Functions Of A Complex Variable

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FUNCTIONS OF A
COMPLEX VARIABLE

WITH APPLICATIONS

UNIVERSITY MATHEMATICAL TEXTS

GENERAL EDITORS

ALEXANDER C. AITKEN, D.SC., F.R.S.
DANIEL E. RUTHERFORD, D.SC. DR.MATH.

Determinants and Matrices A. C. Aitken, D.Sc., F.R.S.

Statistical Mathematics A. C. Aitken, D.Sc., F.R.S.

The Theory of Ordinary

Differential Equations J. C. Burkill, Sc.D., F.R.S.

Waves C. A. Coulson, D.Sc., F.R.S.

Electricity C. A. Coulson, D.Sc., F.R.S.

Protective Geometry T. E. Faulkner, Ph.D.

Integration R. P. Gillespie, Ph.D.

Partial Differentiation R. P. Gillespie, Ph.D.

Infinite Series J. M. Hyslop, D.Sc.

Integration of Ordinary Differential Equations E. L. Ince, D.Sc.

Introduction to the Theory of
Finite Groups ... ... ... ... W. Ledermann, Ph.D., D.Sc.

German-English S. Macintyre, Ph.D., and E. Witte, M.A.

Mathematical Vocabulary

Analytical Geometry of
Three Dimensions W. H. McCrea, Ph.D., F.R.S.

Topology E. M. Patterson, Ph.D.

Functions of a Complex Variable ... E. G. Phillips, M.A., M.Sc.

Special Relativity W. Rindler, Ph.D.

Volume and Integral ... W. W. Rogosinski, Dr.Phil., F.R.S.

Vector Methods D. E. Rutherford, D.Sc., Dr.Math.

Classical Mechanics D. E. Rutherford, D.Sc., Dr.Math.

Fluid Dynamics D. E. Rutherford, D.Sc., Dr.Math.

Special Functions of Mathematical

Physics and Chemistry I. N. Sneddon, D.Sc.

Tensor Calculus* B. Spain, Ph.D.

Theory of Equations H. W. Turnbull, F.R.S.

FUNCTIONS OF A
COMPLEX VAEIABLE

WITH APPLICATIONS

BY

E. G. PHILLIPS, M.A., M.So.

SENIOR LECTURER IN MATHEMATICS IN THE UNIVERSITY COLLEGE OF

NORTH WALKS, TUNGOR: FORMERLY SENIOR SCHOLAR OF

TRINITY COLLKGB, CAMBRIDGE

With 17 Figure*

OLIVER AND BOYD
EDINBURGH AND LONDON
YORK: INTERSCIENOE PUBLISHERS, INC.

FIRST EDITION . . 1940
EIGHTH EDITION . . 1957
REPBINTED . . . 1961

PRINTED AND PUBLISHED IN GREAT BRITAIN BY
OLIVER AND BOYD LTD., EDINBURGH

PREFACE TO THE EIGHTH EDITION

CHANGES that have been made in recent editions include
a set of Miscellaneous Examples at the end of the book
and an independent proof of Liouville's theorem has been
given. In this edition, the proof of the Example on
page 60 has been altered.

Limitations of space made it necessary for me to confine
myself to the more essential aspects of the theory and its
applications, but I have aimed at including those parts of
the subject which are most useful to Honours students.
Many readers may desire to extend their knowledge of the
subject beyond the limits of the present book. Such
readers are recommended to study the standard treatises of
Copson, Functions of a Complex Variable (Oxford, 1935),
and Titchmarsh, Theory of Functions (Oxford, 1939). I
take this opportunity of acknowledging my constant
indebtedness to these works both in material and
presentation.

I have presupposed a knowledge of Real Variable
Theory corresponding approximately to the content of
my Course of Analysis (Cambridge, Second Edition,
1939). References are occasionally given to this book in
footnotes as P.A.

I wish to express my thanks to all those friends who
have made helpful suggestions. In particular, I mention
two of my colleagues, Mr A. C. Stevenson, of University
College, London, who read the proofs of the first edition,
and Prof. H. Davenport, F.R.S., who very kindly suggested
a number of improvements for the second edition. I desire
also to express my gratitude to the publishers for the careful
and efficient way in which they have carried out their
duties. - E Q p

BANGOR, October 1956

vii

CONTENTS

PREFACE p. vii

FUNCTIONS OF A COMPLEX VARIABLE p. 1

PAGB

< Complex Numbers ....... I

Sets of Points in the Argand Diagram ... 7
Functions of a Complex Variable .... 9

Regular Functions . . . . . . .11

Conjugate Functions . . . . . .14

^JPower Series 17

The Elementary Functions . . . . .20

Many -valued Functions . . . . . .26

Examples I ........ 29

CONFORMAL REPRESENTATION p. 32

Isogonal and Conformal Transformations . . .32
Harmonic Functions . . . . . .37

The Bilinear Transformation ..... 40

Geometrical Inversion ...... 43

The Critical Points 45

Coaxal Circles 46

Invariance of the Cross-Ratio 49

Some special Mdbius' Transformations . . .51
Examples II . , . . .54

x CONTENTS

SOME SPECIAL TRANSFORMATIONS p. 67

PAGH

The Transformations 10 = 2". . . .68
w = z* 60

w = \/z 62

w = tan 2 (frr\/z) 64

Combinations of w == z* with Mobius' Transformations 66
Exponential and Logarithmic Transformations . . 70
Transformations involving Confocal Conies . . 71

z = c sin w ........ 74

Joukowski's Aerofoil . . . . . .76

Tables of Important Transformations . . .78
Schwarz-Christoffel Transformation .... 80

Examples III ........ 81

THE COMPLEX INTEGRAL CALCULUS p. 85

Complex Integration ...... 85

Cauchy's Theorem . . . . . . .89

The Derivatives of a Regular Function ... 93

Taylor's Theorem 95

Liouville's Theorem . . . . . .96

Laurent's Theorem . . . . . . .97

Zeros and Singularities ...... 98

Rational Functions . . . . . . .103

Analytic Continuation . . . . . .103

' Poles and Zeros of Meromorphic Functions . .107
Rouch6's Theorem . . . . . . .108

The Maximum-Modulus Principle . . . .109

Examples IV Ill

THE CALCULUS OF RESIDUES p. 114

The Residue Theorem ......

Integration round the Unit Circle 117

CONTENTS xi

THE CALCULUS OF RESIDUES continued.

PAGE
Evaluation of Infinite Integrals . . . .119

Jordan's Lemma . . . . . . .122

Integrals involving Many -valued Functions . ,126
Integrals deduced from Known Integrals . . .128
Expansion of a Meromorphic Function . . .131
Summation of Series . . . . . .133

Examples V ........ 135

MISCELLANEOUS EXAMPLES 138

INDEX 143

OH APTBB I

FUNCTIONS OF A COMPLEX VARIABLE

1. Complex Numbers

This book is concerned essentially with the application
of the methods of the differential and integral calculus
to complex numbers. A number of the form a+ij8, where
t is V(~~ 1) an( l a an d j3 are real numbers, is called a
complex number ; and, although complex numbers are
capable of a geometrical interpretation, it is important
to give a definition of them which depends only on real
numbers. Complex numbers first became necessary in the
study of algebraic equations. It is desirable to be able to
say that every quadratic equation has two roots, every
cubic equation has three roots, and so on. If real numbers
only are considered, the equation a +l = has no roots
and x 3 l = has only one. Every generalisation of
number first presented itself as needed for some simple
problem, but extensions of number are not created by
the mere need of them ; they are created by the definition,
and our object is now to define complex numbers.

By choosing one of several possible lines of procedure,
we define a complex number as an ordered pair of real
numbers. Thus (4, 3), (\/2, e), (J, TT) are complex numbers.
If we write z = (x, j/), x is called the real part, and y
the imaginary part, of the complex number z.

(i) Two complex numbers are equal if, and only if,
their real and imaginary parts are separately equal. The
equation z = z' implies that both x = x' and y = y'.
I

2 FUNCTIONS OF A COMPLEX VARIABLE

(ii) The modulus of z, written | z |, is defined to be
+ \/(x*-\-y*). H> follows immediately from the definition
that | z | = if, and only if , x = and y = 0.

(iii) The fundamental operations.
If z = (z, y), z' = (#', y') we have the following definitions :

(1) z+z' is (*+*', y+y').

(2) -zis(-*, -t/).

(3) z-z' = z+(-z') is (z-a;', y-y').

(4) 22' is (xx'yy' 9 xy'+x'y).

If the fundamental operations are thus defined, we easily
see that the fundamental laws of algebra are all satisfied.
(a) The commutative and associative laws of addition hold :

(b) The same laws of multiplication hold :

(o) TAe distributive law holds :

As an example of the method, we show that the com-
mutative law of multiplication holds. The* others are
proved similarly.

We have thus seen that complex numbers, as defined
above, obey the fundamental laws of the algebra of real
numbers : hence their algebra will be identical in /orm,
though not in meaning, with the algebra of real numbers.

We observe that there is no order among complex
numbers. As applied to complex numbers, the phrases
" greater than " or " less than " have no meaning. In-
equalities can only occur in relations between the moduli
of complex numbers.

FUNCTIONS OF A COMPLEX VARIABLE 3

(iv) The definition of division.

Consider the equation z = z', where z = (x, y),
= (, i)) 9 z' = (x' 9 y') t then we have

so that xyr\ = z'

and, on solving for and 77,

_ yy'+xx' _ xy'x'y

provided that | z | ? 0. Hence, if | ^ | ^ 0, there is a
unique solution, and = (f , TJ) is the quotient z'/z.

Division by a complex number whose modulus is zero
is meaningless ; this conforms with the algebra of real
numbers, in which division by zero is meaningless.

The abbreviated notation.

It is customary to denote a complex number whose
imaginary part is zero by the real-number symbol a?. If
we adopt this practice, it is essential to realise that x may
have two meanings (i) the real number x, and (ii) the
complex number (x, 0). Although in theory it is important
to distinguish between (i) and (ii), in practice it is legitimate
to confuse them ; and if we use the abbreviated notation,
in which x stands for (x, 0) and y for (y, 0), then

x+y = (*, 0)+(y, 0) = (x+y, 0),
xy = (x, 0) . (y, 0) = (x. y-0.0, x . 0+0 . y) = (xy, 0).

Hence, so far as sums and products are concerned, complex
numbers whose imaginary parts are zero can be treated
as though they were real numbers. It is customary to
denote the complex number (0, 1) by . With this
convention, t 2 = (0, 1) . (0, 1) = ( 1, 0), so that i may
be regarded as the square root of the real number 1.
On using the abbreviated notation, it follows that

(x, y) = x+iy,

FUNCTIONS OF A COMPLEX VARIABLE

for, since % = (0, 1), we have

= (*,0)+(0.y-1.0,0.0+l.y)

= (a?, 0)+(0, y) = (a+0, 0+y) = (x, y).

In virtue of this relation we see that, in any operation
involving sums and products, it is allowable to treat
x, y and i as though they were ordinary real numbers,
with the proviso that i* must always be replaced by 1.

2. Conjugate Complex Numbers

If z = x+iy, it is customary to write x = Rz, y = Iz.
The number x iy is said to be conjugate to z and is
usually denoted by z. It readily follows that the numbers
conjugate to z l -\-z 2 and 2 t 2 2 are Zi+z^ and 2^23 respectively.

Proofs of theorems on complex numbers are often
considerably simplified by the use of conjugate complex
numbers, in virtue of the relations, easily proved,

| Z | f == 22, 2R2 = Z+Z, 2ilz = 22.

To prove that the modulus of the product of two complex
numbers is the product of their moduli, we proceed as follows :

2 =

and so, since the modulus of a complex number is never
negative,

I *i*a H I *i I I * !

Theorem. The modulus of the sum of two complex
numbers cannot exceed the sum of their moduli.

2 =

and so

FUNCTIONS OF A COMPLEX VARIABLE 5

a result which can be readily extended by induction to
any finite number of complex numbers.

In .a similar way we can prove another useful result,
viz.

K-*il>l(l*iHil)l-

We have

hence

= (l*i H*i I) 2 ;

\*i~**\ > 1(1 *i H * 1)1-

3. Geometrical Representation of Complex
Numbers

If we denote (# 2 +y 2 )* by r, and choose so that
r cos 6 = x, r sin = y, then r and are clearly the radius

FIG. 1.

vector and vectorial angle of the point P, (x, y), referred
to an origin and rectangular axes Ox, Oy. It is clear
that any complex number can be represented geometrically
by the point P, whose Cartesian coordinates are (x, y)
or whose polar coordinates are (r, 0), and the representation
of complex numbers thus afforded is called the Argand
diagram.

By the definition already given, it is evident that r
is the modulus of z = (x, y) ; the angle is called the
argument of z, written = arg z. The argument is not
unique, for if be a value of the argument, so also is

FUNCTIONS OF A COMPLEX VARIABLE

2/17T+0, (n = 0, 1 2, ...) The principal value of arg z
is that which satisfies the inequalities ?r<arg Z^TT.

Let P! and P 2 (in fig. 1) be the points z l and z 2 , then
we can represent addition in the following way. Through
P l9 draw PjP 8 equal to, and parallel to QP 2 . Then P 8
has coordinates (a^+^a* yi+^a) an d 8O -Ps represents the
point z t +z 2 .

In vectorial notation,

QP 3 =

Similarly, we have, if P 8 is the point 2 3 ,

It is convenient to write cis0 for cos^+fsin^. If
z l = r l cis0 l9 z 2 = r 2 cis^ 2 , ..., z n = r n cis0 n , then, by
de Moivre's theorem,

which readily exhibits the fact that the modulus and
argument of a product are equal respectively to the
product of the moduli and the sum of the arguments of
the factors. In particular, if n be a positive integer and
z = r cis 0, z n = r n cis n$.
Similarly,

^ = ^1 cis ^-fla).

Z 2 r 2

If n is a positive integer, there are n distinct values of
z l i*. If m is any integer, since

j cis 1 = cis0,

\ /

it follows that r*-t n cis{(0+2w7r)/n} is an nth root of z=r cis0.
If we substitute the numbers 0, 1, 2, ... n 1 in succession
for m, we obtain n distinct values of z l l n ; and the sub-
stitution of other integers for m merely gives rise to
repetitions of these values. Also, there can be no other

FUNCTIONS OF A COMPLEX VARIABLE 7

values, since z l l n is a root of the equation u n = z which
cannot have more than n roots.

Similarly, if p and q are integers prime to each other
and q is positive,

where m = 0, 1, 2, ..., ql.

By considering the modulus and argument of a complex
number, the operation of multiplying any complex number
x+iy by i is easily seen to be equivalent to turning the line
OP through a right-angle in the positive (counter-clockwise)
sense. We have just seen that

(- 1 )

\z % /

zj) = arg Zx+arg z 2> arg - = arg z l arg z a ,

\z % /

so that the formal process of " taking arguments " is similar
to that of " taking logarithms." Hence, if arg (x+iy) = a,

arg i(x+iy) = arg i+a,Tg(x+iy) = JTT+OU
Since | i \ = 1, multiplying by i leaves | x+iy \ unaltered.

4. Sets of Points in the Argand Diagram

We now explain some of the terminology necessary
for dealing with sets of complex numbers in the Argand
diagram. We shall use such terms as domain, contour,
inside and outside of a closed contour, without more precise
definition than geometrical intuition requires. The general
study of such questions as the precise determination of
the inside and outside of a closed contour is not so easy
as our intuitions might lead us to expect.* For our
present purpose, however, we shall find that no difficulties
arise from our relying upon geometrical intuition.

By a neighbourhood of a point z in the Argand
diagram, we mean the set of all points z such that |z--z |<,
where is a given positive number. A point z is said

* For further information, see e.g. Dienee, The Taylor Seriet
(Oxford, 1931), Ch. VI.

8 FUNCTIONS OF A COMPLEX VARIABLE

to be a limit point of a set of points S 9 if every neighbour-
hood of z contains a point of S other than z . The
definition implies that every neighbourhood of a limit
point z contains an infinite number of points of S. For,
the neighbourhood | z z |< contains a point z 1 of S
distinct from z , the neighbourhood | z z |<| z l z |
contains a point z 2 of S distinct from z and so on
indefinitely.

The limit points of a set are not necessarily points of
the set. If, however, every limit point of the set belongs
to the set, we say that the set is closed. There are two
types of limit points, interior points and boundary points.
A limit point z of S is an interior point if there exists
a neighbourhood of z which consists entirely of points
of S. A limit point which is not an interior point is a
boundary point.

A set which consists entirely of interior points is said
to be an open set.

It should be observed that a set need not be eithet open
or closed. An example of such a set is that consisting of the
point z = 1 and all the points for which | z | <1.

We now define a Jordan curve.

The equation z = x(t)+iy(t), where x(t) and y(t) are
real continuous functions of the real variable t, defined
in the range a <J<j8, determines a set of points in the
Argand diagram which is called a continuous arc. A
point z x is a multiple point of the arc, if the equation
z l = x(t) +iy(t) is satisfied by more than one value of t in
the given range.

A continuous arc without multiple points is called a
Jordan arc. If the points corresponding to the values
a and j3 coincide, the arc, which has only one multiple
point, a double point corresponding to the terminal values
a and j8 of t, is called a simple closed Jordan curve.

A set of points is said to be bounded if there exists
a constant K such that | z | ^.K is satisfied for all points

FUNCTIONS OF A COMPLEX VARIABLE 9

z of the set. If no such number K exists the set is
unbounded.

A domain is defined as follows:

A set of points in the Argand diagram is said to be
connex if every pair of its points can be joined by a
polygonal arc which consists only of points of the set.
An open domain is an open connex set of points. The
set, obtained by adding to an open domain its boundary
points, is called a closed domain.

The Jordan curve theorem states that a simple closed
Jordan curve divides the plane into two open domains which
have the curve as common boundary. Of these domains
one is bounded and it is called the interior, the other,
which is unbounded, is called the exterior. Although
the result stated seems quite obvious, the proof is very
complicated and difficult. When using simple closed
Jordan curves consisting of a few straight lines and circular
arcs, geometrical intuition makes it obvious which is the
interior and which is the exterior domain.

For example, the circle | z \ = R divides the Argand
diagram into two separated open domains | z \<R and
| z | >7?. Of these the former is a bounded domain and is
the interior of the circle | z \ = R ; the latter, which is
unbounded, is the exterior of the circle | z \ = R.

In complex variable theory we complete the complex
plane by adding a single point at infinity. This point is
defined to be the point corresponding to the origin by the
transformation z' = 1/2.

5. Functions of a Complex Variable. Continuity

If u?(= u+iv) and z(= x-}-iy) are any two complex
numbers, we might say that w is a function of z, w = /(z), if,
to every value of z in a certain domain D, there correspond
one or more values of w. This definition, similar to that
given for real variables, is quite legitimate, but it is futile
because it is too wide. On this definition, a function of

10 FUNCTIONS OF A COMPLEX VARIABLE

the complex variable z is exactly the same thing as a
complex function

u(x, y)+iv(x,y)

of two real variables x and y.

For functions defined in this way, the definition of
continuity is exactly the same as that for functions of a
real variable. The function /(z) is continuous at the
point Z Q if, given any e, >0, we can find a number 8 such that

for all points z of D satisfying | z z 1 <8. The number 8
depends on e and also, in general, upon z . If it is
possible to find a number h(e) independent of z , such that
\f( z )f( z o) |< holds for every pair of points z, z of the
domain D for which |z z |<A, then /(z) is said to be
uniformly continuous in D. It can be proved that a
function which is continuous in a bounded closed domain
is uniformly continuous there.*

It is easy to show that this definition of continuity is
equivalent to the statement that a continuous function
of z is merely a continuous complex function of the two
variables x and y, for, if

/(z) = u(x, y)+iv(x, y),

when /(z) is continuous on the above definition, so are
u(x, y) and v(x y y) ; and conversely, if u and v are continuous
functions of x and y, /(z) is a continuous function of z.

The only class of functions of z which is of any practical
utility is the class of functions to which the process of
differentiation can be applied.

6. Differentiability

We next consider whether the definition of the
derivative of a function of a single real variable is applicable

* For a proof of this theorem for a closed interval, see Phillips,
A Course of Analysis (Cambridge, 1939), p. 73. This will be referred
to subsequently as P.A.

FUNCTIONS OF A COMPLEX VARIABLE II

to functions of a complex variable. The natural definition
is as follows : Let f(z) be a one-valued function, defined in
a domain D of the Argand diagram, then f(z) ia differentiate
at a point Z Q of D if

tends to a unique limit as z-^- z , provided that z is also a
point of D.

If the above limit exists it is called the derivative
of /(z) at z = z and is denoted by /'(z ). Restating the
definition in a more elementary form, it asserts that, given
>0, we can find a number S such that

(*)-/(*o) ,

Z-Z

o

<

for all z, z in D satisfying 0< |z z |<8. That continuity
does not imply differentiability is seen from the following
simple example :

Let/(z) = | z )'. This continuous function ia differentiable
at the origin, but nowhere else. For if z ^ we havo

= +z (cos 2< i sin 2<f>)

where <f> = arg (z 2 ). It is clear that this expression does
not tend to a unique limit as z-> z .

If Zg the incrementary ratio is z t which tends to zero
as z-> 0.

7. Regular Functions

A function of z which is one-valued and differentiable
at every point of a domain D is said to be regular * in the
domain D. A function may be differentiable in a domain
save possibly for a finite number of points. These points
are called singularities of /(z). We next discuss the
necessary and sufficient conditions for a function to be
regular.

* The terms analytic and holomorphic are sometimes used as
ynonymoua with the term regular as defined above.

12 FUNCTIONS OF A COMPLEX VARIABLE

(1) The necessary conditions for f(z) to be regular.

If f(z) =s u(x, y)+iv(x, y) is differentiable at a given
point 2, the ratio {/(z+ Az) f(z)}jAz must tend to a definite
limit as Jz->0 in any manner. Now Az = Ax+iAy.
Take A z to be wholly real, so that Ay = 0, then

u(x+Ax, y)u(x y y) . v(x+Ax, y)v(x, y)
Ax + * Ax

must tend to a definite limit as Ax-* 0. It follows that
the partial derivatives u x , v 9 must exist at the point
(x 9 y) and the limit is u 9 -\-iv x . Similarly, if we take
Az to be wholly imaginary, so that Ax = 0, we find that
t/,, v y must exist at (x, y) and the limit in this case is
v y iuy. Since the two limits obtained must be identical,
on equating real and imaginary parts, we get

UX = Vy , Uy = V 9 . . . (1)

These two relations are called the Cauchy-Riemann
differential equations.

We have thus proved that for the function f(z) to be
differentiable at the point z it is necessary that the four partial
derivatives u x , v x , u y , v y should exist and satisfy the Cauchy-
Riemann differential equations.

We thus see that the results of assuming differentiability
are more far-reaching than those of assuming continuity.
Not only must the functions u and v possess partial
derivatives of the first order, but these must be connected
by the differential equations (1).

That the above conditions are necessary, but not
sufficient, may be seen by considering Examples 6 and 7
at the end of this chapter.

(2) Sufficient conditions for f(z) to be regular.
Theorem. The continuous one-valued function f(z) is

regular in a domain D if the four partial derivatives u x , v xt
u y , v v exist, are continuous and satisfy the Cauchy-Riemann
equations at each point of D.

FUNCTIONS OF A COMPLEX VARIABLE 13

Now

Au = u(x+Ax y y+Ay)u(x, y),

= u(x+Ax, y+Ay) u(x+Ax, y) +u(x+ Ax, y) u(x, y),
= Ay . u v (x+Ax y y+OAy)+Ax . u x (x+6'Ax, y) ;

where 0<0<1, 0<0'<1, by the mean- value theorem.*
Since u x , u v are both continuous, we may write
Au = Ax{u x (x, y)+<i}+Ay{u v (x, y)+c'} 9

where and e' both tend to zero as | Az |-> 0.
Similarly,

Av = Ax{v x (x, y)-i- r rj}+Ay{v 1/ (x, y)+^'},
where rj and rf both tend to zero as | Az |-> 0.

Hence Aw = Au+iAv

= AX(UB +iv x ) +Ay(u v +iv v ) +a)Ax+a)'Ay 9

where a* and aj f tend to zero as | Az \-> 0.

On using the Cauchy-Riemann equations we get

Aw = (Ax+iAy)(u x +iv x )-\-a>Ax+a>'Ay

and, on dividing by Az and taking the limit as \Az |-> 0,
dw
dz

atAx+oj'Ay . . .
since ^ I co I ~p I co I

We notice that the above sufficient conditions for the
regularity of f(z) require the continuity of the four first
partial derivatives of u and v.

If w = tt-f-iv, where u and v are functions of x and y,
since

..!,., -!,-*

* See P.A., p. 101.

14 FUNCTIONS OF A COMPLEX VARIABLE

u and t; may be regarded formally as functions of two
independent variables z and z. If u and v have continuous
first-order partial derivatives with respect to x and y, the
condition that w shall be independent of z is that dw/8S = 0.
This leads to the result

du dx
that is

du dy /<9v dx dv dy\ __
8y'8i\8x'di~dy8s)~~ '

I du 1 du t dv 1 dv __ ^

and, on writing f for 1/f and equating real and imaginary
parts, we get

du dv dv du

which are the Cauchy-Riemann equations.

Hence, in any analytical formula which represents a
regular function of z, x and y can occur only in the combina-
tion x+iy. For example, it is clear at a glance that

sin (x+3iy) = sin (2zz)

cannot be a regular function.

If u+iv =f(x-\-iy) where f(z) is a regular function,
then the real functions u and v of the two real variables
x and y are called conjugate functions.

Since the partial derivatives of u and v are connected
by the relations

du _ 8v 8v _ _te

a5~V8i~~V ()

if the derivatives concerned are assumed to exist and
satisfy the relation (f> Xv = </> vX , it follows by partial
differentiation that

_ _

"" "" an

FUNCTIONS OF A COMPLEX VARIABLE 15

Hence both u and v satisfy Laplace's equation in two
dimensions

This equation occurs frequently in mathematical physics.
It is satisfied by the potential at a point not occupied by
matter in a two-dimensional gravitational field. It is
also satisfied by the velocity potential and stream function
of two-dimensional irrotational flow of an incompressible
non-viscous fluid.

By separating any regular function of z into its real
and imaginary parts, we obtain immediately two solutions
of Laplace's equation. It follows that the theory of
functions of a complex variable has important applications
to the solution of two-dimensional problems in mathe-
matical physics. It also follows from equations (1) that

The geometrical interpretation of (2) is that the families
of curves in the (x, y)-plane, corresponding to constant
values of u and v, intersect at right angles at all their
points of intersection. For if u(x, y) = c l9 then du = 0,
and so

8 dx + ^dy = 0. . . . (3)
ox oy

Similarly, if v(x, y) = c 2 , we have

The condition that these families of curves intersect at
right angles is

*

16 FUNCTIONS OF A COMPLEX VARIABLE

where the suffixes 1 and 2 refer to the u and v families
respectively. On using (3) and (4), it is easy to see that
(5) reduces to (2).

It is possible to construct a function /(z) which has a
given real function of x and y for its real or imaginary
part, if either of the given functions u(x, y) or v(x, y) is
a simple combination of elementary functions satisfying
Laplace's equation. A very elegant method of doing this
is due to Milne-Thomson.*

Since x = g (2+2), y = - (z-z),

zz . (z+z zz

We can look upon this as a formal identity in two inde-
pendent variables z, z. On putting z = z we get

Now /'(z) = UX+WB = UxiUy by the Cauchy-Riemann
equations. Hence, if we write <f>i(x t y) and <f>%(x 9 y) for
u 9 and u y respectively, we have

/'(z) ^^(z, y)-^ 2 (* y)=<f>i(*> 0)-i^ 2 (z, 0).
On integrating, we have

i(*> 0)-^ 2 (z, 0)}dz+C,

where C is an arbitrary constant.

Similarly, if v(x, y) is given, we can prove that

I 2 (z, 0)}dz+C,

where ift^x, y) = v y and 2 (#, y) = v 9 .

Math. Gazette, xxi. (1937), p. 228. See also Misc. Ex. !, p. 138.

FUNCTIONS OF A COMPLEX VARIABLE 17

As an example, suppose that u(x,y) = e*(x ooa y y sin y),

du
Here fa = = e*(x cos y y sin y-fcos y),

c/a?

ti
^ t = ~- = e*(x sin t/ sin yy cos y).

<?2/
Hence f'(z) = ^(z, 0)-t> 2 (z, 0) = e (2+!),

and so f(z) = J e (s+l)efa+(7 = ze +O.

8. Power Series. The Elementary Functions

00 00

Consider the series S a n z n or 2 a n (z z ) w , where the

n=0 n

coefficients a n and z, z may be complex. Since the latter
series may be obtained from the former by a simple change
of origin, the former may be regarded as a typical power
series. It is assumed that the reader is already familiar
with the theory of real power series.*

So far as absolute convergence is concerned, everything
that has been proved for absolutely convergent series of
real terms extends at once to complex series, for the series
of moduli

Kl +lilN + KIN 2 +-

is a series of positive terms. The most useful convergence
test for power series is Cauchy's root test, which states
that a series of positive terms Zu n is convergent or
divergent according as liin (u n ) l l n is less than or greater
than uriity.f If we write lim | a n \ l l n = 1/jR, then we
easily see that the power series Za n z n is absolutely con-
vergent if | z \<R, divergent if | z |>J2, and if | z | = R
we can give no general verdict and the behaviour of the
series may be of the most diverse nature. The number B
is called the radius of convergence, and the circle,
centre the origin, and radius R, is called the circle ol

* See P.A., Ch. XIII.
t See P.A., p. 124.

18

FUNCTIONS OF A COMPLEX VARIABLE

convergence of the power series. Clearly there are three
cases to consider (i) R = 0, (ii) R finite, (iii) R infinite.
The first case is trivial, since the series is then convergent
only when 2 = 0. In the third case the series converges
for all values of z. In the second case the radius of the
circle of convergence is finite and the power series is
absolutely convergent at all points within this circle,
and divergent at all points outside it.
We now prove an important theorem.

00

If f(z) = S a n z n , then the sum-function f(z) is a regular

o
function at every point within the circle of convergence of

the power series.

Suppose that Sa n z" is convergent for | z \<R. Then,
if 0</><J?, a^ n is bounded, say |anp n |</f. Let

$(z) = 2 na n z*~ l .

n-l

We write, for convenience, | z \ = r, | h \ = 77 : then, if
r<p and

n-o

^-|

-nan-* .

JJ

Now

-. = (r+^-r* _ ,-!

Hence

/(+*)-/(*)
A

#(*)

<

KZ 1 -( (r ^ )n - rn -nr->}

n-op n ( -n }

l -(-r P-} /L_l

r) \p-r-r) p-rj (p-r) 2 J

FUNCTIONS OF A COMPLEX VARIABLE 19

which tends to zero as TJ-> 0. Hence f(z) has the derivative

(f)(z). This proves that /(z), which is plainly one-valued,

is also differentiable : hence f(z) is regular within | z \ = B.

Since n l l n -*\ as n->oo, lim|n^ tt | 1 / n = limlaj 1 /" = 1/J2,

and so the series <f>(z) = Z na n z n ~ l has the same radius of

n-l

convergence as the original series. Thus, if <f>(z) = /'(z)
is regular in | z \<R, we can show similarly that its
derivative is Sn(n l)a n z n ~ 2 , and so on. In other words,
we thus prove that a power series can be differentiated
term by term as often as we please at any point within its
circle of convergence.

The above theorem, which is the analogue of a well-known
theorem in real variable theory, can be superseded by a
more general theorem which is one of the characteristic
achievements of complex variable theory. This theorem is
as follows :

Let f(z) _ u l (z)+ui(z) + . . . +u n (z)+ . . . ;

if each term u n (z) is regular within a region D and if the series
is uniformly convergent throughout every region D' interior to
D, then f(z) is regular within D and all its derivatives may be
calculated by term-by -term differentiation.

For a proof of this theorem, the reader is referred to
larger treatises on complex variable theory. The simpler
theorem proved above will suffice for the purposes of this
book.

In a later chapter ( 34) we prove Taylor's theorem
that a function /(z) can be expanded in a power series

00

2a n (za) n about any point a, provided that/(z) is regular

n-O

in | za |<p. By combining Taylor's theorem with the
theorem proved above, we see that the necessary and
sufficient condition that a function /(z) may be expanded
in a power series is that it should be regular in a region.
The Weierstrassian development of complex variable

20 FUNCTIONS OF A COMPLEX VARIABLE

theory begins by defining an " analytic function " of z aa
a function expansible in a power series. (See 39.)

We now consider briefly the definitions of the so-called
elementary functions of a complex variable.

I. Rational functions.

A polynomial in z, a Q -\-a l z+...+a m z m 9 may be regarded
as a power series which converges for all values of z.
Since such functions are regular in the whole plane, rational
functions of the type

are regular at all points of the plane at which the
denominator does not vanish. If we choose a point z ,
at which the denominator does not vanish, and replace
* by Z +(z--z ), the function /(z) becomes

A +A l (z-z )+...+A rn (z-z )<

in which B Q ^ 0. It readily follows that /(z) may be

00

expanded in a power series of the form Hc n (zz ) n .

o

II. The exponential function.

For the exponential function of a real variable, one
method of development is to define exp a: as the sum-
function of the power series

and, on using the multiplication theorem for absolutely
convergent series, we prove that *

exp x . exp x 9 = exp (x+x').
* See P.A., p. 246.

FUNCTIONS OF A COMPLEX VARIABLE 21

In the same way we can define eipz as the sum-
function of the series of complex terms

Since the series converges for all values of z, it defines a
function regular in the whole z-plane. Such functions are
called integral functions.

When x is rational, exp x is identical with the function
e* of elementary algebra, and when x is irrational we
define e 9 to be identical with the function exp x, the
sum-function of the power series (1) above. In the same
way, when z is complex, we find it convenient conventionally
to write e* for exp z. Since the formula exp z . exp
= exp(z + f) can be proved by multiplication of series,
whether z be real or complex, the real number e with a
complex exponent obeys the formal law of indices of
elementary algebra

e'. ef = e*+f .

Thus we may define the power e*, without ambiguity,
by the equation

z 2 z 8
e = l + z +2\ + 3! + *

and, if a is any positive number, a* denotes the value
unambiguously determined by the formula

a * ^ gi log*,
where log a is the real natural logarithm of a.

The reader should notice how far this definition is removed
from the elementary definition " x k is the product of k factors
equal to a?." At first sight there is no knowing what value
belongs to a number of the form 2', but its value is uniquely
determined by our definition.

22 FUNCTIONS OF A COMPLEX VARIABLE

For real values of y we have

00 (iij} n & f/2* oo

e" - 27 ( -ff - 2; (-1)* ITT-, + t(-l) --y- (2)
n-o * ! t o ( 2k ) ! *~o (2&+1) !

= cos y+ i sin y ;

since the cosine and sine of the real variable y are defined
by the two power series on the right of (2).

Hence e* = e** iv = eV* = e* cis y.

We also see that, since

| & | = | cisy | = 1 , | e | = | e|| e<* \ = e,

since e*>0. Similarly, arg e f = Iz = y. TAe function e 9
has the period 2ni ; in other words, if k is any positive
or negative integer, or zero,

for, when we increase z by 27rt, y increases by 2?r and
this leaves the values of sin y and cos y unchanged. Every
value which e* is able to assume is therefore taken in the
infinite strip 7r<j/<7r, or in any strip obtainable from
this by a parallel translation.

It is easy to show that e 9 has no other period. If
e f = ef, this necessarily implies that z = f +2kiri. This
follows at once, because e f ~f = 1 and so

e*~f cis(y 77) = 1.

Hence x % = 0, cos (yrj) = 1, sin (yrj) = ; and this
leads to y7) = Zkn, so that z = 2&?rt.

Finally, e* never vanishes, for e**e-*i = 1, and, if
e* =s 0, this equation would give an infinite value for
e~*>, which is impossible.

Since z = z+iy = r cos 0+ir sin 0, any complex number
may be written in the form z = re^, where | z \ = r,
arg z = #, since we have now assigned a meaning to e'0.

FUNCTIONS OF A COMPLEX VARIABLE 23

By term-by-term differentiation of the power series
defining e* t we readily see that

T *>* = *

dz

III. The trigonometrical and hyperbolic functions.

We define sin z and cos z, when z is complex, as the
sum-functions of power series, just as we do for sin x
and cos x when x is real. Thus

sin Z = -

(2n+l)I*

and, since each of these power series has an infinite radius
of convergence, sin z and cos z are integral functions.

By term-by-term differentiation of these power series,
we deduce at once that the derivatives of sin z and cos z
are cos z and sin z respectively.

The other trigonometrical functions are then defined by

sin z A 1 1 1

tan z = , cot z = , sec z = , cosec z = - .

cosz tanz cosz sinz

If we denote exp iz by e**, according to our agreed con-
vention, we readily obtain the results

cos z+i sin z = e**> cos z i sin z == er** ;
leading to Euler's formulae

cos z = - (e' +*-*) , sin z = - (et'-e-**).

& +->i

From these formulae, and the addition formula for e*,

we find that . , ,

sm 2 z+cos 2 z =5 1 ;

and the addition theorems

sin(z ) = sin z cos cos z sin ,
cos(z J) s=s cos z cos S^sin z sin ,

also hold for complex variables. As all the elementary
identities of trigonometry are algebraic deductions from

24 FUNCTIONS OF A COMPLEX VARIABLE

these fundamental equations, all such identities also hold
for the trigonometrical functions of a complex variable.

The hyperbolic functions of a complex variable are
also defined in the same way as for real variables. The
two fundamental ones, from which the others may be
derived, are

sinh z = |(e f e~*), cosh z = \(e* +e~ f ).

These two functions are clearly regular in any bounded
domain.

The important relations

sin iz = i sinh z, cos iz = cosh ,
sinh iz = i sin z, cosh iz = cos z,

are easily proved and are of great usefulness for deducing
properties of the hyperbolic functions from the correspond-
ing properties of the trigonometrical functions.
If we write z = x+iy,

sin z = sin x cosh y +i cos x sinh y,

and we see that sin z can only vanish if

sin x cosh y = 0, cos x sinh y = 0.

Now coshy>l, and so the first equation implies that
sin x is zero. Hence x = mr, (n = 0, 1, ^2, ...). The
second then becomes sinh y = 0, and this has only one
root y = 0. Hence sin z vanishes if, and only if,
z = ttTr, (n = 0, 1, 2, ...). Similarly, we can show that
cos z vanishes if, and only if, z = (n-\-%)ir.

IV. The logarithmic function.

When x is real and positive, the equation e" = x has
one real solution u = log x. If z is complex, however, but
not zero, the corresponding equation exp w = z has an
infinite number of solutions, each of which is called a
logarithm of z. If w = u+iv we have

e"(cos t>+*' sin t;) = z.

FUNCTIONS OF A COMPLEX VARIABLE 25

Hence we see that v is one of the values of arg z and e u = \z\.
Hence u = log | z |. Every solution of exp w = z is thus
of the form

w = log |z|+iargz.

Since arg z has an infinite number of values, there is an
infinite number of logarithms of the complex number z,
each pair differing by 2iri. We write

Log z = log \z | + i arg z,

so that Log z is an infinitely many- valued function of z.

The principal value of Log z, which is obtained by
giving argz its principal value, will be denoted by logz,
since it is identical with the ordinary logarithm when z
is real and positive. We refer again to the logarithmic
function in the next section, where many-valued functions
are discussed in more detail.

V. The general power f .

So far we have only defined a 1 when a>0. If z and
denote any complex numbers we define the principal
value of the power *, with ^ as the only condition,
to be the number uniquely determined by the equation

where log is the principal value of Log . By choosing
other values of Log we obtain other values of the power
which may be called its subsidiary values. All these are
contained in the formula

Hence f has an infinite number of values, in general,
but one, and only one, principal value.

Example, t* denotes the infinity of real numbers
exp{i(logi+2kiri)} = exp {i(%7Ti+2kiri)}
= exp ( ^TrZk-rr).
exp (Jw) is the principal value of the power i 4 .

If 0, Rz>0, we define f to be zero.

26 FUNCTIONS OF A COMPLEX VARIABLE

9. Many-valued Functions

In the definition of a regular function given in 7, we
note that a regular function must be one-valued (or
uniform). Quite a number of elementary functions, such
as 2 (a not an integer) log z, arc sin z are many- valued.
To illustrate the idea of many-valuedness, let us consider
the simple case of the relation w 2 = z. On putting
z = re*Q, w = Re i( t> 9 we get

For given r and 9(<2rr), two obvious solutions are
Wl = | \/r | e** 8 and w 2 = | y> I e^ e +^ = -| y> I* 4 **,

and these are the only continuous solutions for fixed 0,
since |yV | and | \/r \ are the only continuous solutions
of the real equation x 2 = r, r>0.

In particular, for a positive real z, that is when = 0,
w = | \/r | and u> 2 = | \/ r l> an d both w l and tu 2 are
one-valued functions of z defined for all values of z.

If we follow the change in w l as varies from to 27r,
in other words, as the variable z describes a circle of radius r
about the origin, w l varies continuously and we see that
the final value of w l is | \Jr \ e^ ' 27rt = | ^/r \ = w 2 .

Hence the function w l is apparently discontinuous
along the positive real axis, since the values just above
and just below the real axis differ in sign and are not
zero (except at the origin itself). If, however, z describes
the circle round the origin a second time, the values of
w l continue those of w 2 and at the end of the second
circuit we have w 2 = w l along the positive real axis.

We thus see that the equation w 2 = z has no continuous
one-valued solution defined for the whole complex plane,
but w 2 = z defines a two-valued function of z. The two
functions w l = | \/r \ e W and w? 2 = | \/r \ e^ are
called the two branches of the two-valued function
w 2 = z. Each of these branches is a one- valued function

FUNCTIONS OF A COMPLEX VARIABLE 27

in the z-plane if we make a narrow slit, extending from
the origin to infinity along the positive real axis, and
distinguish between the values of the function at points
on the upper and lower edges of the cut. If OA = x , in
fig. 2, the value of w l at A(6 = 0) is | \/# | and the value

B *

FIG. 2.

of w l at B(6 = 2n) is | \/a? I- Since the cut effectively
prevents the making of a complete circuit about the origin,
if we start with a value of z belonging to the branch w v
we can never change over to the branch w%. Thus w l (and
similarly u> 2 ) is one-valued on the cut-plane.

There is an ingenious method of representing the two-
valued function w 2 = z as a one-valued function, by
constructing what is known as a Riemann surface.
This is equivalent to replacing the ordinary z-plane by
two planes P l and P 2 : we may think of P l as superposed
on P 2 . If we make a cut, as described above, in the
two planes, we make the convention that the lower edge
of the cut in P l shall be connected to the upper edge of
the cut in P 2 and the lower edge of the cut in P 2 to the
upper edge of the cut in P v Suppose that we start with
a value z of z, t#J being the corresponding value of w l
on the plane P 1 , and let the point z describe a path,
starting from z , in the counter-clockwise sense. When
the moving point reaches the lower edge of the cut in Pj
it crosses to the upper edge of the cut in P 2 , then describes
another counter-clockwise circuit in P 2 until it reaches
the lower edge of the cut in this plane. It then crosses
again to the upper edge of the cut in P 1 and returns to its
starting point with the same value W with which it started.
This corresponds precisely to the way in which we obtain
the two different values of \/z, and so \/z is a one-valued
function of position on the Riemann surface.

28 FUNCTIONS OF A COMPLEX VARIABLE

There is no unique way of dividing up the function
into branches, and we might have cut the plane along
any line extending from the origin to infinity, but the
point z = is distinguished, for the function w = \/z 9
from all other points, as we shall now see.

We observe that, if z describes a circle about any point
a and the origin lies outside this circle, then arg z is not
increased by 2n but returns to its initial value. Hence
the values of w l and u> 2 are exchanged only when z turns
about the origin. For this reason the point z = is called
a branch-point of the function w = \/z and, as we have
already stated, w^z) and w 2 (z) are called its two branches.

Since turning about z = oo means, by definition,
describing a large circle about the origin, the point z = oo
is also a (conventional) branch-point for w = <\/z.

The relation w n = z defines an n- valued function of z,

i
since z* has n, and only n, different values

87T

n

, (s = 0, 1, 2, ..., ft 1).

The point z = is a branch-point, and the Riemann
surface appropriate to this function consists of n sheets
P 19 P 2 , ..., P n - Plainly z = 1 is a branch-point for
w = \/(z 1) and the cut is made from z = 1 to z = oo.

For w = Log z, since w = log r+i(0-f 2for), every
positive and negative integer k gives a branch, so Log z
is an infinitely many- valued function of z. The Riemann
surface consists of an infinity of superposed planes, each
cut along the positive real axis, and each edge of each
cut is joined to the opposite edge of the one below. The
points z = and z = oo are branch points.

For w = <\/{(za)(zb)} we make a cut on each plane
along the straight line joining the points z = a and z = 6,
and join the planes P v and P f cross- wise along the cut.
In this case infinity is not a branch-point.

If u>= \/{(*-~ a i)(*~ a i)---( z "~ a *)}; when k is even,

FUNCTIONS OF A COMPLEX VARIABLE 29

we make cuts joining pairs of points a r , a f ; and when
k is odd, one of these points must be joined to oo, as in
the case k = 1. The edges of the various cuts on the
two planes are joined cross- wise as in the case w = ^/z.

Considerable ingenuity is required in constructing
Riemann surfaces for more complicated functions, but it
is beyond our scope to pursue this question further.

Note on notation. In what follows we shall frequently
use w, z and to denote complex numbers, and, whenever
they are used, it will be understood, without further
explanation, that

w = u+iv, z SB x+iy, ss +1*77.

Other symbols, such as t, r, s, may occasionally be used
to denote complex numbers, but we do not specify any
special symbols to denote their real and imaginary parts.

EXAMPLES I

1. Prove that | z l -z 9 1 2 + | z^z 9 | = 2\ z l \*+2 \ z 2 | ; and
deduce that

all the numbers concerned being complex.

2. Prove that the area of the triangle whose vertices are
the points z l9 z s , z 3 on the Argand diagram is

{(* 2 -* 8 )|Zl |*/*l}.

Show also that the triangle is equilateral if

3. Determine the regions of the Argand diagram defined by
|s-3|<l; \z-a\ + \z-b\^k(k>0); \ z*+az+b \<r*.

In the last case, show that, if z l9 z a are the roots of z*+az+b = 0,
we obtain two regions if r< J | z l z t |.

4. A point P(a+ib) lies on the line AB, where A is z = p
and B is z = 2ip. If Q is p a /(a +ib), find the polar coordinates

30 FUNCTIONS OF A COMPLEX VARIABLE

of P and Q referred to the origin as pole and the real axis
as initial line. Indicate the positions of P and Q in an
Argand diagram. If P move along the line AB and G is the
point z = p, prove that the triangles OAP, OQC are similar,
and that the locus of Q is a circle.

5. If /<) - ' <**0),/(0) = 0,

prove that {/(z) /(0)}/z-> as z-$~ along any radius vector,
but not as z-> in any manner.

6. Prove that the function u+iv = /(z), where

is continuous and that the Cauchy-Riemann equations are
satisfied at the origin, yet/'(0) does not exist.

7. Show that the function /(z) = \f\ xy \ is not regular
at the origin, although the Cauchy-Riemann equations are
satisfied at that point.

8. If /(z) is a regular function of z, prove that

9. If w ^/(z) is a regular function of z such that/'(z) ^ 0,
prove that

If \f'(z) \ is the product of a function of x and a function oft/,
show that

f(z)

where a is a real and )3 and y are complex constants.
10. Prove that the functions

(i) u t
(ii) u = sin a; cosh y +2 cos a; sinh y -\-x* t/ f

both satisfy Laplace's equation, and determine the corre*
spending regular function u+iv in each caae,

FUNCTIONS OF A COMPLEX VARIABLE 31

11. If w = arc sin z, show that w = nrr ii

according as the integer n is even or odd, a cross-cut being
made along the real axis from 1 to oo, and from oo to 1
to ensure the one-valuedness of the logarithm.

12. If w =* V{(1-*)(1 +**)>, ^ the point (2, 0) and P
a point in the first quadrant, prove that, if the value of w
when z = is 1, and z describes the curve OP A, the value
of w at A is i\/5.

13. If w = V(2 2z+z f )> and z describes a circle of
centre z = 1 +i and radius V2 in the positive sense, determine
the value of w (i) when z returns to O, (ii) when z crosses the
axis of y, given that z starts from O with the value + \/2 of w.

14. Prove that log(l +z) is regular in the z-plane, cut along
the real axis from oo to 1, and that this function can be
expanded in a power series

z* z* z*

'-2 + 3-4 +
convergent when | z \ < 1.

15. Prove that the function

-"< n+1)

,

n~l "I

is regular when | z \ < I and that its derivative is o/(z)/(l -fa).
Hence deduce that/(z) = (l+z) a .

16. (i) Prove that the exponential function e 9 is a one-
valued function of z.

(ii) Show that the values of z = a*", when plotted on the
Argand diagram for z, are the vertices of an equiangular
polygon inscribed in an equiangular spiral whose angle is
independent of a.

17. If 0<a <a 1 <...<a ll , prove that all the roots of the
equation

lie outside the circle | z | = 1.

18. Show that, if 6 is real and sin sin <f> = 1, then

where n is an integer, even or odd, according as sin 0>0
or sin 0<0. [If ^ = o+t'0, we have sin a cosh ft = cosec 0,
cos a sinh = 0. Solve for a and ft.]

CHAPTER II

CONFORMAL REPRESENTATION

10. Isogonal and Gonformal Transformations

The equations u = u(x, y), v = v(x, y) may be regarded
as setting up a correspondence between a domain D of the
(x, t/)-plane and a domain D f of the (u, v)-plane. If the
functions u and v are continuous, and possess continuous
partial derivatives of the first order at each point of D t
then any curve in D, which has a continuously turning
tangent, corresponds to a curve in D' possessing the same
property, but the correspondence between the two domains
is not necessarily one-one.

For example, if u = x* 9 v = t/*, the circular domain
f +2/ f <l corresponds to the triangle formed by the lines
u = 0, v = 0, w+v = 1, but there are four points of the
circle corresponding to each point of the triangle.

If two curves in the domain D intersect at the point P,
(X Q , y ) at an angle 0, then, if the two corresponding curves
in D' intersect at the point (U Q , v ) corresponding to P
at the same angle 0, the transformation is said to be isogonal.
// the sense of the rotation as well as the magnitude of the
angle is preserved, the transformation is said to be conf ormal.

Some writers do not distinguish between isogonal and
conf ormal, but define conformality as the preservation of the
magnitude of the angles without considering the sense.

If two domains correspond to each other by a given
transformation u = u(x, y), v = v(x 9 y), then any figure in
D may be said to be mapped on the corresponding figure
n

CONFORMAL REPRESENTATION 33

in D' by means of the given transformation. We have
already defined isogonal and conformal mapping, but it
should be observed that, if one domain is mapped isogonally
or conformally upon another, the correspondence between
the domains is not necessarily one-one. If to each point
of D there corresponds one, and only one, point of D',
and conversely, the mapping of D on Z)', or of D 9 on Z),
is said to be one-one or biuniform.

Suppose that w = f(z) is regular in a domain D of the
z-plane, z is an interior point of Z), and C l and C 2 are
two continuous curves passing through the point z , and
let the tangents at this point make angles a l9 a a with the
real axis. Our object is to discover what is the map of
this figure on the uj-plane. For a reason which will
appear in a moment, we suppose thatf(z Q ) ^ 0.

Let Zj and z a be points on the curves C l and (7 a near to
z and at the same distance r from z q , so that

z = re'i , z a z c =

then, as r-> 0, 1 -> a l and 2 -> a a .

The point z corresponds to a point t0 in the u?-plane
and z l and z 2 correspond to points w l and u? a which describe
curves /\ and F 2 . Let

then, by the definition of a regular function,

and, since the right-hand side is not zero, we may write
it JRe<A. We have

and BO lim (^j QI) == A or

lim <f> l = a x -f A.

34 FUNCTIONS OF A COMPLEX VARIABLE

Thus we see that the curve F l has a definite tangent
at WQ making an angle 04 +A with the real axis.

Similarly, F% has a definite tangent at u> making an
angle a 2 +A with the real axis.

It follows that the curves F l and F 2 cut at the same angle
as the curves C l and C 2 . Further, the angle between the
curves F l9 F^ has the same sense as the angle between O f 1 , (7 2 .
Thus the regular function w=f(z), for which /'(z ) ^ 0>
determines a conformal transformation. Any small figure
in one plane corresponds to an approximately similar
figure in the other plane. To obtain one figure from the
other we have to rotate it through the angle A = arg (f'(z Q )}
and subject it to the magnification

It is clear that the magnification is the same in all directions
through the same point, but it varies from one point to
another.

If is a regular function of w and w is a regular function
of z, then is a regular function of z, and so, if a region
of the z-plane is represented conformally on a region of
the w-plane and this in its turn on a region of the -plane,
the transformation from the z-plane to the {-plane will
be conformal.

There exist transformations in which the magnitude
of the angles is conserved but their sign is changed. For
example, consider the transformation

w = x iy ;

this replaces every point by its reflection in the real axis
and so, while angles are conserved, their signs are changed.
This is true generally for every transformation of the
form

. (I)

CONFORMAL REPRESENTATION 35

where f(z) is regular ; for it is a combination of two
transformations

(i){~, (ii) *

In (i) angles are conserved but their signs are changed,
and in (ii) angles and signs are conserved. Hence in the
given transformation, angles are conserved and their signs
changed. Thus (I) gives a transformation which is isogonal
but not conformal.

We have seen that every regular function t0=/(z),
defined in a domain in which /'(z) is not zero, maps the
domain in the z-plane conformally on the corresponding
domain in the u;-plane. Let us now consider the problem
from the converse point of view. Given a pair of differenti-
able relations of the type

u = u(x 9 y), v = v(x, y) . . (2)

defining a transformation from (x y y) -space to (u, v)-space
does there correspond a regular function w = /(z) ?

Let da and ds be elements of length in the (u, v) -plane
and (x, y) -plane respectively. Then da 1 = du z -{-dv 2 9
ds 2 = dx^+dy 2 and so, since

du du 8v dv

du = dx + ~ dy , dv = dx + dy ;
8x dy 8x 8y

da 2 = Edx*+2Fdxdy+Gdy\
where

l-Y F=- 8 - + -
\dx) ' 8x By "*" 8x 8y '

E = 4-

dx

Then the ratio do : ds is independent of direction if

E F O

36 FUNCTIONS OF A COMPLEX VARIABLE

On writing A* for E (or Q), where h depends only on
x and y and is not zero, the conditions for an isogonal
transformation are

'- 1 * (*)'-*

du du dv 8v _
'dx dy ~8x 8y ~~

The first two equations are satisfied by writing u 9 = h cos a,
v m = h sin a, u y = A cos /?, v, = A sin , and the third is
plainly satisfied if

Hence the correspondence is isogonal if either

(a) u 9 = v v , v x = u v or (b) u m = v v , v x = u v .

Equations (a) are the Cauchy-Riemann equations and
express that u+iv = f(x-{-iy) where f(z) is a regular
function of z. Equations (b) reduce to (a) by writing
v for v, that is, by taking the image figure found by
reflection in the real axis of the u;-plane. Hence (b)
corresponds to an isogonal, but not conformal trans-
formation, and so it follows that the only conformal
transformations of a domain in the z-plane into a domain
of the u?-plane are of the form w = /(z) where f(z) is a
regular function of z.

The casef'(z) = 0.

We laid down above the condition that /'(2 ) = 0.
Suppose now that/'(z) has a zero * of order n at the point z 0f
then, in the neighbourhood of this point,

where a = 0. Hence

or Pl f "* = | a | r n + l

See 36,

CONFORMAL REPRESENTATION 37

where A = arg a. Hence

Similarly, lim< 2 = A+(n+l)a 2 .

Thus the curves F l9 F 2 still have definite tangents at
w , but the angle between the tangents is

Also the linear magnification, lim (pjr) 9 is zero. Hence
the conformal property does not hold at such a point.

For example, consider w = z*. In this arg w = 2 arg z
and the angle between the line joining the origin to the point
t# and the positive real axis is double the angle between
the line joining the origin to the corresponding point z and
the positive real axis in the z-plane. Corresponding angles
at the origins are not equal because, at z = 0, dwjdz = 0.

Points at which dw/dz = or oo will be called critical
points of the transformation defined by w =/(z)- These
points play an important part in the transformations.

11. Harmonic Functions

Solutions of Laplace's equation, V 2 F = 0, are called
harmonic functions ; and, in applications to mathematical
physics, an important problem to be solved is that of
finding a function which is harmonic in a given domain
and takes given values on the boundary. This is known as
Dirichlet's problem. In the three-dimensional case, if we
make a transformation from (#, y, z)-space to (, T?, )-space,
it will, in general, alter Dirichlet's problem. If F(x, y, z)
is harmonic, and we make the transformation

* = <i( >?> y = < a ( *n> 0. ^ = <&,(, 7), )>

the function F 1 (^ : , 77, f ), into which F(z, y, z) is transformed
is not, in general, harmonic in (, TJ, )- 8 P ace - In two-
dimensional problems, however, if the transformation is

38 FUNCTIONS OF A COMPLEX VARIABLE

conformal, Laplace's equation in (x, y)-space corresponds to
Laplace's equation in (u, v)-space, and the problem to be
solved in (u, v)-space is still Dirichlet's problem. To prove
this, consider a transformation

u = u(x, y), v = v(x, y) . . . (1)

where w = u+iv is a regular function of z = x+iy, say
w = /(z). Let Z) be a domain of the (x, y) -plane throughout
which /'(z) T 0, and let J be the domain of the (u, v)-plane
which corresponds to D by means of the given trans-
formation.

If x and y are the independent variables, and V any
twice-differentiable function of x and y, we have *

8 2 V 8 2 V 8 2 V 8V 8V

8u . 8u _ . 8v . , 8v .

and du = dx + dy, dv = dx + dy, . (3)
ox cy ox oy

. . (4)

with a similar expression for d 2 v.

On substituting for du, dv, d z u, d 2 v the expressions
(3) and (4), the expression (2) for d 2 V becomes a quadratic
expression in the differentials of the independent variables
dx and dy and, on selecting the coefficients of dx 2 and dy 2 9
we get

4. 2 i
a/ "*"

^\ f a. 2

+

..

dy dy av 2 \8y du dy* dv dy*'

But we have already seen that u and v satisfy the Cauchy-
Riemann equations

du __ dv dv _ 8u

lfa~ 8y' fa~~ ~"8y 9
* See P.A., p. 233.

CONFORMAL REPRESENTATION

39

and also Laplace's equation. On addition we therefore get
8 2 F 8 2 F\

M 2 l
J '

;rr + ^~r =

3 2 F A

^r + -T-T = 0.

Now f'(z) = tt+u;, and so the last bracket is equal
to |/'(z)| 2 , the square of the linear magnification of the
transformation. Since this is not zero, it follows that if

then

12. Superficial Magnification

We have already seen that the linear magnification
at any point in a conformal transformation w = f(z) is
|/'(z)|, it being supposed that f'(z) ^0. We now prove
that the superficial magnification is |/'(2)| 2 . If A be the
closed domain of the uj-plane which corresponds to a
closed domain D of the z-plane, the area A of A is given by

'-///"*-//.

dxdy,

by the well-known theorem for change of variables in a
double integral.* Now

d(u, v) _ dtt dv dv 8u _
d(x, y) ~~ 8x By 8x dy ~~
by the Cauchy-Riemann equations : but, as we have just seen,

du . dv 2
dx 8x

and so m C C , , X|9J ,

\f'(z)\*dxdy.

dx) \8y)

I/'WI 1

This proves the theorem.

* P.A., p. 302; or Gillespie, Integration, p.
cited as Q.I.)

40. (Hereafter

40 FUNCTIONS OF A COMPLEX VARIABLE

13. The Bilinear Transformation

We have seen that a regular function wf(z), for
which /'(z ) 7^ 0, gives a continuous one-one representation
of a certain neighbourhood of the point 2 of the z-plane
on a neighbourhood of a point W Q of the u?-plane, and
that this representation is conformal. It may be expressed
in another way by saying that by such a transformation
infinitely small circles of the z-plane correspond to infinitely
small circles of the t0-plane. There are, however, non-
trivial conformal transformations for which this is true
of finite circles : these transformations will now be
investigated.

Let A, J3, G denote three complex constants, A, 5,
their conjugates and 2, a complex variable and its
conjugate ; then the equation

(A+I)zz+Bz+Bz+C+D = Q . . (1)
represents a real circle or a straight line, provided that

BB>(A+A)(C+C). ... (2)

For, if we write A = a+ia' 9 B = b+ib', C~c+ic',
z = x+iy, (1) becomes

a(&+y*)+bxb'y+e =

which is the equation of a circle. It reduces to a straight
line if a = \(A +A) = 0. If r be the radius of this circle,

_

4a a 4a 2 a'

and the circle is real provided that

which is the same as condition (2).

Conversely, every real circle or straight line can, by
suitable choice of the constants, be represented by an
equation of the form (1) satisfying the condition (2).

CONFORMAL REPRESENTATION 41

The transformation

"-jSl < 3 >

where a, j8, y, 8 are complex constants is called a bilinear
transformation. It is the most general type of trans-
formation for which one and only one value of z corresponds
to each value of w, and conversely. Since the bilinear
transformation (3) was first studied by Mobius (1790-1868)
we shall, following Carath6odory,* call it also a M6bius'
transformation.

The expression a8 j8y, called the determinant
of the transformation, must not vanish. If aS /?y = 0,
the right-hand side of (3) is either a constant or meaning-
less. For convenience it is sometimes agreed to arrange
that aS /?y = 1. The determinant in the general case
can always be made to have the value unity if the
numerator and denominator of the fraction on the right-
hand side of (3) be divided by \/( a <$~~/ty)-

If we write

^ = (a/y) +(j8y-a8)(a>/y), a> = l/, { = yz+8 . (4)

it is easily seen that (3) is equivalent to the succession of
transformations (4).

Now w = z+a corresponds to a translation, since the
figure in the w?-plane is merely the same figure as in the
z-plane with a different origin.

Consider next w = pz, where p is real. The two figures
in the z-plane and the w-plane are similar and similarly
situated about their respective origins, but the scale of
the w-figure is p times that of the z-figure. Such a trans-
formation is a magnification.

In the third place we consider w = ze*0. Clearly
| w | = | z | and one value of arg w is 0+arg z, and so
the w-figure is the z-figure turned about the origin through

* Conformed Representation (Cambridge, 1932).

42 FUNCTIONS OF A COMPLEX VARIABLE

an angle in the positive sense. Such a transformation
is a rotation.

Finally, consider w = 1/z. If | z \ = r and arg z = 0,
then | w | = 1/r and arg w = 0. Hence, to pass from
the t^-figure to the z-figure we invert the former with respect
to the origin of the w-plane, with unit radius of inversion,
and then construct the image figure in the real axis of the
w-plane.

The sequence of transformations (4) consists of a
combination of the ones just considered ; of these, the
only one which affects the shape of the figures is inversion.
Since the inverse of a circle is a circle or a straight line
(circle with infinite radius), it follows that a Mobius'
transformation transforms circles into circles.

The transformation inverse to (3) is also a Mobius'
transformation

z = f (-8)(-a)-y ^ . . (5)

yw a

Further, if we perform first the transformation (3),
then a second Mobius' transformation

the result is a third Mobius' transformation

_ Az+B
{ ~ Fz+A

where AA-BF = (08 j8y)(a'8'-j8y) ^ 0.

Since the right-hand side of (3) is a regular function
of z, except when z = 8/y, Mobius' transformations are
conformal.

If we write the equation of a circle (1) in the form

P=0. . . (6)

CONFORMAL REPRESENTATION 43

where d and ( are real, and then substitute

yw> a yw a

in (6) we get an expression of the form

Dww+Ew+Ew+F = . . . (7)
where

D
and

are both real, and it is easy to verify that the coefficients
of w and w are conjugate complex numbers. Hence (7)
represents a circle in the u?-plane, since it is of the same
form as (1).

14. Geometrical Inversion

From what we have just seen it would be natural to
expect that there would be an intimate relation between
Mobius' transformations and geometrical inversion.

Let S be a circle of centre K and radius r in the z-plane.
Then two points P and P lf collinear with K, such that
KP . KP l = r 2 , are called inverse points with respect to
the circle 8, and it is known from geometry that any circle
passing through P and P l is orthogonal to S. In the case of
a straight line 8, P and P l are inverse points with respect
to s, if P! is the image of P in 8. If P, P l9 and K are the
points z, Zi, and k we have

\(z^k)(z^k)\ = r 2 , arg fo-fc) = arg (z-&), . (8)

the second equation expressing the collinearity of the
points K y P y P v The two equations (8) are satisfied, if,
and only if,

(z l -k)(z-Jc) = r 2 . . . . (9)
If 8 is the circle

dzz+Bz+Bz+ Q = 0, . . . (10)

44 FUNCTIONS OF A COMPLEX VARIABLE

which may be written

we see that (10) is a circle with centre B/^ and radius

BB-

Hence equation (9) becomes

BB-AQ

which on simplification is

dz l z+Bz l +Bz+(S=0. . . . (11)

We thus get the relation between z and its inverse z l
from the equation of S by substituting z l for z and leaving
z unchanged. On solving (11), the transformation is

We now prove the theorem :

The bilinear transformation transforms two points which
are inverse with respect to a circle into two points which are
inverse with respect to the transformed circle.

If z and z 1 are inverse with respect to the circle (10)
then (11) is satisfied. Make the transformation (3) and
let w and w l be the transformed points. We have

.

z = - , z = n: r
l a ywa

and, on substituting these values in (11) we get an
expression

=

where the coefficients D, E, $, F are the same as those of
(7) ; in fact we get (7) with w replaced by w v But this
is the condition that w and w l are inverse points with

CONFORMAL REPRESENTATION 45

respect to the transformed circle (7). The theorem is
therefore proved.

The inversion (12) can be written as a succession of
two transformations

-w-(S

t0 = Z, Zi=* -77-.

1 Aw+B

The first is a reflection in the real axis and the second is
a Mobius* transformation. The first preserves the angles
but reverses their signs ; the second is conformal. Hence
inversion is an isogonal, but not conformal, transformation.
Since inversion is a one-one isogonal, but not conformal,
transformation, it is clear that the result of two, or of an
even number of inversions, is a one-one conformal trans-
formation, since both the magnitude and sign of the angles
is preserved. In other words, the successive performance
of an even number of inversions is equivalent to a Mobius*
transformation .

15. The Critical Points

If the z-plane is closed by the addition of the point
z = oo, then (3) and (5) show that every Mobius' trans-
formation is a one-one transformation of the closed z-plane
into itself.

If y ^ the point w = a/y corresponds to z = oo and
w = oo to z = S/y ; but, if y = 0, the points z = oo,
w = oo correspond to each other. Since, from (4),

dw aS /?y
'dz^ (yz+S) 2 '

the only critical points of the transformation are z = oo
and z = S/y.

These two critical points cease to be exceptional if we
extend the definition of conformal representation in the
following manner. A function w = /(z) is said to transform
the neighbourhood of a point z conformally into a neigh-
bourhood of w = oo, if the function i = l//(z) transforms

46

FUNCTIONS OF A COMPLEX VARIABLE

the neighbourhood of z conformally into a neighbourhood
of t = 0. Also w = f(z) is said to transform the neighbour-
hood of z = oo conformally into a neighbourhood of W Q
if w = <() =/(!/) transforms the neighbourhood of
= conformally into a neighbourhood of w . In this
definition w may have the value oo.

With these extensions of the definitions we may now
say that every Mobius 9 transformation gives a one-one
conformal representation of the whole closed z-plane on the
whole dosed w-plane. In other words, the mapping is
biuniform for the complete planes of w and z.

16. Coaxal Circles

Let a, b y z be the affixes of the three points A, B, P of
the z-plane. Then

2 ft *
arg = APB,
6 z a

if the principal value of the argument be chosen. Let
A and B be fixed and P a variable point.

FIG. 3,.

If the two circles in fig. 3 are equal, z 1? z 2 , z 8 are the
affixes of the points P lt P 2 , P 8 and APB = 0, we see that

Z 2 * /i 2 i ft * z*~~b

arg - - = 7T0 , arg = , arg

z a a

z, a

z 8 a

CONFORMAL REPRESENTATION 47

The locus defined by the equation

z 6

arg = 0, . . . (1)

z a

when is a constant, is the arc APB. By writing 0,
^0, 7T+0 for we obtain the arcs AP+B, AP^B 9 AP 3 B
respectively. The system of equations obtained by varying
from 77 to TT represents the system of circles which can
be drawn through the points A, B. It should be observed
that each circle must be divided into two parts, to each
of which correspond different values of 0.

Let T be the point at which the tangent to the circle
APB at P meets AB. Then the triangles TPA, TBP are
similar and

PB BT TP *'

Hence TAJTB = k 2 and so T is a fixed point for all positions
of P which satisfy

-k (2)

z-b ~ ' * * * ( '

where k is a constant. Also TP 2 = Tu4 . TB and so is
constant. Hence the locus of P is a circle whose centre
isT.

The system of equations obtained by varying k
represents a system of circles. The system given by
(1) is a system of coaxal circles of the common point
kind, and that given by (2) a system of the limiting point
kind, with A and B as the limiting points of the system.
If k-+ oo or if k-* then the circle becomes a point circle
at A or B. All the circles of one system intersect all the
circles of the other system orthogonally.

The above important result is of frequent application
in problems involving bilinear transformations. It may
be used to prove that the bilinear transformation transforms
circles into circles.

48

FUNCTIONS OF A COMPLEX VARIABLE
Suppose that the circle in the t0-plane is
w-X

W[JL

1 we substitute for w in terms of z from the bilinear
relation

w =

(3)

we obtain

z-A'

where

A' = -

a Ay '

,,__-?.,_

a -Ay

k,

and so the locus in the z-plane is also a circle.
We may write (3) in the form

a z+B/a
*i\ _

_ _ _

y z+3/y '

and since

z+8/y

represents a circle in the z-plane, the circle in the w-plane
corresponding to it is plainly

By taking special values of a, j8, y, S and k the boundaries
in the u?-plane corresponding to given boundaries in the
z-plane are easily determined.

For example, let 0/a = t", 8/y *= * and k = 1 : since the
locus \(zi)/(z+i)\ =* 1 is plainly the real axis in the z-plane,
this axis corresponds to the circle | w \ = | a/y | in the t^-plane :
this will be the unit circle if, in addition, | a | = | y |.

CONFORMAL REPRESENTATION 49

17. Invariance ol the Gross-Ratio

Let z v z 2 , z a , z 4 be any four points of the z-plane and
let w lf u> 2 , w s , u> 4 be the points which correspond to them
by the Mobius* transformation

d)

If we suppose that all the numbers z r , w r are finite,
we have

'~' ~ y Zf +8 y 2 .+8 ~ (y

and hence it follows that

(u>i w> 4 )(w 8 w 2 ) (! z 4 )(z 3

(2 1 -Z 2 )(2 3 -2 4 )

(2)

The right-hand side of (2) is the cross-ratio of the four
points z 1? z 2 , z s , z 4 , and so we have the result that the
cross-ratio is invariant for the transformation (1).

If equation (2) be suitably modified, it is still true if
any one of the numbers z r or one of the numbers w f is
infinite. For example, let z 2 = oo and w l = oo, then

tt>3-w 2 _ z 1 z i

(o)

Now suppose that z f , w f (r = 1, 2, 3) be two sets each
containing three unequal complex numbers. Suppose
first that these six numbers are all finite. Then the
equation

- -~ 2 )

'

when solved for w leads to a Mobius' transformation which
transforms each point z r into the corresponding point w r .
The determinant of the transformation has the value

-u?^^

50 FUNCTIONS OF A COMPLEX VARIABLE

It is also clear that (4) is the only Mobius' transformation
which does so. The result still holds, if (4) be suitably
modified, when one of the numbers z r or w r is infinite.

The equation (4) above may be used to find the
particular transformations which transform one given
circle into another given circle or straight line. A circle
is uniquely determined by three points on its circumference
and so we have only to give special values to each of the
tliree sets z r , w r (r = 1, 2, 3) and substitute them in (4).

Example. Let z t = 1, z a = i, z 3 = 1 and w l = 0,
u>t = 1, w 3 = oo then we get, after substitution in (4),

which transforms the circle | z \ = 1 into the real axis of
the w-plane and the interior of the circle | z \ < 1 into the upper
half of the u>-plane.

The easiest way to prove this is as follows. Equation (5)
is equivalent to

w i

2 ; _ .

w+i'

The boundary | z \ = 1 corresponds to | w i \ = | w+i |,
which is the real axis of the w-plane, since it is the locus of
points equidistant from w = i.

Since the centre z = of the circle corresponds to the
point w = i, in the upper half of the w-plane, the interior
of the circle | z \ = 1 corresponds to the upper half of the

Similarly, since w = i corresponds to z = oo, the outside
of the circle | z \ = 1 corresponds to the lower half of the
K;-plane.

It may be observed that although this use of the
invariance of the cross-ratio will always determine the
Mobius' transformation which transforms any given circle
into any other given circle (or straight line), it is not
necessarily the easiest way of doing so.

CONFORMAL REPRESENTATION

51

Thus, in the previous example, since z = 1 and 1
correspond to w =* and oo, the transformation must take
the form

fc(z-l)

ITT'

W =3

t corresponds to w = 1, we can determine k : thus
fc(t-l)

Since z

from which it readily follows that k =* i, and we obtain (5)
above.

18. Some special Mdbius' Transformations

I. Let us consider the problem of finding all the Mobius*
transformations which transform the half -plane I(z)^0 into
the unit circle \w\ ^1.

We observe first that to points z, z symmetrical with
respect to the real axis correspond points w, l/w inverse
with respect to the unit circle in the w-plane. (See
14.) In particular, the origin and the point at infinity
in the w?-plane correspond to conjugate values of z. Let
the required transformation be

w =

yz+8"

Clearly y^O, or the points at infinity in the two planes
would correspond. Since w = 0, w = oo correspond to
z = j8/a, z = S/y we may write j8/a = a, 8/y = a
and

a 2 a

w = -- 1
y z a

The point z = must correspond to a point on the circle
| w | = 1, so that

a

y

a

a

a

52 FUNCTIONS OF A COMPLEX VARIABLE

hence we may write a = ye*0, where is real, and obtain

w =

za

(1)

Since z = a gives to = 0, a must be a point of the upper
half-plane, in other words, Ia>0. With this condition,
(1) is the transformation required.

II. To find all the Mobius* transformations which
transform the unit circle \ z (^ 1 into the unit circle \w\^l.

Let

w =

yz+S*

In this case w = and w = oo must correspond to inverse
points z = a, z = I/a, where | a |<1. Hence j8/a = a,
I/a, and so

aa za

w = -

y 2 I/a y dz 1*
The point z = 1 corresponds to a point on | u; | = 1, and so

aa 1 a

aa

y

= 1.

It follows that, if is real, aa = ye*0 and so

za .

w - i v

This is the desired transformation ; for, if z = e*V, a
then

If z = re*V, where r<l, then

r 2 ~2r6 cos (-

-r cos ~

CONFORMAL REPRESENTATION

53

hence | w \<1 ; in other words, the interiors of the circles
correspond.

The identical transformation w = z is a special case of
the above : if the point 2 = corresponds to w = 0, then
a = and the transformation reduces to

w = zeiO.

If, in addition, dw/dz = 1 when z = we get

w = z.

III. The reader should find it easy to verify that the
transformation

w== p(z-a)

maps the circle \ z \ = p on the unit circle \ w \ = 1. If
| a |<p it maps | z \<p on | w |<1 and | z \>p on | w
If | a | >/> it maps | z \ >p on | w |< 1 and | z \<p on | w \

IV. Representation of the space bounded by three circular
arcs on a rectilinear triangle.

Consider three circles in the z-plane intersecting at
the point 2 = a.

B' (w-o)

z-plane

Fio. 4.

w-plane

The angles of the curvilinear triangle BCD of fig. 4
are such that a+jS-f y = TT. Consider the transformation

7

w = K

a

/I

(1
x

54 FUNCTIONS OF A COMPLEX VARIABLE

The point A(z = a) corresponds to w = oo and B(z = 6)
corresponds to w = 0. Equation (1) may be written

. k(a-b)

wk = ,

za

or w 9 = A/2', where w 9 = wk, z f = za and A = k(a b).
Since the changes of variable from w to w' and from z to z'
are mere translations, (1) is a pure inversion and reflexion.
Since z = a corresponds to w = oo and each of the three
circular arcs BC, CD, DB passes through A they correspond
to three straight lines in the u;-plane. The two arcs BC, BD
which pass through B correspond to two straight lines
passing through w = 0, and the arc CD to a straight line
which does not pass through w = 0.

It readily follows from (1) that the shaded areas of
fig. 4 correspond.

With the usual convention of sign, we regard a motion
round a closed simple contour, such as a circle, in the
clockwise sense as positive for the area outside and negative
for the area inside the contour.

If, by any conformal transformation, three points
A, B, C on a closed contour in the z-plane correspond to
the three points A', B', C' in the w- plane lying on the
corresponding closed contour, then the interiors correspond
if the points A', B', C 9 occur in the same (counter-clockwise)
order as the points A, B, C.

We can see in this way that the shaded areas in fig. 4
correspond. It also follows that the curvilinear triangle
formed by the arcs AC, CD, DA in the z-plane corresponds
to the portion of the u>-plane A'C'D'A' where A 9 is the
point at infinity.

EXAMPLES II

1. (i) Prove that, if u = a? 1 y 1 , v = y/(#*+y f ), both
u and v satisfy Laplace's equation, but that u+iv is not a
regular function of z.

CONFORMAL REPRESENTATION 55

(ii) Show that the families of curves u = const, v = const,
cut orthogonally if u = x*/y, v = # 2 +2?/ 2 but that the
transformation represented by u+iv is not conformal.

2. Prove that, if w = x+iby/a, 0<a<6, the inside of the
circle x* +y* = a 1 corresponds to the inside of an ellipse in
the to-plane, but that the transformation is not conformal.

3. Prove that, for the transformation w* = (z a)(z /J),
the critical points are z = o, z = jS, z = J(a + j3), w = 0,
w = $i(a-p).

Show also that the condition that z = oo is not a critical
point of the transformation w = /(z) is that lim z 2 /'(z) must

Jh>00

be finite and not zero.

4. If, by the inversion transformation x = k*(/p* 9

y = fcV P J , * = 2 //> 2 where r P = k *> f2 = z 2 +</ 2 +z*>
p* = ^-(-^-[-^ ^ e twice-differentiable function V(x, y, z)
becomes V^f, TJ, ) prove that if

(d a /df 2 + a 2 /^ 2 + av^D FI=O, then

(d*/dx* + B*ldy* + B*/dz 2 ) (V/r) = 0.

5. If w = cosh z, prove that the area of the region of the
u;-plane which corresponds to the rectangle bounded by the
lines x = 0, x = 2, y = 0, y = JTT is (TT sinh 4 8)/16.

6. If a is real and O<C<TT, find the area of the domain
in the u;-plane which corresponds by the transformation
w = e* to the rectangle a c^.x^.a+c 9 c<t/<c. Find
the ratio of the areas of the two corresponding domains and
prove that the ratio ->- e 20 as c -> 0.

7. Show that, if the function w =/(z), regular in | z \< R,
maps the circle | z \ = r<R on a rectifiable curve G in the
u>-plane, then the length of C is given by

r

I

J

\f'(reie)\rd0.
o

Show that the length of the curve into which the semi-
circular arc | z | = 1, Jn"^argz<i7r is transformed by
w = 4/(l+z) 2 is 2V2+2 log(l + \/2). (See 22, equation (4)).

8. Find the Mobius' transformations which make the
sets of points in the z-plane (i) a, 6, c, (ii) 2, 1 -ft,
to the points 0, 1, oo of the u^ane. In case (ii)

56 FUNCTIONS OF A COMPLEX VARIABLE

sketches the domains of the u?-plane and 2 -plane which
correspond.

9. Find a M6bius* transformation which maps the circle
1*1^1 on | w 1 | ^ 1 and makes the points z = , 1
correspond to w = , respectively. Is the transformation
uniquely determined by the data ?

10. Find the transformation which maps the outside of
the circle | z \ = 1 on the half -plane Ru;>0, so that the
points 2 = 1, i, I correspond to w = i, 0, i respectively.
What corresponds in the w- plane to (i) the lines arg z = const.,
| z | ^ 1, (ii) the concentric circles | z \ = r, (r>l) ?

11. Prove that w = (l+iz)/(i+z) maps the part of the
real axis between 2 = 1 and z = 1 on a semicircle in the
ttf-plane.

Find all the figures that can be obtained from the originally
selected part of the axis of x by successive applications of this
transformation.

12. Find what regions of the w-plane correspond by the
transformation w = (z i)/(z+i) to (i) the interior of a circle
of centre z = t, (ii) the region t/>0, x>0 9 \z+i\ < 2.
Illustrate by diagrams. Show that the magnification is
constant along any circle with z = i as centre.

13. Let GI, | zz l | =5 T! and C 8 , | z z % \ = r 8 be two
non -concentric circles in the 2-plane, C l lying entirely within
C f . Show that, if z = a, 2 = 6 are the limiting points of
the system of coaxal circles determined by C l and <7 a , then
w = k(z 6)/(2 a) transforms C l and (7, into concentric
circles in the w -plane with centres at w = 0. If the radii
of these concentric circles are p l and p 2 , show that, although
there is an infinite number of such representations, p t : p f
is a constant.

14. Prove that, if w = (oz + 0)/(y2 + 8) and a8 0y = 1,
then the linear and superficial magnifications are | y2 + S j" 1 ,

Show that the circle | y2 + 8 | = 1 (y ^ 0) is the complete
locus of points in the neighbourhood of which lengths and
areas are unaltered by the transformation. Prove that
lengths and areas within this circle are increased and lengths
and areas outside this circle are decreased in magnitude by
the transformation. (This circle is called the isometric circle.)

CHAPTER III

SOME SPECIAL TRANSFORMATIONS

19. Introduction

The fundamental problem in the theory of conformal
mapping is concerned with the possibility of transforming
conforrnally a given domain D of the z-plane into any
given domain D' of the w-plane. It is sufficient to consider
whether it is possible to map conforrnally any given
domain on the interior of a circle. For if = f(z) maps
D on | C|<1 and w = F(Q maps D' on | |<1, then
w = F{f(z)} provides a conformal transformation of D
into D'.

The fundamental existence theorem of Riemann states
that any region with a suitable boundary can be conforrnally
represented on a circle by a biuniform transformation. Rigor-
ous proofs of this existence theorem are long and difficult,
and it is beyond our scope to discuss the question here.

In the applications of conformal transformation to
practical problems, the problem to be solved is as follows :
given two domains D and D' with specified boundaries,
find the function w = f(z) which will transform D into D'
so that the given boundaries correspond. Although, by
Riemann 's existence theorem, we can infer the existence of
the regular function /(z), the theorem does not assist
us to find the particular function f(z) for each problem
whose solution is desired. We have seen that when the
two domains D and D' are bounded by circles, it is fairly
easy to find the Mobius' transformation which maps D
biuniformly on Z)'. Since for any arbitrary boundary
curves there is no general method of finding the appropriate

58 FUNCTIONS OF A COMPLEX VARIABLE

regular function f(z), it is important to know the types of
domain which correspond to each other when f(z) is one
of the elementary functions or a combination of several
such functions.

In this chapter we discuss some of the most useful
transformations which can be effected by elementary
functions. The reader, who is mainly interested in the
application of these transformations to practical problems,
will find the special transformations discussed here of
great value, but he must refer to other treatises for the
details of the practical problems to which they can be
applied.

Many useful transformations are obtained by combining
several simple transformations.

For example, the transformation

seems at first sight somewhat complicated, but on examination
it is seen to be a combination of the successive simple trans-

formations, Z = z 8 , == - - , t = *, w == .

1 6 t-\-l

It can be shown that (1) maps the circular sector | z j< 1,
0<arg z< JTT, conformally on the unit circle * | w |< 1.

20. The Transformations w = z n

Let w = u+iv = pety, z = x+iy = re^, then it follows
at once that p = r n , < = n6, so that

u+iv = r n (cos nO+i sin n0).

From the equations

u = r n cos ri0, v = r n sin n0, . . (1)
* CJompare 24 IV, equation (8), with a }.

SOME SPECIAL TRANSFORMATIONS 59

either or r may be eliminated, giving

f . . (2)

or tan n0 = - . . . (3)

u ^ '

Equation (2) shows that the circles r = c of the z-plane
and the circles p = c n correspond, and in particular, that
points on the circle r = | z \ = 1 are transformed into
points in the to-plane at unit distance from the origin.
The lines = const., radiating from the origin of the
z-plane, are transformed into similar radial lines <j> = const.
It should be noticed, however, that the line whose slope
is in the z-plane is transformed into the line whose slope
is n0 in the w-plane. Since z = is a critical point of the
transformation, the conformal property does not hold at
this point.

In the simple case w = z 2 , the angle between two radial
lines in the z-plane is doubled in the w-plane. The case
w = z 2 is typical, and we shall now consider it in greater
detail.

We consider first the important difference between the
transformation w = z 2 and the Mobius' transformations
discussed in the preceding chapter. In the latter, points
of the z-plane and of the t0-plane were in one-one
correspondence. For w = z 2 , to each point z there
corresponds one and only one point W Q = z^, but to a
point w Q there correspond two values of z, z= JV^ol*
z = \\/w \. If we wish to preserve the one-one corres-
pondence between the two planes, we may either consider
the w-plane as slit along the real axis from the origin to
infinity, or else construct the Riemann surface in the
t0-plane corresponding to the two-valued function of w
defined by w = z 2 . The method of constructing the
Riemann surface was described in 9.

If we use the cut w-plane, then the upper half of the
-plane corresponds to the whole cut u?-plane. There is

60 FUNCTIONS OF A COMPLEX VARIABLE

a one-one correspondence between points of the upper
half of the z-plane and points of the whole u?-plane, and
a one-one correspondence between points of the lower
half of the z-plane with points of the whole t0-plane ; but
if we choose one of the two branches of w = z 2 , say w v
the cut plane effectively prevents our changing over,
without knowing it, to the other branch w 2 . The positive
real z-axis corresponds to the upper edge and the negative
real z-axis to the lower edge of the cut along the positive
real axis in the u;-plane.

If we use the two-sheeted Riemann surface in the
w?-plane, the sheet P l corresponds to the upper half of the
z-plane for the branch w l9 and the sheet P 2 corresponds
to the lower half of the z-plane for the branch w 2 . Thus
there is a one-one correspondence between the whole
z-plane and the two-sheeted Riemann surface in the
u>-plane.

For w = z n , where n is a positive integer, a wedge of
the z-plane of angle 2n/n corresponds to the whole of the
t0-plane. If we divide up the z-plane into n such wedges,
each of these corresponds to one of the n sheets of the
n-sheeted Riemann surface in the w-plane.

If, for w = z 2 , we cut the u?-plane along the negative
real axis, then the sheet P l of the Riemann surface
corresponds to the half-plane Rz^O, and the sheet P 2 to
the half-plane Rz<0.

21. Further Consideration ol w = z 2
From the equations

w = u-}-iv = (x-\-iy) 2 = x 2 - y z +2ixy,
we have

u = x 2 i/ 2 , v = 2xy. . . (I)

By regarding u and v as curvilinear coordinates of points
in the z-plane, the transformation w = z 2 can be examined
from a knowledge of the curves in the z-plane which

SOME SPECIAL TRANSFORMATIONS

61

correspond to constant values of u and v. This method
is frequently used in applying the theory of conformal
transformation to practical problems.

Equations (1) show that the curves u = const., v = const,
in the z-plane are two orthogonal families of rectangular
hyperbolas.

The reader will easily verify that the shaded area in
the z-plane of fig. 5 between two hyperbolas 2xy = v lt

z-plane

FIG. 5.

u;-plane

2xy = v 2 corresponds to the infinite strip of the t^-plane
shaded in the figure. Hence w = z a maps the region
between two hyperbolas on a parallel strip.

If B l is at infinity, the point B\ is also at infinity,
and the interior of the hyperbola ABC is transformed
into the part of the upper half-plane above the line
A' B' C'.

We also observe that the transformation w = z* makes
circles \ za \ = c, (a, c real) 9 in the z-plane correspond to
limaqons in the w-plane.

Consider the circle

2 a = ce*"*, ... (2)
then

t/* -a 2 + c 2 == 2c(c cos 9 f

62 FUNCTIONS OF A COMPLEX VARIABLE

Hence, on writing w a 2 -j-c a = Be^ 9 so that the pole in
the w-plane is at w = a 2 c 2 , the polar equation of the
curve into which (2) is transformed is the Iiina9on

B = 2oc+2c 2 cos 0.

When a = c the Iima9on becomes a cardioid. This is
the case if the circle (2) touches Oy at the origin,

22. The Transformation w = y'*

Prom the equations

u*v* = x, 2uv = y, . * (1)
we get

. . (2)

By means of the first of the equations (2), to the straight
lines u = const, correspond parabolas with vertex at
x = w a and focus at the origin of the z-plane. To the
orthogonal system of straight lines v = const., we see,
by the second of the equations (2), there corresponds
another system of confocal parabolas with vertex at
x = v 2 .

Consider the particular parabola of the first system
corresponding to the value u = 1,

y 2 = 4(l-*). ... (3)

Its transform in the w-plane is the line through the point
w = 1 parallel to the t;-axis. The points, A, B, C in
fig. 6 correspond, in that order, to the points A', B', C' .
The reader can easily verify that the shaded areas
correspond, the two parabolas drawn corresponding to
values u = 1, and u = ^ (>1).

If w -> oo, the region developed in the 2-plane is the
area outside the parabola (3), which accordingly corresponds
to that part of the t#-plane to the right of the line u = 1.

If the parameter u tends to zero, the parabola
y 2 = 4u 2 (u 2 x) narrows down until it becomes a slit
along the negative real axis OX 19 which is a branch-line.

SOME SPECIAL TRANSFORMATIONS

63

Hence the portion of the w-plane between the line u = 1
and the line u = corresponds to the portion of the
z-plane between the parabola ABC, y* = 4(1 x), and
the cut along the negative real axis OX l from the origin
to --00.

2-plane

FIG. 6.

c f
plane

Hence we see that the portion of the w-plane
corresponds to the whole z-plane cut along the negative real
axis from to -co.

The simple w-plane is associated in a one-one corre-
spondence with a two-sheeted Riemann surface covering
the z-plane. The two sheets of the Riemann surface
would be connected along the edges of the cuts along the
negative real axis of the z-plane in the usual way.

The line u = 1 plainly corresponds to the same
parabola y a = 4(l x) as does the line u = 1. Hence
the portion of the w-plane to the left of the line u =* 1
corresponds to the region outside the parabola ABC
which lies on the second sheet of the Riemann surface.

If we combine w = y'z with a Mobius' transformation
by writing = (2/w)-~ 1 we see that the transformation

C---1 (4)

fc / * v*;

transforms the region outside the parabola (3) into the interior
of the unit circle in the {-plane. The points z = 1, z = 4,

64 FUNCTIONS OF A COMPLEX VARIABLE

z = oo correspond to the points = 1, = 0, {=1.
The focus of the parabola (3), z = 0, lies outside the region
of the z-plane which is under consideration, and it
corresponds to the point = oo outside the unit circle

Kl-i.

The reader should observe, however, that the preceding
transformation cannot be used to represent the inside of
the parabola ABC on the inside of the unit circle.

The transformation w = \/z just considered, illustrates
an important point in the use of many-valued functions for
solving problems in applied mathematics. The transformation
w = ^z could be used to deal with a potential problem in which
the field was the region outside the parabola ABC of fig. 6, but
it could not be used for a problem in which the field was the
space inside this parabola, since two points close to each
other, one on each edge of the cut along the branch-line OX l9
will transform into two points on the axis of v, one in the upper
and the other in the lower half -plane. Since these points
are not close together in the t-0-plane, they would correspond
to different potentials. It is important to realise that we
cannot solve potential problems by using transformations
which require a branch -line to be introduced into that part
of the plane which represents the field.

23. The Transformation w = tan 2 (j7r\/z)

We have just seen that w = (2/\/z) l cannot be used
to map the region inside the parabola t/ 2 = 4(1 x) on the
unit circle |t0|^l. We now consider a transformation
which enables us to do this.

The transformation can be considered as a combination
of the three transformations

w =

where w = u+iv, = +^'77, t = a-ftY, z = x+iy.
The first transformation can be written

Icos
"*" l+cos*

SOME SPECIAL TRANSFORMATIONS

65

If we consider the infinite strip between the lines = 0,
= J^TT of the -plane, we see that, by writing f = far -\-irj,
cos = i sinhrj and | w \ = 1. Thus, as y goes from
oo to oo along the line g = far, w describes the unit-
circle once. By writing f = irj , cos = cosh rj and w; is
real. Thus as 77 goes from +00 to 0, w goes from 1 to ;
and as 77 goes from to oo, w retraces its path from
to 1. Thus the strip = 0, = \n corresponds to
the cut-circle as illustrated in fig. 7. It is easy to verify

to-plane

f-plane

J-plane

z -plane

FIG. 7.

that the interiors correspond. The strip in the 2-plane
is plainly that between the lines a = and a = 1. As we
have already seen, t = \/z transforms the strip in the -plane
into the region inside the parabola ABC, f/ 2 = 4(l x),
with a cut from the origin to infinity along the negative
real axis. In fact, as or -> the parabola y 2 = 4a 2 (or 2 x)
becomes a very narrow parabola which is the slit illustrated
in the z-plane in fig. 7.

The transformation w = tan 2 (7r\/z) represents the
region inside the parabola ABO on the inside of the unit
circle | w \ = 1 in a one-one correspondence, for the real
axis of the w-plane between 1 and corresponds to the
real axis of the z-plane between oo and 0. The cuts in
the z-plane and w-plane are not needed for the direct trans-
formation from the w>-plane to the z-plane, but they are
needed for the subsidiary transformations used in order to
show how the boundaries of the various regions correspond.

66 FUNCTIONS OF A COMPLEX VARIABLE

Since dw/dz = 77 tan(}rr\/z) sec 2 ( j7r\/ 2 )/4 \/ z > w ^ich tends
to a finite non-zero limit as z-> 0, the points z = and
u> = are not critical points of the transformation, and
so the representation is conformal as well as one-one.

24. Combinations of w = z a with Mobius' Trans-
formations

I. Semicircle on half-plane or circle.

Consider the transformation
/ z _ j c \ 2

' (creal) - (1)

This is clearly a combination of

w = 2 and = (z ic)/(z+ic).

The second of these may be written *

. +1

for which it is clear that the circle | z | = c corresponds to
the imaginary axis of the -plane |-fl| = | 1|.

The boundary of the semicircle in the z-plane A DCS A
plainly corresponds to A'D'C'B'A' in the -plane, C' being
the point = -oo. The sense of description of the
two boundaries shows that the shaded areas correspond.

Now consider w = a : if w = pety, = re*0, we have

The shaded domain of the -plane corresponds to
7r<0<37r/2 and so the domain of the w-plane corresponding
to this is 27r<<<37r, which is of course the same as
r, or the upper half of the w-plane.

* The use of the results of 16 is frequently simpler than the
procedure of splitting up the transformation into its real and
imaginary parts.

SOME SPECIAL TRANSFORMATIONS

67

Hence the interior of the shaded semicircle of fig. 8
corresponds to the upper half of the w-plane.

z-plane

FIG. 8.

It is easy to verify that the upper half of the w-plane
corresponds to the interior of the respective semicircles
BAEDB, AECDA, ECBDE by the transformations

{z+c\*
w = I 1 , w

_
w -

Also, by combining (1) with the transformation
the transformation

iw

t ==

i+w'

t =

\z+icj

i + (*!=*]*

. z

a

2 a C 2 2CZ

conformally represents the interior of the z-semicircle ABCDA
on the interior of the unit circle 1 1 1 = 1.

II. Wedge or sector on half -plane.
By the transformation

w = 2 1 / ,

(2)

68 FUNCTIONS OF A COMPLEX VARIABLE

the area bounded by the infinite wedge of angle na with its
vertex at z = and one arm of the angle along the positive
x-axis is transformed into the upper half of the w-plane.
The reader will find this quite easy to verify.

The sector cut off from this wedge by an arc of the unit-
circle | z | = 1 is transformed by (2) into the unit semicircle
in the upper half of the w-plane.

This is also easy to verify.

III. Circular crescent or semicircle on half -plane.

We readily see, from 18, IV, that the circular crescent
with its points at z = a and z = b and whose angle is
TTOL can be transformed into the wedge mentioned in II
above by

z ~ a

if the constant k be suitably chosen. Hence the crescent
can be transformed into the w-half-plane by

-a\l/a

=} i) ()

A semicircle may be regarded as a particular case of a
crescent in which a = J. The semicircle of radius unity
and centre 2 = lying in the upper half-plane is trans-
formed into the first quadrant of the -plane by

The quarter-plane becomes a half-plane by * w = 2 and
so the semicircle is transformed into the upper half of the
w-plane by

=<=()' < 6)

* See I above.

SOME SPECIAL TRANSFORMATIONS

69

IV. Sector on unit circle.

Consider the sector in the z-plane, shaded in fig. 9.
Let us find the transformation which represents this sector
on a unit circle.

o j

Fia. 9.

By means of (2) the sector is transformed into the unit-
semicircle in the upper half of the w-plane. By means
of (5) we see that this unit semicircle is transformed into
the upper half of the J-plane by

(6)

Again, the upper half of the -plane is transformed into
the interior of the unit circle in the -plane, | f | = 1, by

.... (7)

l-fz 1/a \ 2

and on combining these, the transformation which represents
the shaded area of fig. 9 on the unit circle in the {-plane is *

V. By combining w = \/z with a Mobius' transforma-
tion we find in a similar way the transformation which
represents the z-plane, cut from to oo along the positive
real axis on the unit circle \ | < 1 in the form

When a = J, this transformation is the same as (1) of 19.

70 FUNCTIONS OF A COMPLEX VARIABLE

VI. Transformations of the cut-plane.
Consider the two transformations

z a za

W = r , W = r

zb zb

where a and b are real and a> 6. By means of the first
of these, the z-plane, cut along the real axis from z = a
to +00 and from z = b to oo, is transformed into the
usplane cut from w = to w = oo, the cut passing through
the point w = 1 which corresponds to z = oo. By means
of the second, the 2-plane cut from z = a to z = 6 is
transformed into the u;-plane cut along the positive real
axis from to oo. The cut in this case does not pass
through w = 1, the point corresponding to z = oo.

25. Exponential and Logarithmic Transformations

Most of the transformations so far considered have
been Mobius' transformations, w = z a and combinations
of these two types. We now observe that the relation

t* = e . . . . (1)

gives rise to two important special transformations.

If we use rectangular coordinates x, y and polar co-
ordinates />, <f> in the to-plane we get
p = e * , <f> = y.

The horizontal strip of the 2-plane bounded by the
lines y = y l and y = y 2 where | y^y^ |<27r is transformed
into a wedge-shaped region of the t^-plane, the angle of
the wedge being a = |< 2 ~~^i I ^ I V^^Vi I- The repre-
sentation is conformal throughout the interior of these
regions since dw/dz is never zero. In particular, if y l = 0,
y 2 = TT, so that | y^y\ \ = TT, the wedge becomes a half-
plane. The semi-infinite strip oo<#<0, 0<y<7r is
readily seen to correspond to unit semicircle in the upper
half of the u>-plane.

If | yiy^ |>2rr, the wedge obtained covers part of the

SOME SPECIAL TRANSFORMATIONS 71

w-plane multiply. We may in this case make use of the
cut w-plane. If y l = , y a = 27r, the strip of width 2ir
in the z-plane corresponds to the w-plane cut along the
positive real axis. When | s^ 1/ 2 \ is an integral multiple
of 277 the strip is transformed into a Riemann surface.
Each strip of the z-plane of breadth 2rr corresponds to one
sheet of the oo-sheeted Riemann surface.

A second special transformation is obtained from (1)
by considering an arbitrary vertical strip bounded by the
lines x = x l , x = x 2 , (x^x^. This strip is represented
on a Riemann surface which covers the annulus between
the concentric circles \w\=p l ,\w\=p 2 an infinite
number of times. If we keep # 2 constant and let x l -^~ oo,
the strip x l <x<x 2 becomes in the limit the portion of
the z-plane to the left of the line x = # 2 , and we obtain
in the w-plane a Riemann surface which covers the circle
| w |</> 2 except at the point w = 0, where it has a
logarithmic branch-point.

The inverse function

w = Log z . . . (2)

gives, on interchanging the z-plane and u;-plane, exactly
the same transformations as (1).

It should be remembered that although Log z is an
infinitely many- valued function of z, e* is one-valued.

Since z a = e a Log * the transformation w = z a may be
regarded as a combination of the two transformations

w = ef , = a Log z.

26. Transformations involving Confocal Conies

Consider the transformation

(1)

If to = reW, we get

2x = |(a 6)r + ^tH cosl9, 2y = j(a-6)r- a \ sin 0,

n FUNCTIONS OF A COMPLEX VARIABLE

and so the curves in the z-plane, corresponding to concentric
circles in the w-plane having the origin for their centre,
are confocal ellipses, the distance between the foci being
2<v/(a 2 6 2 ). The curves in the z-plane corresponding to
straight lines through the origin in the u;-plane are the
confocal hyperbolas, a result to be expected, since the
two families of curves in each plane must cut orthogonally.
Clearly there is no loss of generality by taking a = 1,
6 = 0, and so we may consider the transformation

2z = w + - . . . (2)

w

as typical. Clearly z becomes infinite when w = 0, and
since

dw ~~" :

the points w = 1, at which the derivative vanishes, are
critical points of the transformation. We now have

2x = I r -f- ~ I cos , 2y = I r I sin

and on eliminating 0, we get the ellipse in the z-plane

rr 2 */ 2

= 1 , . . (3)

corresponding to each of the two circles | w \ r , | w \ = 1/r.
As r->l, the major semi-axis of the ellipse tends to 1,
while the minor semi-axis tends to zero. As r-> 0, or as
r-> oo, both semi-axes tend to infinity. From this it is
plain that the inside and the outside of the unit circle in
the w-plane both correspond to the whole z-plane, cut
along the real axis from 1 to 1. The unit circle | w \ | = 1
itself corresponds to a very narrow ellipse, which is the
cut along the real axis enclosing the critical points 1 and 1.

SOME SPECIAL TRANSFORMATIONS 73

On solving equation (2) for w we get

w = zA/(z 2 -l)

and the inverse function is a two-valued function of z.
If we choose the lower sign, the transformation

w = z -V(z*-l) ... (4)

transforms the area outside the ellipse (3) conformaHy into
the inside of the circle \ w \ = r. The lower sign is the
correct one to select, since the point w = inside the circle
must correspond to the point z = oo ; the other sign of
the square root would make the points w = oo, z = oo
correspond. The region between two confocal ellipses in
the z-plane is transformed into the annulus between two
concentric circles in the w-plane.

The function (4) gives a transformation of the z-plane,
cut along the real axis from 1 to 1, on the interior of the
circle \ w \ = 1.

If we take the other sign, it is clear that the trans-
formation

w =

gives a transformation of the cut z-plane on the outside of
the unit circle \ w \ = 1. The relation (2) is remarkable
in that it represents the cut z-plane not only on the interior
but also on the exterior of the unit circle | w \ = 1.

The ambiguity can be removed from this transformation
by replacing the z-plane by a Riemann surface of two
sheets, each cut from 1 to 1 and joined crossways along
the cut. Then, of course, the interior of the unit circle
| w | = 1 corresponds to one sheet and the exterior of the
unit circle to the other sheet of this Riemann surface.

If r>l, the transformation (2) maps the exterior of
the circle | w \ = r, or the interior of the circle | w \ = 1/r, on
the exterior of the ellipse (3). It should be observed,
however, that the interior of the ellipse cannot be
represented on the interior of the unit circle by any of the

74 FUNCTIONS OF A COMPLEX VARIABLE

elementary transformations so far employed. We may
remark, however, that the upper half of the ellipse (3)
is represented by (2) on the upper half of an annular
region cut along the real axis : this last area, and hence
the semi-ellipse also, can be transformed into a rectangle
by a method similar to that described in 25. The
transformation which maps the interior of an ellipse on
a unit circle involves elliptic functions.

27. The Transformation z = c sin w
From the relation

z = csin;, (c real) . . (1)

we get, on equating real and imaginary parts,
x = c sin u cosh v, y = c cos u sinh v,

so that, when v is constant, the point z describes the
curves

2 2

i /o\

' * * l '

C 2 cosh 2 t; c 2 sinh 2 i;

which, for different values of t>, are confocal ellipses.
Consider a rectangle in the w-plane bounded by the lines
u = i^j v = A. For all values of u, cos u is positive ;
hence when v = A, y is positive and x varies from c cosh A
to c cosh A, that is, the half of the ellipse on the positive
side of the axis of x is covered.

Let u = 77-, then y = and x = c cosh v. Hence
as v varies from A through zero to A along the side of the
rectangle, x passes from A' to the focus H (see fig. 10)
and back from H to A'.

When v = A then z describes the half of the ellipse
on the negative side of the axis of x. When u = far then
y = and x = c cosh v, so that z moves from A to the
focus S and back from 8 to A.

SOME SPECIAL TRANSFORMATIONS

75

Hence the curve in the z-plane corresponding to the
contour of the rectangle in the w-plane is the ellipse with
two slits from the extremities of the major axis each to
the nearer focus. It is easy to see that the two interiors
correspond.

FIG. 10.

Since sin w = COS(JTT w), the transformation given by
z = c cos w

can be dealt with in a similar way. The details are left
to the reader.

The function inverse to (1),

w = arc sin (z/c),

is an infinitely many- valued function of z. If we use the
cut plane, the cuts must be from S to infinity along the
positive real axis and from H to infinity along the negative
real axis. The Riemann surface of an infinite number of
sheets in the z-plane, which would secure unique corres-
pondence between every z-point and every w-poini, would
have the junctions of its different sheets along the above-
mentioned cuts.

76 FUNCTIONS OF A COMPLEX VARIABLE

28. Joukowski's Aerofoil
The transformation

WKC lzc\*

I I 4 t ^ \-*-J

UJ+/CC \2+<V

is important in the practical problem of mapping an
aeroplane-wing profile on a nearly circular curve. If the
profile has a sharp point at the trailing edge and we write
jg = (2 JC)TT, then /? is the angle between the tangents to
the upper and lower parts of the profile at this point. If a
circle is drawn through the point c in the 2-plane, so that
it just encloses the point z = c and cuts the line joining
2 = c and 2 = c at 2 = c+* where is small, this circle
is mapped by (1) on a wing-shaped curve in the w-plane.

A special case of (1) when c = 1, K = 2 will now be
discussed in detail.

In practical problems on the study of the flow of air
round an aerofoil, the transformation desired is one which
maps the region outside the aerofoil on the region outside
a circle or nearly circular curve. The special case of
(1) when c = 1, *= 2,

transforms a circle in the 2-plane, passing through the
point z = 1 and containing the point 2 = 1, into a
wing-shaped curve in the w?-plane, known as Joukowski's
profile.

We readily see that (2) is the same as the trans-
formation, already discussed,

w = z + - . . . . (3)
z

If C is a circle in the 2-plane passing through the point
2 as 1, such that the point z = 1 is within (7, the trans-
formation (3) maps the outside of C conformally on the

SOME SPECIAL TRANSFORMATIONS

77

outside of Joukowski's profile F. The shape of the curve F
can easily be obtained from the circle G by making the
point z trace out this circle, and adding the vectors z and
1/z. See fig. 11.

FIG. 11.

We may also consider (2) as a combination of the three
transformations

2+1

/

2-1

= *, 10:

By the first of these, the circle C is transformed into a
circle F in the -plane passing through t == 0. By the
second, the circle F is transformed into a cardioid * in the
-plane with cusp at = 0. The third transformation
then maps the cardioid on the wing-shaped curve F in
the w-plane. Since z = 1 corresponds to J = oo, the
outside of C is mapped on the interior of F. The interior
of the cardioid corresponds to the interior of F. Since
= 1 corresponds to w = oo, the outside of F corresponds
to the inside of the cardioid, and so to the outside of C.

In fig. 11, C is the given circle, C' the circle obtained
from C by the transformation l/z and Q is the unit circle.

* See 21.

78

FUNCTIONS OF A COMPLEX VARIABLE

The critical points of (3) are z = 1 and z = 1, and since
the point z = 1 is inside C, the mapping of the outside of
C on the outside of F is conformal.

If the point z = 1 is outside (7, then the inside of F
corresponds to the inside of (7, and the figure corresponding
to this case is the same as fig. 11 with the circles G and C'
interchanged.

29. Some Important Transformations Tabulated

In Table 1 we tabulate for convenience a number of
examples of domains in the 2-plane which can be mapped
conformally on the interior of the unit circle | w |

TABLE 1

Domain in the z-plane

Domain in the u?-plane

Transformation.

1,

Unit circle |f |<1

Unit circle | w |^1

iX *~

W ~ C dz-1

2.

Upper half -plane

Unit circle | w\^.l

\z

3.

Infinite strip of finite
breadth oo^/^oo,

Unit circle | w|^l

w ~ l _ .iz
w+l **

4.

Area outside the ellipse

Unit circle | u>|<l

w

6.
6.
7.

Area outside the para-
bola r cos 2 J0 = 1
Area within same para-
bola
Semicircle

Unit circle | w |^1
Unit circle | u;|^l
Unit circle | t0|^l

w = tan 2 (i77\/2)

w ~~~ i

Some useful conformal transformations in which the
domain in the to-plane is not a circle are given in Table 2.
When the domain in the u>-plane is either the upper half-
plane or a semicircle it can of course be transformed into
a circle by either 2 or 7 of Table 1.

SOME SPECIAL TRANSFORMATIONS

79

i

"*"

> +

J?

1 >

^> > I
I II II II

* 3 S S

_j_

+

+

S S

II
S

lane

Domain

S o O

< V A

Upper half -plan
Upper half -plan

l

S

:

o

V

V*

C5

w |<

cj a o

2 ^ |

o

1 A

^

i *

.2 II

OQ

I

s

plane.

e angle 0<0<27r/n
icircle | z \ < 1, t/>

V

ii :A.S-XT;
% J39^

- "So-s^-c

"K o H I

2 $ 3 : >

.
5 O

*O
t^QO

80 FUNCTIONS OF A COMPLEX VARIABLE

It is impossible, in the limited space at our disposal,
to discuss all the transformations which are of practical
importance. It is important, however, to mention briefly
the Schwarz-Christoffel transformation, which has
numerous important applications.

Let a, 6, c, ... be n points on the real axis in the to-plane
such that a<6<c<... ; and let a, jS, y, ... be interior
angles of a simple closed polygon of n vertices so that
a+j3+y+... = (n-2)7T.

Then the transformation of Schwarz-Christoffel is a
transformation from the w-plane to the 2-plane defined by

dz ?! ft + y ,

~ ~

It transforms the real axis in the w-plane into the boundary
of a closed polygon in the 2-plane in such a way that the
vertices of the polygon correspond to the points a, 6, c, ...
and the interior angles of the polygon are a, j8, y, ....
When the polygon is simple, the interior is mapped by
this transformation on the upper half of the to-plane.
The number K is a constant, which may be complex.

If we write K = Ae*\ where A and A are real, one
vertex of the polygon can be made to correspond to the
point at infinity on the to-axis. If a-* oo we can choose

(X

A to be of the form B(a)~" +l and since, as a-> oo,

--i
{(w a)/ a}^ -> 1, the transformation becomes

dz > P . y_,

t - l

The reader who desires further information about this
important transformation is referred to larger treatises.*

* See e.g. Copson, Functions of a Complex Variable (Oxford,
1935), p. 193 seq.

SOME SPECIAL TRANSFORMATIONS 81

EXAMPLES III

1. Prove that, by the transformation
c tza z+a

w -

two sets of coaxal circles are transformed into sets of confocal
conies. What region of the t0-plane corresponds to the
inside of the circle \(z a)/(z+a)| = J ?

(z-\~
- -. +
Z I

the real axis in the z-plane corresponds to a cardioid in the
to-plane. Indicate the region of the 3-plane which corres-
ponds to the interior of the cardioid.

3. If w = ic cot \z y where c is real, show that the rectangle
bounded by x = 0, x = IT, y = 0, y = oo, is confonnally repre-
sented on a quarter of the to-plane. Find a transformation
=/(z) which maps this infinite rectangle on the semicircle
||<a, >0.

4. If w = tan z, prove that

cot 2a?-l = 0, u'+v'-^v coth 2y+l = 0.

Hence show that the strip %TT<X<%TT corresponds to the
whole to-plane. To obtain a Riemann surface in the to-plane
so as to secure unique correspondence between every t0-point
and every z-point, show that the to-plane must be cut along
the imaginary axis from i to oo and from t to oo.
Investigate to = tan z as a combination of

iw = (-!)/(+!), = 6 .

6. Find the curves in the z-plane corresponding to | to | = 1
if

w =

6. Show that w = 2z/(l z 1 ) maps two of the four domains,
into which the circles | z 1 | = V2, | z + l \ = V2 divide
the 2-plane, confonnally on | w \ < 1.

7. Prove that, if 3z a 2w + l = 0, the annulus
l/\/3<| z \<l is mapped confonnally on the interior of the

82 FUNCTIONS OF A COMPLEX VARIABLE

ellipse u*4-4y | = 4 cut along the real axis between its foci.
Discuss what corresponds in the t0-plane to the curves
(i) | z | = r, (ii) arg z = a.

8. Find the transformation which maps the outside of the
ellipse | s2 |+| z+2 \ = 100/7 on the circle | w |<1.

9. Show that w = $(z + l/z) maps the upper half of the
circle |z|<l on the upper half of the w-plane. At what
points of the z-plane is the linear magnification equal to ?
At what points is the rotation equal to 77 ? Prove that
the magnification is greater than unity throughout the interior
of the semicircle 3 | z | 2 = 1 in the upper half -plane.

10. By considering the successive transformations
= (z + l/z), w = l/ a prove that w = 4z 2 /(l+z 2 ) a maps
the upper half of the circle | z |< 1 on the w-plane, cut along
the positive real axis, so that the points z = 0, 1, i correspond
to w = 0, 1, oo respectively.

What points of the t0-plane correspond to z = 1 ?

11. Show that by w = e 77 " 2 /* an equiangular spiral in the
w -plane corresponds to a straight line in the z -plane.

12. Discuss the transformation

showing that the lines u = const., v = const, correspond to
sets of confocal conies with foci at z = a, z = j8.

13. Show that 2w = log{( l+z)/( lz)} represents |z|<l
on the strip of the w-plane JTT<V< JTT.

14. Show that iw = log{V(^/)~ 1} represents the strip
v = 0, v = oo, u = TT, u = TT on the interior of the cardioid
r = 2a(l+cos 0) in the z -plane cut along the real axis from
the cusp to x = a.

15. Show that, if c is real,

-=?

z+c

conformally transforms the strip v= oo, t>= oo, w = 0,
u s=s IT, into the circle | z \ ^ c.

16. Show that, by the relation w 2 = l+e f , the linos
x = const, are transformed into a series of confocal lemniscatea
(Cassini's ovals) in the w-plane.

SOME SPECIAL TRANSFORMATIONS 83

If a>l and Z 2 (a 8 +t0* 1) = aw*, show that the interior
of the circle | z \ = 1 is transformed into the interior of the
Cassini's oval pp = a, where p and p are the distances of a
point from the foci (1, 0) and ( 1, 0).

17. If z = x+iy, prove that the inside of the parabola
y* = 4c*(a;+c a ) is mapped on the upper half of the u>-plane by

_

u? = t cosh

2c

18. Show that the transformation
Z

C ~ l-W

transforms the inside of the circle | w \ = 1 with two semi-
circular indentations, of centres 1 and 1, drawn so as to
exclude these points from the circular area and boundary,
into the annulus between two circles in the z -plane, of centre
the origin and radii ce a , ce~, with a single slit along the real axis.

19. If 0<a< 0<27T, show that

w ^ (ze-

maps the region a<argz</J on the K;-plane cut along the
positive real axis. Hence find the transformation w =/(z)
which maps the circular sector a<argz<0, |2J<1, on the
circle | w |<1.

20. Use the successive transformations

C = (z+iM s = e*", t = - - , r = t\ w = -.,

18 r+i

to form the single transformation w =/(z) which maps the
strip <#<$, J/5^0 of the z-plane on | w |^1.

21. Use the transformations = y% ^ = sin TT, w = - ,

to show that

sin irr\/ 2 1

W s -

sin

maps the inside of the parabola r = 2/(l -fcos 0) in the z-plane,

84 FUNCTIONS OF A COMPLEX VARIABLE

out from the focus (z = 0) to the point z = oo, on the unit
circle in the t0-plane cut from w = to w = 1.

22. Find the equations of the curves in the s-plane which
correspond to constant values of u and v if z = w+e w . What
corresponds to the lines v = 0, v = ir ? Sketch some of
the curves v = const, for values of v between TT and IT.

23. Show that the transformation

w/a = i sinh \(z ij8)/cosh (z+if))

transforms one of the regions bounded by the orthogonal
circles | w \ = a and | w a cosec ft \ = a cot ft into the infinite
strip 0<t/<j7T.

24. If w = tanhz, show that the lines x = const.,
y = const., correspond to coaxal circles in the w-plane.

Prove that this transformation maps the strip
conformally on the upper-half of the w-plane.

25. Prove that, if 0<c< 1, the transformation

z(z-c)

w

cz-1

transforms the unit circle in the z -plane into the unit circle,
taken twice, in the u;-plane, and the inside of the first circle
into the inside, taken twice, of the second.
26. Prove that, if a>0,

maps the upper-half of the z-plane on the positive quadrant
of the ttf-plane with a slit along the line v = TT, u^h 9 where
w = h+in when z = I/a. (See p. 80.)
27. Show that by the transformation

dz __ 1
dw ~~"

the upper half of the w-plane can be mapped on the interior
of a square, the length of a side of which is

r

J n

CHAPTER IV

THE COMPLEX INTEGRAL CALCULUS

30. Complex Integration

The development of the theory of functions of a
complex variable follows quite a different line from that
of functions of a real variable. In the latter theory,
having discussed functions which possess a derivative,
we proceed to consider the more special class of functions
which possess derivatives of the second order ; then,
from among those functions which possess derivatives of
all orders, we select those which can be expanded in a
power series by Taylor's theorem. In complex variable
theory, on the other hand, we begin by dealing with
regular functions, and, by virtue of the definition of
regularity, the class of functions is so restricted that a
function which is regular in a region possesses derivatives
of all orders at every point of the region and the function
can be expanded in a power series about any interior
point of the region.

By following Cauchy's development of complex variable
theory, everything depends upon the complex integral
calculus, and, in order to prove that a regular function
possesses a second derivative, we must first of all express
f(z) as a contour integral round any closed contour
surrounding the point z.

In order to develop the subject further we must now
consider the definition of the integral of a function of a
complex variable along a plane curve.

The equations x = <f>(t), y = iff(t) 9 where a<J<j8, define
the arc of a plane curve. If we subdivide the interval

85

86 FUNCTIONS OF A COMPLEX VARIABLE

(a, jS) by the points a = * , t lt J 2 , ..., t r , ..., t n = j8, then
the points on the curve corresponding to these values
of t may be denoted by P , P l9 P 2 , ..., P n . The
length of the polygonal line PoP v ..P n , measured by

r~ yr-i) 2 }*> depends on the particular

r-l

mode of subdivision of (a, jS). We call this summation
the length of an inscribed polygon. If the arc be such that
the lengths of all the inscribed polygons have a finite
upper bound A, the curve is said to be rectifiable and A is
the length of the curve.

It can be shown that the necessary and sufficient
condition that the arc should be rectifiable is that the
functions <(), tfj(t) should be of bounded variation in (a, j3).
If <f>'(t) and 0'() are continuous, it can be proved that the
curve defined by x = <f>(t), y = *fj(t), a<tf<j8, is rectifiable
and that its length s is given by *

= f

J

If we consider an arc of a Jordan curve whose equation
is z = (j)(t)-{-iifj(t), where a^^jS, we define a regular
arc of a Jordan curve to be one for which <f>'(t), */*'(t) are
continuous in a^^jS. From the above theorems we
see that the length of this regular Jordan arc is

By a contour we mean a continuous Jordan curve
consisting of a finite number of regular arcs. Clearly a
contour is rectifiable.

We now define the integral of a function of a complex
variable z along a regular arc L defined by x = <f>(t),

y = <!>(*), a<*<

* For proofs and further details, see P.A., p. 205 acq., or G.I.,
p. 113.

THE COMPLEX INTEGRAL CALCULUS 87

Let f(z) be any complex function of z, continuous along
L y a regular arc with end-points A and J3, and write
f(z) = u(x, y)+iv(x,y). Let z , z l9 ..., z n be points on
L, z being A and z n being JB. Consider the sum

(Zr-*r-l)}, . . . (1)

where f is any point in the arc z r _ l9 z r . If f = r 4-^ f
we write u f = u( r , 7j r ), v f = v(^ r , 7j r ), and (1) may be
written

Now, by the mean-value theorem,

X r X r _i =^(^)-^(^r-l)

where ^ r _i<r r <^ f , ^ f -i<r f '<^ f . Hence the sum may be
written

( Ur +iv r ){<j>'(r T ) +i^'(r r ')}(t r -t r _ l )]. . (2)

r-1

Since all the functions concerned are continuous, and
therefore uniformly continuous, we can, given e, find 8(e)
so that

for every r, provided that each | t r t r _ l |<8. Also

f-1

It follows that, as and 8 tend to zero,

r-1

tends to the same limit as

{(* y,)
r-1

88 FUNCTIONS OF A COMPLEX VARIABLE

that is to the limit

J

Similarly the other terms of (2) tend to limits, and we find
that the whole sum tends to the limit

(3)

This limit (3) is taken as the definition of the complex
integral of f(z) along the regular arc L, and it is written

f(z)dz.

L

The integral of /(z) along a contour (7, consisting of a finite
number of regular arcs L r , is given by

f f(z)dz=Z f

J C r J

f(z)dz.

C r J L r

31. An Upper Bound for a Contour Integral

I. // f(z) is continuous on a contour L, of length I, on
which it satisfies the inequality \ f(z) \ ^M, then

f(z)dz

!,

It suffices to prove this theorem for a regular arc L.
Since the modulus of any integral of a function of a
real variable cannot exceed the integral of the modulus
of that function, we have

U/w* -|J

<f.

a

= ML

If C is a closed contour we make the convention that the
positive sense of description of the contour is anti-clockwise.

THE COMPLEX INTEGRAL CALCULUS 89

32. Cauchy's Theorem

The elementary proof of Cauchy's theorem, which
depends on the two-dimensional form of Green's theorem,
requires the assumption of the continuity of f'(z). We
first give a proof with this assumption, but, on account
of the fundamental importance of Cauchy's theorem in
complex variable theory, we shall also prove the theorem
under less restrictive assumptions.

II. The elementary proof of Cauchy's theorem.
If f(z) is a regular function and if f'(z) is continuous
at each point within and on a closed contour C f , then

(2)dz = (1)

Let D be the closed domain which consists of all points
within and on C. Then by 30 (3) we can write the
integral (1) as a combination of curvilinear integrals

I f(z)dz= I (udx vdy)-\-i\ (vdx -\-udy).
J c J c J c

We transform each of these integrals by Green's theorem,*
which states that, if P(x, y), Q(x, y), dQ/8x, 8P/dy are all
continuous functions of x and y in D, then

Since f'(z) = u x -\-iv x = v y iu v , and, by hypothesis,
f'(z) is continuous in D, the conditions of Green's theorem
are satisfied and so

= o,

by virtue of the Cauchy-Riemann equations.
* See P.A., pp. 290-1, or G.I., p. 64

90 FUNCTIONS OF A COMPLEX VARIABLE

It was first shown by Goursat that it is unnecessary to
assume the continuity of f'(z) and that Cauchy's theorem
holds if we only assume that/'(z) exists at all points within
and on C. In fact the continuity of f'(z) 9 and indeed its
differentiability, are consequences of Cauchy's theorem.

Second proof of Cauchy's theorem.

If f(z) is regular at all points within and on the closed
contour C then

f f(z)dz = 0.
J c

The integral certainly exists, for a regular function f(z)
is continuous and a continuous function is integrable.
We observe also that, if we construct a network of squares,
by lines parallel to the axes of x and y, having the contour
C as outer boundary, then C is divided into a network
of meshes, either squares or parts of squares, such that

f

J

f(z)dz,
y

where y denotes the boundary of a mesh described in the
same sense as C.

If z lies inside a square contour 8 of side a, then

l/j

\z-z \ \dz\

<4v/2a 2 = 4V2(Area of S).

This follows at once from 31, for | z z \<a\/2 and the
length of the contour S is 4a. *^

We now prove two lemmas.

Lemma 1. IfC is a closed contour, I dz = 0, zdz = 0.

J c J c

These results both follow from the definition of the
integral, for

J,

n

dz = lim E {{z f z f _i).\} = 0, as max | z r z r _ l \-+ 0.

7 -1

THE COMPLEX INTEGRAL CALCULUS 91

Also

zdz = lim 2{z r (z r z r _J} = lim {z r -i(z f z r _i)}
J c

= 0,

Lemma 2. Goursat's Lemma. Given , then, by suitable
transversals, we can divide the interior of C into a finite
number of meshes, either complete squares or parts of
squares, such that, within each mesh, there is a point z
such that

for all values of z in the mesh, where \ e y |<.

Suppose the lemma is false ; then, however the interior
of C is subdivided, there will be at least one mesh for
which (1) is untrue. We shall show that this necessarily
implies the existence of a point within or on C at which
f(z) is not differentiate.

Enclose C in a large square F 9 of area A, and apply
the process of repeated quadrisection. When F is
quadrisected there is at least one of the four quarters of
F for which (1) does not hold. Let F l be the one chosen.
Quadrisect F l9 choose one quarter of F 19 and so on. We
thus obtain an unending sequence of squares F l9 F 2 , ...,
r n , ..., each contained in the preceding, for which the
lemma is untrue. These squares determine a limit-point f ,
and it is clear that must lie within C.

Since f(z) is differentiate at ,

where, for sufficiently small values of [2 |, |^|<.
Now all the F r , from one particular one onwards, lie within
a circle, of centre , for which | z - | is so small that
| f |<. This gives a contradiction, for by taking to
be z , (1) is satisfied. This proves Goursat's lemma.

92

FUNCTIONS OF A COMPLEX VARIABLE

Proof of the theorem. Some of the meshes y obtained
by the subdivision of the interior of G will be squares,
others will be irregular, since we are not concerned with
the exterior of G.

Integrate (1) round the boundary of each mesh. By
virtue of lemma 1, we get

f(z)dz = y | z Z Q | dz\

J Y J y

and so, by addition,

f(z)dz = H\ Y \z-z Q \dz,

J C J Y

where

If y is not a complete square, divide it into two parts,
y x consisting of straight pieces, y a consisting of parts of (7.
Since I >!<,

.

A being the area of the large square F surrounding G.
Also, the sum of the lengths of the portions y a cannot
exceed the length I of the rectifiable curve (7, and so

f eylz-zJ
I J Y*

where K is the length of the diagonal of JT, since | z Z Q
We deduce that

U.

f(z)dz

wtere B is a constant, and, since e is arbitrary, the theorem
is proved.

THE COMPLEX INTEGRAL CALCULUS 93

33. Cauchy's Integral, and the Derivatives of a
Regular Function

By means of Cauchy's integral we can express the
value of a regular function /(z) at any point within a closed
contour C as a contour integral round C.

III. If f(z) is regular within and on a closed contour G 9
and if be a point within (7, then

Describe about z = a small circle y of radius S lying
entirely within (7. In the region between C and y the

FIG. 12.

function (f>(z) = f(z)/(zt > ) is regular. By making a
cross-cut joining any point of y to any point of C we
form a closed contour F within which <}>(z) is regular, so
that, by Cauchy's theorem,

<l>(z)dz = 0.

In traversing the contour F in the positive (counter-
clockwise) sense, the cross-cut is traversed twice, once in

aarh a^naa nnrl an if. fnllnwa f.hnf.

each sense, and so it follows that

Jo Jy

Now

J_ f , (z)dz I f f(*)dz = J_ f f(?)dz + JL

94 FUNCTIONS OF A COMPLEX VARIABLE

Now on y, z J = 8e*0, and so the first of the two
terms on the right becomes

and, by theorem I, the modulus of second term on the
right of (2) cannot exceed

JL max |/(z)-/() | . 2778.

Since /(z) is continuous at z = J this expression tends to
zero as S-> : this proves the theorem.

The next theorem shows how to find the value of
/'() as a contour integral.

IV. // /(z) is regular in a domain D, its derivative is
given by

o \ z )

where C is any simple closed contour in D surrounding the
point z = J.

We have, by III,
1

f(z)dz

If we now prove that | / |-> as | ^ [-> 0, the required
result is established. Since /(z) is regular in and on
it is bounded, so that |/(z) [^M on (7. Let d be the

THE COMPLEX INTEGRAL CALCULUS 95

lower bound of the distance of from G : suppose h chosen
so small that | h \<\d, then

** 2n d*.$d 9

where I is the length of (7. It is now clear that the term
on the right tends to zero as | h |-> 0.

V. // f(z) is regular in a domain D, then f(z) has, at
every point of D, derivatives of all orders, their values
being given by

If we assume the theorem proved for = m and
consider the expression

h

we can readily prove that it is equal to
(m+1)! f f(z)dz ,

i)i r

" Jo

2ni Jc(*-) m+a ' '

and the proof that | / | tends to zero as | h \~> follows the
same lines as in IV. The details are left to the reader.

34. Taylor's Theorem

VI. If f(z) is regular in \ za |<p, and if is a point
such that | a | = r(<p) then

n-0

where a n =/ (n >(a)/n !.

Let C be a circle of radius />', centre z = a, where
r<p'<p, and consider the identity

^- a (t-g) 1 ^ (g-a) n

- a (2-a) 2 '" (z-a)

96 FUNCTIONS OF A COMPLEX VARIABLE

Multiply each term by /(z)/27rt and integrate round
we clearly obtain

\n i;i
where

f(z)dz

This is Taylor's theorem with remainder.
Since | /(z) | < Jf on (7 we readily see that

r"

where J? is a constant independent of n. Since r</>' we
see that | R n |-> as n->oo.

GO

It therefore follows that the series 27a n (f a) n is con-

o
vergent and has/(^) as its sum-function. If f(z) is regular

in the whole z-plane, the expansion is valid for all .

Corollary. If [/() | has a maximum M(r) on
| a | = r</> <Aen, t/a n =/ (n) (a)/n !, w;e Aave <Ae inequality

For, if C be the circle | za \ = r we have

1 M(r) n M(r)

35. The Theorems of Liouville and Laurent

VII. Liouville's Theorem. // /(z) is regular in the
whole z-plane and if |/(z) | <K for all values of z, then
/(z) must be a constant.

Let z l9 z 2 be any two points and a circle of centre z l

THE COMPLEX INTEGRAL CALCULUS 97

and the radius p>2 |2 1 z 2 1, so that, when z is on (7,
. By III,

so that

- * i r fa-sj/tt* < i r

- 0^1 J c (z_ Zl)(2 _ 22 ) < 27 r J

Keep 2, and z 2 fixed and make p->oo, then it follows that
/(Zj) = /(z 2 ) ; in other words, /(z) is a constant.

VIII. Laurent's Theorem . Let C l and C 2 be two circles of
centre a with radii p l and p 2 (p 2 <Pi) 5 th en > tff( z ) be regular
on the circles and within the annulus between (/j and <7 a ,

J being any point of the annuliis. The coefficients a n and
b n are given by

By making a cross-cut joining any point of O x to any
point of C t , we readily see that

(z)dz

Consider the two identities

JL s J_j._iz^^. - (C-^)"- 1 , (C-) n i

- z-a' t "(2-a) a " f "*"" r " ( 2 -o) "" (z-a)" *-

z - 2 -""- 1 z - n i

98

FUNCTIONS OF A COMPLEX VARIABLE

O^n it Mows that

where

(z-a)(z--)' 27*

Now P n is precisely the same remainder as in Taylor's
theorem and we can prove, in the same way as in VI,
that | P n |-> as

Also we have

f f^Y&^

Jc.l^-a/ -{

where r = | ^ a | and |/(z)
it follows that | Q n |-> as U-+CQ
If, therefore, we write

' on (7 2 . Since /) a <r,

where ^(f) = a n (-a) and /,() = MC-a)-, we

o o

see that/(^) converges forp 2 ^| ^ a |^/)j.

It also follows that/ 1 (^) is regular and converges for
| fl |^PI and that / a () is regular and converges for

36. Zeros and Singularities

If f(z) is regular within a given domain Z), we have seen
that it can be expanded in a Taylor series about any
point z = a of D and

THE COMPLEX INTEGRAL CALCULUS 99

If a Q = a l = ... = a m-1 = 0, a m 4= 0, the first term in
the Taylor expansion is a m (z a) m . In this case f(z) is
said to have a zero of order m at z = a.

A singularity of a function f(z) is a point at which
the function ceases to be regular.

If /(z) is regular within a domain D, except at the point
z = a, which is an isolated singularity of /(z), then we can
draw two concentric circles of centre a, both lying within
D, The radius of the smaller circle p a may be as small
as we please, and the radius p l of the larger circle of any
length, subject to the restriction that the circle lies wholly
within D. In the annulus between these two circles,
/(z) has a Laurent expansion of the form

/(z) = f a n (z-a)+f b n (z-a)~\
o i

The second term on the right is called the principal part
of /(z) at z = a.

It may happen that b m * while 6 m+1 = 6 m + 2 ... = 0.
In this case the principal part consists of the finite number
of terms

and the singularity at z = a is called a pole of order m
of /(z) and the coefficient b l9 which may in certain cases
be zero, is called the residue of /(z) at the pole z = a.
If the pole be of order one, b l = lim{(z a)/(z)}.

If the principal part is an infinite series, the singularity
is an isolated essential singularity.

(1) If z = a is a zero of order m of /(z), we now prove
that this zero is isolated : in other words, there exists a
neighbourhood of the point z = a which contains no other
zero off(z).

Clearly we can write /(z) = (z a) m ^(z), where ^(z) is
regular in | a \<p and <j>(a) * 0, since </>(a) = a m .

100 FUNCTIONS OF A COMPLEX VARIABLE

Write ^(a) =s 2c, then, since <f>(z) is continuous, there
exists a region | z a |<8 in which |^(z) ^(a) | < | c \.
Hence

where | za |<8, and so ^(z) does not vanish in | z a |<8.

(2) If z = a is a pole of order m of /(z) it follows, from
the definition of a pole by means of Laurent's theorem,
that poles are isolated, for, the small circle, of centre z = a
and radius p a , encloses the only singularity of /(z) within
the domain D which contains the annulus between the
two circles of radii p l and p a .

(3) ///(z) has a pole at z = a, ften |/(z) [->oo a$ z-* a
tn ant/ manner. For, if the pole be of order m,

f(z) = (-a)--{6 m +6 m . 1 (z~o)+...+6 1 (-ar-Hra n (2~ar^}

o

and, since 6 m * 0, we may write /(z) = (z a)- m ^r(z),
where ^(z) is regular in | za [</>, and 0(a) = 6 m ( 4 s 0).
Hence, by (1), we can find a neighbourhood | za |<8
of the pole in which | ^r(z) |>J | 6 m |, from which it follows
that

l/(*)l>ilMI*-*|- m .

Hence |/(z) |-> co as z-> a in any manner.

(4) Limit points of zeros and poles.

Let a x , a 2 , ..., a n , ... be a sequence of zeros of a function
/(z) which is regular in a domain D. Suppose that these
zeros have a limit point a which is an interior point of D.
Since /(z) is a continuous function, having zeros as near
as we please to a, /(a) must be zero. Now z = a cannot
be a zero of /(z), since we have proved in (1) that zeros
are isolated. Hence /(z) must be identically zero.

If /(z) is not identically zero in Z), then z = a must
be a singularity of /(z). The singularity is isolated, but
it is not a pole, since |/(z) | does not tend to infinity as
z-+ a in any manner. Hence a limit point of zeros must
be an isolated essential singularity of /(z).

THE COMPLEX INTEGRAL CALCULUS 101

If f(z) be regular, except at a set of points which are
singularities c l9 c a , ..., c n , ..., infinite in number, and
having a limit point y in D, then y must be a singularity
of/(z), since /(z) is unbounded in the neighbourhood of y.
Since y is not isolated it cannot be a pole. We call such
a singularity a non-isolated essential singularity.

Examples, sin 1/z has an isolated essential singularity
at z = 0. It is the limit point of the zeros, z = l/n?r, (n = J^l,
2, ...). tan 1/2 has poles at the points z = 2/nrr, (n = 1,
3, ...)> and so the limit point of the poles, z = 0, is a non-
isolated essential singularity.

Note on the region of convergence of a Taylor series.
Iff(z) be a function which is regular, except at a number
of isolated singularities at finite points of the z-plane,

then we can expand f(z) in a Taylor series Sa n (za) n

o
about any assigned point z = a, and the radius of con-

vergence p of this power series will be the distance from
z = a to the nearest singularity of/(z), since /(z) is clearly
regular in | z a |<p, and cannot be regular in any circle
of centre a whose radius exceeds p.

We see that the radius of convergence of a power
series depends upon the extent of the region within which
the sum-function is regular, and it may be controlled by
the existence of singularities which do not necessarily
lie on the real axis.

If we consider the real function 1/(1 a), the binomial
expansion leads to

the series being convergent if | x \<1 This seems quite
natural, since the sum-function has a singularity at x = 1.
However, on considering the function l/(l+x a ), we have

..... (2)

102 FUNCTIONS OF A COMPLEX VARIABLE

and the series is again convergent only if | x |<1 ; but
if we regard l/(l+# 2 ) as a function of the real variable x
there is nothing in the nature of the function to suggest
the restriction of its range of convergence to |#|<1.
If, however, we consider l/(l+z 2 ), where z is complex,
the restriction on the region of convergence is at once
evident, since l/(l+z 2 ) has singularities at z = , and
the radius of convergence of the series (2), if a; is complex,
is the distance of the origin from the nearest singularity
and this is plainly unity.

37. The Point at Infinity

In complex variable theory we have seen that it is
convenient to regard infinity as a single point. The
behaviour of /(z) "at infinity " is considered by making
the substitution z = l/ and examining /(!/) at = 0.
We say that /(z) is regular, or has a simple pole, or has
an essential singularity at infinity according as /(!/)
has the corresponding property at = 0.

We know that if /(!/) has a pole of order m at f = 0,
near = we have

n-O 2 b

and so, near z = oo,

/(z) = 2 a n z~+b l z+b 2 z*+...+b m z<.

n-O

Thus, when /(z) has a pole of order m at infinity, the
principal part of /(z) at infinity is the finite series in
ascending powers of z.

Since

the function sinz has an isolated essential singularity at
infinity, the principal part at infinity being an infinite series.

THE COMPLEX INTEGRAL CALCULUS 103

38. Rational Functions

Theorem. If a single-valued function f(z] lias no essential
singularities either in the finite part of the plane or at infinity,
thenf(z) is a rational function.

Since the point at infinity is not an essential singularity
of f(z), we can surround it by a region in which f(z) either
is regular or has the point at infinity as its only singularity.
That is, we can draw a circle (7, with centre the origin,
such that the point at infinity is the only singularity
outside G. There can only be a finite number of singularities
within C, since poles are isolated singularities. Suppose
that the poles inside G are at a v a 2 , ..., a n . The principal
part at a f may be written

'28

z-a, ^ (z-a,) 2 ^ " ^ (z-a,)-'

a, being supposed to be a pole of order m. The principal
part at infinity is of the form

Now consider the function

The function ^(2) is plainly regular everywhere in the plane,
even at infinity : hence <f>(z) is bounded for all z, and so,
by Liouville's theorem, <f>(z) is a constant. Hence

f(z) - 0+2

and so /(z) is a rational function of 2.

39. Analytic Continuation

Suppose that / x (z) and / 2 (z) are functions regular in
domains D l and J5, respectively and that D l and D 2 have

104 FUNCTIONS OF A COMPLEX VARIABLE

a common part, throughout which f^z) = / a (z), then we
regard the aggregate of values of f^(z) and / 2 (z) at points
interior to D l or D 2 as a single regular function (f>(z).
Thus <f>(z) is regular in A = D^-\-D^ and <f>(z) = / 1 (z) in
JDj and <f>(z) = / 2 (z) in Z) 2 . The function / 2 (z) may be
regarded as extending the domain in which f^z) is defined
and it is called an analytic continuation of /^z).

The standard method of continuation is the method of
power series which we now briefly describe.

Let P be the point z , in the neighbourhood of which
f(z) is regular, then, by Taylor's theorem, we can expand
f(z) in a series of ascending powers of z z , the coefficients
in which involve the successive derivatives of f(z) at z .
If S be the singularity of f(z) which is nearest to P, then
the Taylor expansion is valid within a circle of centre P
and radius PS. Now choose any point P l within the
circle of convergence not on the line PS. We can find
the values of f(z) and all its derivatives at P l9 from the
series, by repeated term-by-term differentiation, and so
we can form the Taylor series for f(z) with P l as origin,
and this series will define a function regular in some circle
of centre P v . This circle will extend as far as the singularity,
of the function defined by the new series, which is nearest
to P l and this may or may not be S. In either case the
new circle of convergence may lie partly outside the old
circle and, for points in the region included in the new
circle but not in the old, the new series may be used to
define the values of f(z) although the old series failed to
do so.

Similarly, we may take any other point P 2 in the region
for which the values of the function are now known and
form the Taylor series with P 2 as origin which will, in
general, still further extend the region of definition of the
function ; and so on.

By means of this process of continuation, starting from
a representation of a function by any one power series,
we can find any number of other power series, which

THE COMPLEX INTEGRAL CALCULUS 105

between them define the value of the function at all points
of a domain, any point of which can be reached from P
without passing through a singularity of the function.
It can be proved that continuation by two different paths
PQR, PQ'R gives the same final power series provided that
the function has no singularity inside the closed curve
PQRQ'P.

We may now, following Weierstrass, define an analytic
function of z as one power series together with all the
other power series which can be derived from it by analytic
continuation. Two different analytic expressions then
define the same function if they represent power series
derivable from each other by continuation. The complete
analytic function defined in this way need not be a one-
valued function. Each of the continuations is called an
element of the analytic function.

If f(z) is not an integral function there will be certain
exceptional points which do not lie in any of the domains
into which /(z) has been continued. These points are
the singularities of the analytic function. Clearly the
singular points of a one- valued function are also singularities
in this wider sense.

There must be at least one singularity of the analytic
function on the circle of convergence (7 of the power

00

series a n (z z ) w .* For, if not, we could construct, by

o

continuation, a function equal to /(z) within <7 but regular
in a larger concentric circle jT . The expansion of this
function in a Taylor series in powers of z z would then
converge everywhere within F Q . This is impossible, since
the series would be the original series whose circle of
convergence is (7 . If z l is any point within <7 , let C l
be the circle of convergence of the power series

* For a proof of this, and further details, see Titchmarsh,
Theory of Function* (Oxford, 1932), p. 145.

106 FUNCTIONS OF A COMPLEX VARIABLE

If Q l is the circle of centre z l which touches (7 internally,
the new power series is certainly convergent within Q l
and has the sum /(z) there. There are now three
possibilities. Since the radius of C l cannot be less than
that of Q v we have either (i) C l has a larger radius than
Q l9 or (ii) (7 is a natural boundary * of /(z), or (iii) C l may
touch (7 internally, though (7 is not a natural boundary

of/(*).

In case (i) C l lies partly outside (7 and the new power
series provides an analytic continuation of /(z) : we can
then take a point z a within C l and outside C and repeat
the process. In case (ii) we cannot continue /(z) outside
(70 and the circle C l touches (7 internally no matter what
point z l within (7 is chosen. In case (iii) the point of
contact of (70 and G l is a singularity of the analytic function
obtained by continuation of the original power series.
For there is necessarily one singularity on C l and this
cannot be within (7 .

We may illustrate some of the above remarks by the
following examples.

1. The series

represents the function f(z) = l/(az) only for points within
the circle \z\ =* | a |. If 6/a is not real, the series

1 *-b (s-6)

a-6 + (a-6) 1 "*" (a~6) "*" "' f

for different values of 6, represents f(z) at points outside the
circle | z \ = | a |.

2. That there are functions to which the process of
continuation cannot be applied may be seen by considering
the function

g(z) = l+2 2 +z* + ...+* |n + ....

* See Example 2 below.

THE COMPLEX INTEGRAL CALCULUS 107

It is readily shown that any root of any of the equations

2 1 1 2 4 1 * 1 ! I

Z - 1, Z - 1, 1, 9 ... 9

is a singularity of g(z) 9 and hence that on any arc, however
small, of the circle | z \ = 1 there is an unlimited number
of them. The circle | z \ = 1 is in this case a natural boundary
of g(z). This illustrates case (ii) above,

40. Poles and Zeros of Meromorphic Functions

A function /(z), whose only singularities in the finite
part of the plane are poles, is called a meromorphic
function. We now prove a very useful theorem.

// /(z) is meromorphic inside a closed contour C, and
is not zero at any point on the contour, then

*- - <"

where N is the number of zeros and P the number of poles
inside 0. (A pole or zero of order m must be counted m
times.)

Suppose that z = a is a zero of order w, then, in the
neighbourhood of this point

/(z) = (z-a)^(z),
where <f>(z) is regular and not zero. Hence

"*"

__
/(z) z-a "" ftz) '

Since the last term is regular at z = a, we see that /'(z)//(z)
has a simple pole at z = a with residue m. Similarly, if
z = 6 is a pole of order k, we see that /'(z)//(z) has a simple
pole at z = 6 with residue ft. It follows, by 33, III,
that the left-hand side of (1) is equal to ZmSk = NP.
If /(z) is regular in (7, then P = 0, and the integral
on the left of (1) is equal to N. Since

108 FUNCTIONS OF A COMPLEX VARIABLE

we may write the result in another form,

W) dz==Al

where A c denotes the variation of log/(z) round the
contour C. The value of the logarithm with which we
start is immaterial ; and, since

and log |/(z)| w one- valued, the formula may be written
N=-A *Tgf(z).

This result is known as the principle of the argument.

41. Rouch6 's Theorem

If f(z) and g(z) are regular within and on a closed contour
C and | g(z) \ < \ f(z) \ on C, then f(z) and f(z)+g(z) have
the same number of zeros inside C.

We observe that neither f(z) nor f(z) +g(z) has a zero
on (7, and so, if N is the number of zeros of f(z) and N'
the number of zeros of f(z)+g(z),

2<jrN = A c arg/,

27rN' = AC arg (f+g) = A Q arg /+ A arg ( J + y)
The theorem is proved if we show that

+j\ = 0.

Since | g \ < |/|, the point w = l+g/f is always an
interior point of the circle of centre w = 1 and radius
unity : thus, if w = pety, <f> always lies between \n
and \n and so arg (1 +g/f) = ^ returns to its original value
when z describes C. Since <f> cannot increase or decrease
by a multiple of 2rr, the theorem follows.

THE COMPLEX INTEGRAL CALCULUS 109

The preceding theorems are useful for locating the
roots of equations. The method is illustrated by the
following example.

Example. Prove that one root of the equation z 4 +z 8 + l =0
lies in the first quadrant.

The equation z 4 +z 3 + l = plainly has no real roots.
For, if we put z = x, x*+x* + l = has no real positive
roots. If we put z = a; and write $(x) = x* # 3 + 1=0,
we see that

, if x>\ ;
and <f>(x) =x*+(l~x)(x*+x + l)>0 9 if

Hence the given equation has no real negative roots.

The given equation has no purely imaginary roots either,
for, on putting z = iy, we get y 1 iy* + l = and it is plain
that the real and imaginary parts never vanish together.

Consider A arg (z 4 +z 8 + l) round part of the first quadrant
bounded by | z \ = R where R is large. On the arc of the
circle, z = ReiO, and we have

A arg ( 2 4+z 8 + l) = j arg (R* e *iO)+ A arg {1 +O(R~ 1 )},
= 2<jr

On the axis of y we have

arg(z 4 +z s + l) = arc tan

~~ y

The numerator of y*/(y l + l) only vanishes when y = and
the denominator does not vanish for any real y. Hence as
y ranges from oo to along the imaginary axis, the initial
and final values of arg (z 4 +z a + l) are zero. Hence the total
change in arg(z 4 +z 8 + l), where R is large, is 2w. It follows
that one root of the given equation lies in the first quadrant.

42. The Maximum-Modulus Principle

We now establish an important theorem which may
be stated as follows.

// /(z) is regular within and on a dosed contour C 9 then
|/(z)| attains its maximum value on the boundary of G
and not at any interior point.

110 FUNCTIONS OF A COMPLEX VARIABLE

Lemma. If<f>(x) is continuous, (f>(x)^,K and

. (1)

then (f>(x) = K.

Suppose that ^(x 1 )</c, then there is an interval
(x l 8, Zx+S) in which </>(X)^.K and

which contradicts (1).

Theorem. If |/(z)|<Jf on C, then \f(z)\<M at all
interior points of the domain D enclosed by C, unless f(z)
is a constant j in which case \f(z)\ = M everywhere.

Suppose that at an interior point z of D, \f(z)\ has
a value at least equal to its value elsewhere. Let F be
a circle of centre Z Q lying entirely within D. Then by

33, in,

/w _ ' r /w* . . . (2)

2<TTl J r Z-Z Q

Write z z = re^, f(z)/f(z Q ) = petf, so that p and ^ are
functions of 0, then (2) may be written

1 f2

=

^ J o

MM. . . . (3)
Hence 1 < ^- I odd.

By hypothesis p^l, and so, by the lemma, p = 1 for all
values of 0. On taking the real part of (3) we get

THE COMPLEX INTEGRAL CALCULUS II!

and so, by the lemma, cos^i = 1. Hence f(z) =/(z ) a
JT. Since f(z) is a constant at any point a on J*, it follows
by Taylor's theorem that it is constant in a neighbourhood
of a, and hence, by analytic continuation, f(z) is constant
everywhere within and on C.

There is a corresponding theorem for harmonic functions.
A function which is harmonic in a region cannot have a
maximum at an interior point of the region.

EXAMPLES IV

1. The function /(z) is regular in | z a \<R ; prove that,
if 0<r<R,

I /-27T .-

f'(a) - _ P(e)e-* dO,

7TT J Q

where P(0) is the real part off(a+reiO).

2. <f>(z) and $(z) are two regular functions ; z = a is a
once repeated root of $(z) = and <f>(a) ^ 0. Prove that
the residue of <t>(z)l\ff(z) at z = a is

{6 ^'(a)0"(a) -2 #a)0'"(a)}/3 {f' (a)}.

3. The function f(z) is regular when | z \<R'. Prove
that, if | a \<K<R',

where O is the circle | z \ = B. Deduce Poisson's formula
that, if 0<r<K,

1 T 27r # a r*

T, / *-2Kr C o* ( e-t )+ r*

4. By using the integral representation of / (n) (a), ( 33, V),
prove that

ajfl \ t _ 1 f x " e *'
fTl) = 2^ J c nTz*+* *'

where C is any closed contour surrounding the origin. Hence
prove that

112 FUNCTIONS OF A COMPLEX VARIABLE

6. Obtain the expansion

KM}
/V;

- ...

2 J \ 2 / + 2>.3t / \2/^2.6! / \2/^ J

and determine its range of validity.

6. If f(z) = S z a /(4+n 2 z a ), show that f(z) is finite and

w*i

continuous for all real values of z but/(z) cannot be expanded
in a Maclaurin series. Show that /(z) possesses Laurent
expansions valid in a succession of ring spaces.

7. Prove that cosh (z+-j = a + a n (z n H ) , where

\ zl i \ z n l

1 r 27r

o n == cos n6 cosh (2 cos 8)dd.
2n J

8. Find the Taylor and Laurent series which represent
the function (z*-l)/{(z+2)(z+3)} in (i) | z |<2, (ii) 2<| 2 |<3,

9. Find the nature and location of the singularities of the
function l/{z(e f 1)}. Show that, if 0< | z |<27r, the function
can be expanded in the form

and find the values of a and a a .

10. The only singularities of a single-valued function
f(z) are poles of orders 1 and 2 at z = 1 and z = 2, with
residues at these poles 1 and 2 respectively. If /(O) = 7/4,
/(I) = 6/2, determine the function and expand it in a Laurent
series valid in 1< | z |< 2.

11. Classify the points z = 0, z = 1 and the point at
infinity, in relation to the function

/(,).!=? sin jij,

and find the residues of /(z) at z = and at z = 1.

12. Show that, if 6 is real, the series

THE COMPLEX INTEGRAL CALCULUS 113

is an analytic continuation of the function defined by the
series

13. The power series z + z 2 +i^ 8 + and

have no common region of convergence : prove that they
are nevertheless analytic continuations of the same function.

14. If a>e, use Rouch6's theorem to prove that e* = az*
has n roots inside the circle | z \ = 1.

15. The Fundamental Theorem of Algebra. By taking
f(z) = a z m , g(z) = c^z- 1 +a^z m ~ z + ... +a m , use RoucWs
theorem to prove that the polynomial

F(z) = a z"+a 1 z'- 1 + ... +a m

has exactly m zeros within the circle \z\ = R for sufficiently
large R.

Deduce from Liouville's theorem that F(z) has at least
one zero.

16. Prove that 3 8 +3z 8 +7z+6 has exactly two zeros in
the first quadrant.

17. If |/(z) \>m on | z \ = a, /(z) is regular for | z \^a
and |/(0) | <ra, prove that /(z) has at least one zero in | z | <a.
(See 42.)

Deduce that every algebraic equation has a root. (This
is another proof of the Fundamental Theorem of Algebra.)

18. If a domain D of the z -plane is bounded by a simple
closed contour C and w =/(z) is regular in Daiid on C7, prove
that, if /(z) takes no value more than once on C, then f(z)
takes no value more than once in D. (Use the theorem
of 40.)

Prove that the above result holds for the function
w = z a +2z+3, if D is the domain | z \ < 1 and C is the unit-
circle.

CHAPTER V

THE CALCULUS OF RESIDUES

43. The Residue Theorem

We now turn our attention to the residue theorem,
and to one of the first applications which Cauchy made
of this theorem the evaluation of definite integrals. It
should be observed that a definite integral which can be
evaluated by Cauchy's method of residues can also be
evaluated by other means, though usually not so easily.*

We have already defined the residue of a function
f(z) at the pole z = a to be the coefficient of (z a)~ l in
the Laurent expansion of /(z), which, if z = a is a pole
of order m, takes the form

Za n (z-a)+3b n (z-a)-*.
o i

We have also remarked that, when z = a is a pole of
order one, the residue b l can be calculated as lim{(z a)/(z)}.

->a

The residue can also be defined as follows. If the point
2 = a is the only singularity of /(z) inside a closed contour

1 f

C y and if . I f(z)dz has a value, that value is the residue
1m J Q

of /(z) at z = a.

The residue of /(z) at infinity may also be defined.
If /(z) has an isolated singularity at infinity, or is regular

/oo
e~ m * dx, sometimes stated to be an integral which

cannot be evaluated by Cauchy's method, see Courant, Differential
and Integral Calculus, II, p. 661. In this case Cauchy's method
is the more difficult.
114

THE CALCULUS OF RESIDUES 115

there, and if G is a large circle which encloses all the finite
singularities of/(z), then the residue at z = oo is defined to be

taken round O in the negative sense (negative with respect
to the origin), provided that this integral has a definite
value. If we apply the transformation z = l/ to the
integral, it becomes

taken positively round a small circle, centre the origin.
It follows that if

has a definite value, that value is the residue of /(z) at
infinity.

Note that a function may be regular at z = oo but yet
have a residue there.

The function /(z) = A/z has a residue A at z = and a
residue A at z = oo, al though /(z) is regular at z = oo.

Theorem 1. Cauchy's Residue Theorem.

Let f(z) be continuous within and on a closed contour
and regular, save for a finite number of poles, within G.
Then

I f(z)dz = 2mSy%>
J c

where Z72 is the sum of the residues off(z) at its poles within C.
Let a lf a 2 , ..., a n be the n poles within C. Draw a
set of circles y r of radius 8 and centre a r , which do not
intersect and which all lie inside O. Then /(z) is certainly
regular in the region between C and these small circles y f .

116

FUNCTIONS OF A COMPLEX VARIABLE

We can therefore deform C until it consists of the small
circles y r and a polygon P which joins together the small
circles as illustrated in fig. 13.

FIG. 13.

Then

f f( z )dz = f f(z)dz+Z ( f(z)dz = 2 f f(z)dz,

J C J P fl J 7r f = l J Yr

for the integral round the polygon P vanishes because
f(z) is regular within and on P.
If a f is a pole of order m, then

where <f>(z) is regular within and on y f . Hence

f m C b

f(z)d*=Z 7-~T^Z.

J y r 8-1 J y r \ z a r)

On writing z = a r -\-&eW, varies from to 77 as the point z
makes a circuit of the circle y r , and so

f f( z )dz = Zb 8 &-> \ * e(i-'Wid0 = 2irib v
J Yr i J

Hence j f(z)dz = Z \ f(z)dz = 2iri

J C f-l J Yr

which proves the theorem.

THE CALCULUS OF RESIDUES 117

Theorem 2. // lim{(z a)f(z)} = 6, and if C is the

i-^a

arc, #!< arg (2 a) <0 2 , o/ fe circle \ za \ = r,
lim f /(z)dz

r-^Oj

Given c, we can find an 77 (e) such that, if | za |<ij,
| S |<, where (z-a)/(z) = fr+8.

Hence

f // 1 J f 6 + 8 J f *'

/( 2 ) <fe = dz =

Jo Jcz o Jfli

i r

IJo

and so, on taking the limit as r-> 0, the theorem follows.

If 2 = a is a simple pole of /(z), 6 is the residue of f(z)
at z = a, so that if the contour is a small circle surrounding
the pole, a #1 == 2ir and we get

J,

/(Z) dz = 27N&.

44. Integration round the Unit Circle

We consider first the evaluation by contour integration
of integrals of the type

f

, sin0)d0,

where <(cos0, sin#) is a rational function of sin0 and
cos 0. If we write z = e*0, then

f27T f

and so I ^(cos 0, sin 0)d0 = 4t(z)dz,

Jo Jo

118 FUNCTIONS OF A COMPLEX VARIABLE

where 0(z) is a rational function of z and is the unit circle
I z I = 1. Hence

/.<

where Z7c denotes the sum of the residues of ^r(z) at its
poles inside C.

Example. Prove that, if a > 6 > 0,

J = \

Now on making the above change of variable, if C is the
unit circle | z \ = 1,

j = JL J _XI l^ZI = J_ I _JL: i^I_ = 1 f ^( Z )^,

where

are the roots of the quadratic z*+2az/b + l = 0. Since the
product of the roots a, j8 is unity, we have | a || ]8 | = 1 where
| ft \>\ a |, and so z = a is the only simple pole inside C.
The origin is a pole of order two. We calculate the residues
at (i) z = a, and (ii) z = 0.

(a--)'

(i) Residue = lim(z-a)F(z) = lira ~ ^-' = - - ^
z-+a z^a z ( z -P> a ~P

(z I) 1
(ii) Residue is the coefficient of 1/z in - - - ,

Z \Z "T~iC**y(/ "|~ L)

where z is small. Now

z\z* + 2az/b + 1 )
and coefficient of l/z is plainly 2a/6.

THE CALCULUS OF RESIDUES 119

which proves the result.

45. Evaluation of a Type of Infinite Integral

Let Q(z) be a function of z satisfying the conditions :
(i) Q(z) is meromorphic in the upper half-plane ;
(ii) Q(z) has no poles on the real axis ;
(iii) zQ(z)-+ uniformly, as | z |->oo, for 0<arg

(

f [0

iv) I Q(x)dx and Q(x)dx both converge ; then
Jo J oo

where EJ& denotes the sum of the residues of Q(z) at its
poles in the upper half-plane.

Choose as contour a semicircle, centre the origin and
radius R, in the upper half-plane. Let the semicircle be
denoted by I 7 , and choose R large enough for the semicircle
to include all the poles of Q(z). Then, by the residue
theorem,

f Q(x)dx+ f Q(z)dz =
J -R J r

From (iii), if R be large enough, | zQ(z) |< for all points
on JT, and so

I f Q(z)dz = I { V Q(Ref8)R&OidO j <c f" d0 = TIC.

Hence, as jR->oo, the integral round JT tends to zero.
If (iv) is satisfied, it follows that

/)//M\/7/M _ OUM|* ^*f&^

\\X](LX = ^TTl^i//^^.
oo

If Q(z) be a rational function of z, it will be the ratio
of two polynomials N(z)/D(z), and condition (iv) is satisfied

120 FUNCTIONS OF A COMPLEX VARIABLE

if the degree of D(z) exceed that of N(z) by at least two,
for, when x is large, Q(x) behaves like x~* 9 where p^2 and

f dx A f ~ M dx i. *u * *
I and I both exist.*

J A x> J -*x*

Note that condition (iv) is required as well as (iii), for the
condition xQ(x)->0 is not in itself sufficient to secure the

/GO

convergence of I Q(x)dx. This can be seen by taking
Q(x) = (* log *)-*.

Example. Prove that, if a>0,
dx

If 2 4 +a* = 0, we have z 4 == a 4 e 7ri , and the simple poles
of the integrand are at ae 7 ^/ 4 , ae 377 */ 4 , ae 57r ^/ 4 , ae 77ri / 4 . Of these,
only the first two are in the upper half-plane. The conditions
of the theorem are plainly satisfied, and so

dx
= 2iri E {Residues at z = ae 77 */ 4 , ae 37r */ 4 }.

Let k denote any one of these, then A 4 = a 4 and the residue
at the simple pole z = k is lim{(2 fc)(z 4 & 4 )- 1 }. This may

*->*

be evaluated by Cauchy's formula, as applied to the evaluation
of limits of expressions of the indeterminate form f 0/0, and so

z-k

Hence

* For the convergence of infinite integrals, see P. A., p. 193,
or G.I., p. 77.
t P.A., p. 106.

THE CALCULUS OF RESIDUES 121

00 dx

f ax

Hence I . =

j x*+a 4

The theorem can be extended to the case in which
D(z) = has non-repeated real roots, so that Q(z) has
simple poles on the real axis. We now indent the contour
by making small semicircles in the upper half-plane to
cut out the simple poles on the real axis. Suppose that
D(z) = has only one root z = a, where a is real. The
contour is then as shown in fig. 14. The small semicircle

is denoted by y, its centre is the point x = a and its
(small) radius is />. If F is large enough to enclose all
the poles of Q(z) in the upper half-plane, then the integral
round F tends to zero as R-+OO, as before. We therefore
have, if the path of integration be as indicated by the
arrows in fig 14,

a+p

f + P~ P + f + f*
J T J -R Jy Ja

fa-p [R f

As IZ^oo, + Q(x)dx = P Q(x)dx,

J -R J a+p J -oo

and it remains to consider I Q(z)dz. Now, on y f

JY
z = a+/>e l *0, and so

f Q(z)dz= t
J y J

122 FUNCTIONS OF A COMPLEX VARIABLE

*

Since Q(z) contains the factor (2 a)- 1 we may write
Q(z) = <f)(z)I(za) and ^(z) is regular at and near z = a.
Hence

f Q(z)dz = f <(>(a+peiO)id6 = t f

J y J 7T J

since <f>(a+peiO) is regular at and near a and can be expanded
by Taylor's theorem with remainder when n = 1. It
follows that

Q(z)dz-+ 7n^(a) as p-> 0.
y

Since ^(a) is plainly the residue of Q(z) = <f>(z)/(za) at
z = a, we can write the final result in the form

f

J -0

Q(x)dx

where Z7 denotes the sum of the residues of Q(z) at its
simple poles on the real axis, for clearly each pole on the
real axis can be treated in the same way as z = a. The
principal value of the integral is involved, because equal
spaces p are taken on either side of the real poles, and,
by definition,*

lim (~~ P + ( ft f(x)dx = P ( ft f(x)dx.

p-+Q J a J a+p J a

It should be noticed that if a pole be cut out by a small
semicircle, the contribution to the value of the integral is
half what it would be if a small circle surrounded the pole.
(See 43, Theorem 2.)

46. Evaluation of Infinite Integrals by Jordan 's
Lemma

We now prove a very useful theorem which is usually
known as Jordan's lemma.

* See P.A., p. 195, or G.I., p. 81.

THE CALCULUS OF RESIDUES 123

If F be a semicircle, centre the origin and radius R, and
f(z) be subject to the conditions :

(i) f(z) is meromorphic in the upper half -plane,
(ii) /(z)-> uniformly as \ z |-*oo/or 0<arg
(ill) m is positive ; then

J.

e mit f(z)dz-+ as J?->oo.

By (ii), if R is sufficiently large, we have, for all points
on/ 1 , | /(z)|<. Now

| exp miz \ = | exp{wiJ?(cos0+t sin0)}| =
Hence,

j f(z)e*dz = I f(z)e>i*ReiOid6 < J e-Ri

J

2Re *

Now it can be proved, by considering the sign of its
derivative, or otherwise, that sinO/d steadily decreases
from 1 to 2/7T as 6 increases from to Jw. Hence, if

sin0
~W
Hence

I f /(zjgm&dz < 2Re ( e-2M/nd0,

m m

from which the lemma follows.

By using this lemma, we can evaluate another type
of integral. The method may be set out as a theorem
as follows.

124 FUNCTIONS OF A COMPLEX VARIABLE

Let Q(z) = N(z)/D(z), where N(z) and D(z) are poly*
nomials, and D(z) = has no real roots, then if

(i) the degree of D(z) exceeds that of N(z) by at least one,
(ii) m>0,

f00

Q(x)e mi *dx =

I.

where 27S+ denotes the sum of the residues of Q(z)e mi9 at
its poles in the upper half -plane.

If we write f(z) = Q(z)e m< * , we see that f(z) satisfies the

conditions of Jordan's lemma and so I f(z)dz-> as jR->oo.

On using the same contour as before, a large semicircle
in the upper half-plane, by making B-+CQ we get

Q(x)e mix dx = ^mETe*.

30

On taking real and imaginary parts of this result we see
that by this method we can evaluate integrals of the type

I f(x) cos mx dx , f(x) sin mx dx.

J -00 J -00

By a well-known test for convergence of infinite in-

fx

tegrals,* if f(x) decreasesand -> as s->oo, since , c ? 8 mx dx

J a sm

is bounded, the integrals in question converge.

Example. Prove that, if a>0, m>0,
00 cosma;^ ir

f
J

The only pole of the integrand considered, e

in the upper half-plane, is a double pole at z = ai. The

* The test is known as Dirichlet's test. See Titchmarsh,
Theory of Functions (Oxford, 1932), p. 21.

THE CALCULUS OF RESIDUES 125

conditions of the theorem are easily seen to be satisfied
and so

dx =s 27rt{Residue of at z = ai}.

jo (a 2 +a; 2 ) a (a 2 +z 2 ) a

Put z = ai+t then, since t is small,

gtnif e~ mo 6 m ** 6""***

(a a +z a ) a < 2 (2at+0 1 4a 2 * 2

and the residue, which is the coefficient of tr l , is easily seen
to be

te- ma (l+ma)

f e m<a! dx 7re~ tno (l+ma)

Hence = ^ ' .

J -oo (a 2 +x 2 ) 2 2a 8

On equating real parts, and taking one half of the result
for the integral from to oo, we get

r cos mx dx TJ

If there are simple poles of the integrand on the real
axis, we get a modification of this result, similar to that
obtained for the theorem of 45. Thus, if D(z) = has
non-repeated real roots, we get

r 00

> I 0(x

)e mi *dx =

the proof following the same lines as before.

An example of this extension of the theorem is the proof
that

00 *

! dx = JTT, if W>0.

On considering the integrand e mi '/z we see that it has
a simple pole at z = and none in the upper half -plane.
The residue at z = is easily seen to be unity, and so we get

r oo 0m<

j 1

J -at X

126 FUNCTIONS OF A COMPLEX VARIABLE

On equating real and imaginary parts we get

cos mx _
dx = 0,

*

sin mx _
dx = ir i

r

J

the " P '* is not necessary in the second integral, since
sin mx/x^- m as #-> 0, whereas the integrand in the first integral
becomes infinite at the origin. From the second result we get

00 sin mx ,

dx = JTT.

x

47. Integrals Involving Many- Valued Functions

f
A type of integral of the form &- l Q(x)dx, where a

Jo

is not an integer, can also be evaluated by contour
integration, but since z a ~ l is a many-valued function, it
becomes necessary to use the cut plane. One method of
dealing with integrals of this type is to use as contour a
large circle F, centre the origin, and radius B ; but we
must cut the plane along the real axis from to oo and
also enclose the branch-point z = in a small circle y of
radius p. The contour is illustrated in fig. 15.

Fio. 15.

THE CALCULUS OF RESIDUES 127

Let Q(x) be a rational function of x with no poles on
the real axis. If we write f(z) = z?~ l Q(z) and suppose
that z/(z)-> uniformly both as | z \ -> and as | z \ ->oo,
then we get the integral round F tending to zero as
jR->oo and the integral round y tending to zero as />-> ;
for, on F, if -R is large enough, | zf(z) \ <e and so \f(z) \ </R 9

\z)dz
Similarly on y, | z/(z)|< if p is small enough, and so

f(z)dz < - 2np = 27T.
P

Hence on making p-+ and R-+CQ we get

/> fo

&~ l Q(x)dx +

JO Jo

where Sy\* is sum of residues of /(z) inside the contour.
We observe that the values of z*" 1 at points on the upper
and lower edges of the cut are not the same, for, if z = re^,
we have z*"" 1 = r*- V^*" 1 * and the values of z at points
on the upper edge correspond to | z | = r, = 0, and at
points on the lower edge they correspond to | z \ = r ,

= 277.

Since e^wifa"" 1 ) - ^Trtc^ \^e get

f

J

We also observe that, when calculating the residues at
the poles, z a ~~ l must be given its correct value r a
at each pole.

Example. Prove that

f &- l dx ir .. _

-7- = -: , if 0<o<l.

Jo 1 +a? sin aw

128 FUNCTIONS OF A COMPLEX VARIABLE

Here we observe that, when/(z) = z?~ l (l +z)~ l , z/(z)-> aa
| z | ->oo, if a< 1, and z/(z)-> as | z \ -> 0, if a>0. Hence, if
0<a<l, the integral round F tends to zero as .R->oo and
the integral round y tends to zero as p-> 0.

Thus

r

J o

{Residue of z- l (l+z)~ l at z = -1}.

1+3? 1-

At z = 1 we have r = 1, $ = w , and so, for the residue,

lim f , % z - 1 ) , , . .

g -*-l 1 (1+z) f = ( I)*"" 1 = e (a ~ 1)7r< =

Hence

f * &~ l , . f e 1 " 7 '

Jo 1+5* = ~ 27rt I !=*

Bin air

This integral can also be evaluated by integrating
) t using as contour a large semicircle in the upper
half plane and the real axis indented by semicircles at
2 = and at z = 1. In this case the cut plane is
unnecessary. The evaluation of the integral by this
second method is left as an exercise for the reader.* By
this second method one obtains the further result that

f*&-*dx

=77

Jo 1-*

COtOTT.

48. Use of Contour Integration for deducing
Integrals from Known Integrals

The contours used so far have been either circles or
semicircles, and although a large semicircle in the upper
half plane is generally used for integrals of the type
discussed in 45, there is no special merit in a semicircle.
The rectangle with vertices <R, 7J+iJ? could also be

* Bee Copaon, Functions of a Complex Variable, p. 140.

THE CALCULUS OF RESIDUES

129

used in these cases. We now give two examples of
deducing the values of some useful integrals by integrating
a given function round a prescribed contour. We use,
in the first case, a rectangle and in the second a quadrant
of a circle.

Example 1. Prove that f e~ x% cos 2ax dx . \ ^ire-** by

J o

integrating -* round the rectangle whose vertices are 0, R,
~ , ia.

y

C B

Fio. 10.

Let A be (1?, 0) and O be (0, a) in the Argand diagram.
On OA, z = x ; on AB 9 z = R+iy ; on BG 9 z = x+ia ;
and on 0(7, z = iy. Now e"** has no poles within or on this
contour and so, by Cauchy's theorem,

[ R f* f f

'o JQ JR Ja

Hence

0.

p /j

e~ x *dx e a *
Jo 'o

Now

2oa? t sin
t-ij a &'dy=*0. . (1

i f e-^a-zfly+if 1 ^ <e-^ $ . e a *. a
Jo

and so this integral -> as R > oo. On using the result that

I e~ x *dx = i\/7r,
we find, on making !?-> oo and equating real parts in (1),

1 cos

r

e~*

Jo

130

FUNCTIONS OF A COMPLEX VARIABLE

Example 2. By integrating e^'z - 1 round a quadrant of
a circle of radius R, prove that, if 0<a< 1,

_1 cos

The contour required is drawn in fig

y

cos

FIG. 17.

Since the origin is a branch-point for the function z*- 1 ,
we enclose it in a quadrant of a small circle y of radius p.
We integrate round the contour in the sense indicated by
the arrows. On y, z = peiO and we get

If

<p a J dO =
since | e~P sin e \ < 1 when p is small. It follows that

/B->0, i/a>0.

If a< 1, | z a ~ l \ < when J? is large enough, and so by the
same argument as was used in proving Jordan's lemma
( 46) we have

f
J

as #->oo, if a< L

Hence, if 0<o< 1, we get, on making p-> and

f
J

0.

THE CALCULUS OF RESIDUES 131

since there are no poles inside the contour. Hence

r 00 r 00 / TTQ ira\

x*~ 1 (cos x+i sin x)dx = e~*y*~ l (cos +i sin I dy.
JQ J \ 2 27

Since f tr*y*- l dy = r'(a),* on equating real and imaginary
Jo

parts, the required results follow.

49. Expansion of a Meromorphic Function

Let f(z) be a function whose only singularities, except
at infinity, are simple poles at the points z = a l9 z = a a ,
z = a 8 , ... ; and suppose that

Suppose also that we know the residues at these poles :
let them be b l9 6 2 , 6 3 , .... Consider a sequence of closed
contours, either circles or squares, C lt (7 2 , <7 3 , ..., such
that G n encloses a l9 a a , ..., a n but no other poles. The
contours C n must be such that (i) the minimum distance
R n of C n from the origin tends to infinity with n 9 (ii) the
length L n of the contour C n is 0(R n ), (iii) on C n we must
have /(z) = o(R n ). Condition (iii) would be satisfied if
/(z) were bounded on the whole system of contours G n .

When these conditions are satisfied we can prove that,
for all values of z except the poles themselves,

1 1

To prove this, consider the integral

wh^re z is a point within C n . The integrand has poles
at the points a m with residues b m l{a m (a m z)} ; at = z

* G.I., p. 84.

132 FUNCTIONS OF A COMPLEX VARIABLE

with residue f(z)/z ; and at = with residue /(0)/z.
In particular cases the last two residues may be zero.
Hence

If now we can prove that J->Q as w->oo, the theorem is
proved. Here we require the conditions laid down above
on the contours <7 n . On making use of these, we see that

The series is uniformly convergent in any finite region
which does not contain any of the poles.

As an example of this theorem we prove that

1 oo / )n-l

cosec z = 2z S -y-^ - .

Consider the function /(z) = cosec z (z^0),/(0) = 0. Now

z

sin z has simple zeros at the points z=n7r,(n=... 2, 1, 1, 2,...)

and so/(z) = : will have simple poles at those points.

z sin z

The residue at z = HIT becomes, on writing znir = f,

sintf+nrr)} ^ i im

(f +nir) ain (t+nn) f-^0 (f-f nn) cos (f +nir)+ am (f -j-nw)

nir cos WTT

There is no singularity at z = since
z~sinz 0(1 z I 1 }

zamz ~ z*+0{\z\ 4 }

0(\z\).

Let O n be the square with corners at the points (n+ J)( 1 *)ir.
The function 1/z is certainly bounded on these squares. To
prove that cosec z is also bounded, consider separately the

THE CALCULUS OF RESIDUES 133

regions (a) y>^ 9 (b) y<-ln, (c) -fr^y^n. In (a) we
have y>i?r and

coseo z

for | 6 "-a-*| > |{| a"| - | a-* |}| - |{| a-f | - | c'\} |;

and a similar argument applies to (b), writing y = t so that
$> JTT. For (c), let AB be the line joining the points \ir,\Tri.

Since | sin z \ = (cosh 2 !/ cos 2 #)* ; on AB we have, since x = JTT,
| sin z | = cosh y^l, so that | cosec z \ ^1.

Since cosec z has period w, it is bounded on all the lines
joining (n + $i)n and (n + $ + $i)ir. Hence cosec z is
bounded on all the squares (?. The previous theorem
therefore gives

cosec z- -

I 1 \

+ ,

~n7T HTT/

the accent indicating that the term n = is omitted from
the summation. Since the series with n>0 and with n<0
converge separately, we may add together the terms corre-
sponding to n and write the expansion

ft
cosec z -- =* 2z

z - n w~"

50. Summation ol Series by the Calculus ol
Residues

The method of contour integration can be used with
advantage for summing series of the type 2f(n), if /(z)
be a ineromorphic function of a fairly simple kind.

Let be a closed contour including the points m,
m+1, ..., ft, and suppose that /(z) has simj
points a l9 a 2 , ..., a fc , with residues b v
the integral

/

77 COt7TZ/(z)

o

134 FUNCTIONS OF A COMPLEX VARIABLE

Tho function IT cot TTZ has simple poles inside C at the
points z = m, ra+1, -., w, with residue unity at each
pole. The residues at these poles of TT cot 772: /(z) are
accordingly /(m), /(m+1), ...,/(n). Hence, by the residue
theorem,

f f(z)7TCotirz dz = 27Ti{f(m)+f(m+I)+...+f(n)

J

+6x77 cot 7ra 1 +... +6^71 cot TraJ.

If conditions are satisfied which ensure that the
contour integral tends to zero as w->oo, we can find the
sum of the series Ef(ri). Suppose that /(z) is a rational
function, none of whose zeros or poles are integers, such that
z/(z)-> as | z |->oo. Let C be the square with corners
(n+\)(\i). We have seen that cotTrz is bounded

... , I f fl . , dz 7rML

on this square and so zf(z)ir cot TTZ
\Jo z

f
for n

R

large enough, where M is the upper bound of | cot TTZ |
on (7, L is the length of C and R is the least distance of the
origin from the contour. Since L = 8R, the integral tends
to zero as w->oo, and so

n

lim S f(m) = Trtyi cot 7ra 1 + . . . +6 fc cot 7ra k }.

If we use 77 cosec TTZ instead of 77 cot m, we can obtain
similarly the sums of series of the type Z( l) m f(m).

oo I oo ( l) n

Example. Find the sums of the series E -, E -.

For the first, /(z) = and so z/(z)-> as | z |->oo. The

z a +a a

two poles of /(z) are at z = ai and the residues at these
poles are l/2af. Hence

00 1 I ] I "\ _

27 r ; SB w i r . cot Trai -; cot ( nai) \ = - coth we

!.- m 2 +a 2 t^a* 2ai J a

or -^ +2<

THE CALCULUS OF RESIDUES 135

Similarly we get, by using IT cosec irz instead of w cot nz 9

oo / l\m I

Z = T- . -4- r- cosech iro.

m m2 + a 2o 2a

In simple cases we can deal similarly with functions
f(z) which have poles which are not simple. As an example,
consider the series

s i

-.(+)'

Here f(z) = (a+z)~ 2 has a double pole at z = a. By
Taylor's theorem

cot wz cot( 7rGt)4*(w2J+wct){ cosec f ( ira)}-f-...,
and so the residue of cot irz/(z+a) 2 at = a is w cosec'Tra.

oo l

Hence 27 = w 2 cosec 1 wa.

EXAMPLES V

Use the method of contour integration to prove the
following results 1 to 10 :

, C* a dS

L

oos>30

. ( ,.v, ) - f (napositive
+2 cos V5

integer).

ir

(6+2c)

r
' J -

cos a?

136 FUNCTIONS OF A COMPLEX VARIABLE

cos ax dx ir .

f
Jo

73

f<* a^-icfo 2w /27ra + 7r\

J 1+5+? = 73 cos Hi-) coseo - ' (0<0<2) '

++ 3

x a dx nil a)

/ rf z
11. Evaluate I - taken round the ellipse whose equation

r c?2J
is x % xy+y* +x+y = 0. Evaluate similarly - 1 round

J 1+2

the ellipse

12. Show that the function }(z) = z/(ae" iM ) has simple
poles at the points z = i loga+27rn, (n = 0, 1, 2, ...) ;
and by integrating /(z) round a rectangle with corners at
ir, .7r+in prove that, if a>l,

fir xsinxdx n , 1+a

I - - log -
J 1+a 2 2a cos a; a a

13. By taking as contour a square whose corners are
N, N+2Ni, where N is an integer, and making N-+CQ,
prove that

f 00 dx 10

I - = log 2.

Jo (l+# 2 ) cosh (jTTtf) 6

14. By integrating e-*^* 1 " 1 round a sector, of radius R 9
bounded by the lines arg z = 0, arg z = a< JTT (indented at 0),
prove that, if fc>0, n> 0,

r cos cos

J -

r> x*dx 7T\/3

15. Prove that P -3 r = r-
J o a^ l o

16. Prove that ____&- _(J-a+l-ar-), (o>0).
*a*+x* 2o*

THE CALCULUS OF RESIDUES 137

17. By integrating e a< /(e~ 2<1 1) round a suitable contour,
prove that

I : L. = ITJ. coth fact

J e*y-l * * 2a

18. Prove that sees = 47727(--l) n (2n + l)/{(2n-fl)*77 a --42*}.

o

19. Prove that, if 77<o<77,

sin 02 2 *J n sin na

~: = - 27 ( l) n - ,

sin 772 77 n i 2' n*

cos 02 1 22 cos no

__ == -| 2i ( 1)

sm 772 772 77 na ,i z*n*

20. Prove that

* 1 __ 77 sinh(77aV2)+sin (77a\/2)
n--ao nT+a* a 3 V 2 cosh (770 V 2 )~

oo / !\n

and find 27 ~ ^ .

(ii) Prove that

00 1 1 77

27 - - = - i ; (coth 77a+cot na).
nl w o 2a 4 4a 3

77 sin az , . _ _

21. By integrating - - round a suitable contour,

2 3 Sin 772

prove that

1_I + I_I+ ^l 8
P 3 3 5 8 7 3 32'

f 2+1

22. (i) Prove that 2 2 log - d2, taken round the circle

J 2 1

2 | = 2, has the value 477^/3.

f lg x dx 77

(u) Prove that ~ - ~- = -- , using as contour a
J (l+ar) a 4

arge semicircle in the upper half-plane indented at the
>rigin.

MISCELLANEOUS EXAMPLES

1. If u> = tt+tt> is a regular function of z, show that

/ d* d 2 \

I + 1 w = is equivalent to

dzcz
Hence show that

iv\'(z+z) 9 !.(-)} #)-#*) +c f

\& 41 J

where C is an arbitrary constant.*

Use the above relations to find the regular function / (z)
for which u is (i) log (x* +t/ 2 ), (ii) x 2 y 2 +4txy and for which v is

e 2x (y cos 2y+x sin 2y).

2. If x = r cos 0, y = r sin change the independent

/ a 2 d* \
variables in I + 1 <f> = 0, to r, 8.

If u (x 2 +y 2 )*/(x*y*) find the function <f>(u) which
satisfies v a < = 0. Find also the regular function / (z) of
which <f>(u) is the real part.

3. Show that, if y +0, there are two points unaltered by
the transformation

unless (8 a) 2 +4y = 0, in which case there is only one such
point. Show that, if z = I is this point,

-L L+

,_! s-1 +"

where K is a constant.

* This result was communicated to me by Prof. A. Oppenheim-
138

MISCELLANEOUS EXAMPLES 139

Show that z = 1 is the only fixed point of the trans-
formation

l+iz tan A

w

l+i tan A*

(0<A<j7r), and that this transformation maps the inside of
| z | = 1 on the inside of | w \ = 1.

Sketch the curve in the w-plane which corresponds to the
straight line joining 2 = 1 and z = t.

4. Prove that, if fc>0, w = tan (irz/lk) maps lines parallel
to the axes in the z-plane on systems of coaxal circles in the
u?-plane.

Find what corresponds to the infinite strip t = & and
indicate in a figure the region of the w-plane corresponding to
the square Q^x^k, k^y^2k.

5. Show that, if a>0, the relation u> = aicot \z maps
the semi-infinite strip 0^a5^2?r, t/^0 on a half -plane cut from
u = a to w = oo .

Two circles, with real limiting points (a, 0), are drawn
in the cut w-plane with centres (2ka, 0), (to, 0), where Jfe>l.
Show that the space between these circles is mapped on the
interior of a rectangle in the s-plane whose area is

(2k-l)(k + I)
" g

6. Express the transformation

in the form - = k\ - - 1 and hence show that the inside
w ft \z b!

of | z | 1 is mapped on the whole w-plane cut along a segment
of the real axis.

Illustrate by a diagram what corresponds in the u;-plane
to that part of the circle | z i \ = <\/2 which lies in the fourth
quadrant.

7. If a, 6, c, d are real constants, some of which may be
zero, and

az*+bz+o
*= z+d '

140 FUNCTIONS OF A COMPLEX VARIABLE

show that there are two values of w, each of which corresponds
to a pair of equal values of z ; and that these values of w
can only be equal if the transformation is bilinear.
Discuss the transformation

-2s

w--

1-23

in this way, and find the boundaries in the z-plane which
correspond to| w \ = 1.

Show in a diagram the regions of the z-plane corresponding
to |10|<1.

8. Iff (z) is regular within and on the circle | z \ = R and
^ If ( z ) I <M on ^ e circle, prove that, if | z \ = r<R 9

oo

where a is the constant term in / (z) = Z a^s*.

=*o

If R = 1, a = 1 and \f (z) \<k on \z\ = I, prove that
/ (z) does not vanish within the circle | z \ = 1/(1 +&).
9. Show that, if 0<a<7r and

4az cot a

then w f (z) gives a conformal transformation when z lies
in any finite region excluding the points z = t, z = cot a,
z = tan a.

Show that the boundary of the semicircle | z \ = 1, Rz>0
corresponds to an arc of a circle in the w-plane subtending
an angle 4a at the centre.

Find the two points in the z-plane corresponding to the
centre of this circle.

10. Show that, if

u? =

two finite points of the z-plane are mapped on every finite
point of the to-plane, except the origin and w = 1, and
explain why the mapping ceases to be conformal at the
point z = 1.

Show, in a diagram, the two domains of the z-plane which
are mapped on the semi -circular domain | w |<1, Iw>0.

MISCELLANEOUS EXAMPLES 141

oo oo

11. Iff(z) = 2a n z n and <f>(z) = b n z n are both regular within

a domain D which includes the circle | z \ = 1, prove that

o * o

Hence, or otherwise, prove that

<W- " (IT) 5 + dV- " " s; J7 COS (2 8in e)de -

12. A function <(z) is regular over the whole z-plane,
except at z = and at z = oo, and, for all values of z 9
<f>(z) z<l>(pz), where |)5|<1. Find the Laurent expansion
of (f>(z), given that the constant term in the expansion is unity.

Show that <() =0.

13. ABCD is a square whose vertices are 0, t, 1 +i, 1 ;
F t is the line BG and F 2 consists of the other three sides of the
square. Iff (z) = 2 5 +z 2 + 1 prove that (i)/ (z) does not assume
any real non-negative value on JTj (ii) R/ (z) >0 at all points of
r a except B.

Hence, or otherwise, evaluate

A

and deduce that /(z) has just one zero inside the square
ABGD.

14. By integrating e ij>f /cosh a z along the lines I(z) = 0,
I(z) = TT prove that

oo
COS px irp

cosh 2 x 2 sinh

\j

15. By integrating

cosec z

round a square whose vertices are (n + J)(7r^) prove that

cosec t 2et * ( l) n

" /i_iw

142 FUNCTIONS OF A COMPLEX VARIABLE

16. By discussing

dz
L (-7r<a<7r)

cos TTZ z

round the circle | z \ = r, where r is an integer, show that
cos q = 2 (-l)f+l(r-}) CO s(f--})a
~~

17. By integrating z~* sec \itz sech \TTZ round the square
Rz = 2N, Iz 2N, where N is a positive integer, and
making N-+CQ prove that

sech sech - sech 5

16 35 I 55

18. By considering

coth TTZ cot TTZ ,

taken round the square a;= (N+$), y == (^+|) where
W is a large positive integer, prove that

^r 77T 8

n

INDEX

The numbers refer to the pages

Aerofoil, 76

Algebra, the fundamental

theorem of, 113
Analytic continuation, 103-107
Analytic function, 105
Argand diagram, 6
Argument, 6; principle of the, 108

Bilinear transformation. (See

Mobius)
Biuniform mapping, 33 ; by

Mobius* transformations, 46
Boundary point, 8
Bounded set of points, 8
Branch of function, 26
Branch -point, 28

Calculus of residues, 114-137

Cardioid, 62, 77, 81

Cauchy's theorem, 89-92 ;
integral, 93 ; residue
theorem, 116

Cauchy-Riemann equations, 12,
30, 36, 89

Circle of convergence, 1 7

Circular crescent, 53, 68 ; sector,
transformations of, 69

Coaxal circles, 46-48

Complex integration (see Con-
tour integral), 85

Complex numbers, defined, 1 ;
modulus of, 2 ; argument
of, 6 ; real and imaginary
parts of, 1 ; geometrical
representation, 6 ; ab-
breviated notation for, 3

Confocal conies, 71, 74, 81, 82

Conformal transformations,

definition of, 32 ; tables of
special, 78, 79

Conjugate, complex numbers,
4 ; functions, 14

Connex, 9
143

Continuity, 9

Contour, 86 ; integral, definition
of, 88; integration, 117,
128 etseq.

Critical points ol trans-
formations, 37, 45, 65

Cross-cut, 93

Cross-ratio, 49

Curve, Jordan, 8 ; Jordan curve
theorem, 9

Cut plane, 27, 69, 70, 73, 126

Definite integrals, evaluation of,

114-131
Determinant of transformation,

41

Differentiability, 10-11
Domain, 9

Element of an analytic function,

105

Elementary functions, 17-25
Ellipse, 72, 74
Equation, roots of, 1, 31, 109,

113

Equiangular spiral, 31, 82
Expansion, of function in a

power series, 18, 95 ; of

meromorphic functions,

131 ; Laurent's, 97
Exponential function, 20, 31

Function, analytic, 11, 105;
hyperbolic, 23 ; holo-
morphic, 11; integral, 21;
logarithmic, 24 ; many-
valued, 26 ; meromorphic,
107, 131 ; rational, 20,
103; regular, 11

Gamma function, 130
Goursat's lemma, 91
Green'* theorem, 89

144

INDEX

Harmonic functions, 37, 111
Hyperbolic functions, 23

Imaginary part, 1

Infinite integrals, evaluation of,

119-130; principal value

of, 120

Infinity, point at, 9, 102
Integral functions, 21
Interior point, 8
Invariance of cross-ratio, 49
Inversion, 43 ; transformation,

55
Isogonal transformation, 32

Jordan curve, 8
Jordan's lemma, 122

Laplace's equation, 15, 30, 54
Laurent expansion, theorem, 97,

112

Lima9on, 61
Limit point, 8
Liouville's theorem, 96
Logarithm, 21
Logarithmic function, 24

Magnification, 34, 39, 41
Many-valued functions, 26
Mapping, 32
Maximum modulus, principle of,

109

Meromorphic function, 107, 131
Milne -Thomson's construction,

16
M6bius' transformations, 40-56 ;

some special, 51-54
Modulus, 2
de Moivre's theorem, 6

Neighbourhood, 7
One-one correspondence, 33

Parabola, 62, 64
Point at infinity, 9, 102
Poisson's formula, 111
Pole, 99, 107
Power series, 17-25
Principal part at pole, 99

Principal value, of arg z, 6 ;
of log z, 25 ; of infinite
integral, 122 ; of f*, 25

Radius of convergence of power

series, 17

Rational functions, 20, 103
Real part, 1

Rectangular hyperbolas, 61
Rectifiable curve, 86
Regular function, 1 1
Residue, 99; theorem, 114;

at infinity, 114
Riemann surface, 27, 59-60;

63, 71, 73, 75

Roots of equations, 1,31,109, 113
Rouche"'s theorem, 108

Schwarz-Christoffel transforma-
tion, 80

Sets of points, in Argand
diagram, 7-9 ; bounded, 8 ;
closed and open, 8

Singularity, 11, 99; essential,
99; isolated, 99; non-
isolated, 101
of analytic function, 105

Spiral, equiangular, 31, 82

Successive transformations, 58,
83

Summation of series, 133 et seq.

Taylor's theorem, 19, 96-96;

series, 101, 112

Term-by-term differentiation, 19
Transformations, conformal, 32

et seq. ; isogonal, 32 ;

special, 57-83 ; tables of,

78, 79
Trigonometric functions, 23

Uniform continuity, 10, 87
Upper bound for contour
integral, 88

Vectorial representation of com-
plex numbers, 6

Weierstrass' definition of ana-
lytic function, 105

Zero, 98, 107