplace, 82, 136-53, 183.

Legendre, 81, 95, 96, 98, 110, 114, 115,

116, 142.
Liapounoff, 57.
Lindemann, 180, 193-6.
Lindsteclt, 182-7, 189.
Liouville, 2, 12-13, 44, 156, 166, 187,

191.

Love, 82, 134.
Lowry, 31, 32.

Markoff, 113.
Markovi6, 199.
Mathieu, 178-99.
Mellin, 104.
Molk, 116.

Papperitz, 86.
Peano, 2, 11.
Perron, 136.

202

INDEX OF NAMES

Picard, 2, 44, 136, 170-5.
Pidduck, 36.
Pincherle, 104.
Pochhammer, 106.
Poincare, 46, 149, 178.
van der Pol, 32.
P61ya, 95, 153, 158.

Riemann, 46, 83-7, 88-92, 98, 100,

104, 107, 118, 120, 131.
Rodrigues, 111.

Schendel, 111.

Seherk, 152.

Schlafli, 114.

Schlesinger, 2, 44, 104.

Schrodinger, 153.

Schwarz, 118-22, 125, 128, 132, 134,

135, 159.
Smith, 28.

Sonine, 143, 145, 153.
Steiner, 127.

Stieltjes, 158, 193-6.
Strutt, 154, 178.
Sturm, 156, 163, 165, 187, 191.
Szego, 95, 153, 158.

Tannery, 116.
Titchmarsh, 125.
Tschebyscheff, 95, 111.
Turnbull, 27, 60.

van Vleck, 113.

Wangerin, 175, 176.

Watson, 104, 139, 147, 152, 154, 178,

189, 198.
Weber, 133, 140.
Weierstrass, 51, 159, 186.
Whipple, 98.
Whittaker, 104, 139, 141, 152, 154,

175, 178, 189, 197, 198, 199.
Wirtinger, 104, 108, 124.
Wronski, 10-12, 41.

PRINTED IN
GREAT BRITAIN

AT THE

UNIVERSITY PRESS
OXFORD

BY

JOHN JOHNSON
PRINTEli

TO THE
UNIVERSITY'