Hassani S. Mathematical Physics: A Modern Introduction to Its Foundations

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analyticity

and

singularity;

regular

and

singular

points;

entire

functions

9.2

ANALYTIC

FUNCTIONS

233

By the triangle inequality,

IRHSI

lEI

iE21

181

i821.

This follows

from

the fact that Il!.xl/Il!.zl

and

Il!.yl/Il!.zIare both

equal

to at

most

The

and

terms

can

made

as small as desired by

making

l!.z small enough. We have thus

establishedthatwhenthe CoRconditions hold,

the

function f is differentiable. D

Angustin-Louis Cauchy (1789-1857) was

ODe

of themostinflu-

ential

French

mathematicians

of the

nineteenth

century.

Hebegan

his

career

asa

military

engineer,

butwhenhishealth

broke

down

in 1813 he followedhis

natural

inclination and

devoted

himself

wholly

mathematics.

mathematical

productivity

Cauchy

was

surpassed

onlyby

Euler,

andhis collected

works

fill 27 fatvolumes. He madesub-

stantial

contributions

number

theory

anddeterminants; is con-

sideredto bethe originatorof the theoryof finitegroups; anddid

extensive workin

astronomy,

mechanics,

optics,andthe

theory

elasticity.

His

greatest

achievements,

however,

layin thefieldof analysis.

Together

withhis con-

temporaries

Gauss

and

Abel,

hewasa

pioneer

inthe

rigorous

treatment

of limits,

continuous

functions,

derivatives,

integrals,

andinfinite series.Severalofthebasictestsforthe

conver-

genceof series

are

associated withhis

name.

He also

provided

the

first

existence

proof

for

solutions of

differential

equations,

gavethe

first

proofofthe

convergence

of a

Taylor

series,

andwasthe

first

tofeel theneedfora

careful

studyof theconvergence

behavior

Fourier

series(see

Chapter

8).

However,

his most

important

workwas in the

theory

functions

of a complex

variable,

whichinessencehe

created

andwhichhas

continued

tobeone of

the

dominant

branches

of bothpureand

applied

mathematics.

In thisfield,Cauchy's inte-

gral

theorem

and

Cauchy's

integral

formula

are

fundamental

tools

without

whichmodem

analysiscouldhardlyexist (seeChapter 9).

Unfortunately,

his

personality

didnot

harmonize

withthe

fruitful

powerof his

mind.

He was an

arrogant

royalist

in politics anda self-righteous,

preaching,

piousbeliever in

religion-all this in an age of

republican

skepticism-and most of his fellow scientists

disliked him and considered

him

a smug hypocrite. It might be fairer to

put

first things

first

and

describe

him as a

great

mathematician

who

happened

also to be a

sincere

but

narrow-minded

bigot.

9.2.6. Definition. Afunction f : C

--+

C is called analytic at

ifit is differen-

tiable at

and at all other points in some neighborhood

zoo

A point at which

f is analytic is called a regularpoint

A point at which f is not analytic is

called a singularpointor a singularity

A function for which all points in

are regular is called an entire function.

9.2.7.

Example.

DERNATIVES OF SOME FUNCTIONS

(a)

fez)

= z.

Hereu =x andv = y; theC-Rconditions areeasilyshownto hold,andforanyz,wehave

df/dz =

ou/ox

+iov/ox = 1.

Therefore,

thederivativeexists at all pointsof thecornpiex

234 9.

COMPLEX

CALCULUS

plane. (b)

f(z)

Here

u = x

andv = 2xy; the C-R

conditions

hold,

and

for all

points

z of the

complex plane, we have

dfldz

aulax

iavlax

= 2x +

i2y

= Zz. Therefore,

f(z)

differentiable at all points. (c)

f(z)

= zn for n 2:

Wecanuse

mathematical

induction

andthefact

that

the

product

oftwo

entire

functions

isan

entire function to show that

..</..(zn)

nzn-I.

(d)

f(z)

= ao +aJZ+...

+an_JZ

anZ

where

ai are

arbitrary

constants.

That

fez)

entire

follows

directly

from

part(e) andthe

factthatthesum

of two

entire

functions

entire.

(e) fez) =

liz.

The

derivative

canbe

found to be

f'(z)

-Ilz

which does not exist for z =

Thus, z = 0 is a singularity

f(z).

However, any other point is a regular point of

(f)

f(z)

[z[2.

Using the

definition

of the

derivative,

we obtain

!;.f

[z+

!;.z[2

_[z[2

!;.z = !;,z

(

.z:....:,+..:!;.=z)cc(z::.*--,:.+..:!;.=z::.*-')_---..::zz"--*

* A * !;.z*

+uz

+z--.

!;,z !;.z

For z = 0, !;.fl!;.z = !;.z*, which goes to zero as Az -->

Therefore,

dfldz

= 0 at

= 0.

However,

0, the limit of !;.fI!;.z will depend on how z is approached. Thus,

df/dz

does not exist

This shows that

[z[2

is differentiable only at z = 0 and

nowhere else in its neighborhood.

also shows that even

the real (here, u =

;)

and

imaginary

(here,

v = 0)

parts

of acomplexfunction have

continuous

partial

derivatives

of alI orders at a point, the functiun may not he differentiable there. (g)

f(z)

sin z:

This gives

dfldz

coszl

sin

z. Thus, f has infinitely many (isolated) singularpoints

at z = ±nrr forn = 0, 1,

2,....

III

9.2.8. Example.

THE

COMPLEX EXPONENTIAL FUNCTION

In this example, we lind the (unique) function f :

-->

that has the following three

properties:

(a) f is single-valued and analytic for all z,

(b)

dfldz

f(z),

and

(c)

f(ZI

+Z2) =

f(ZI)f(Z2)·

Property(b) shows that

f (z) is well behaved, then

dds

also well behaved. In particular,

f (z) is defined for all values

z, then f must he entire.

For

= 0 = Z2,property (c) yields flO) =

[/(0)]2

f(O) = I, or f(O) =

the other hand,

= lim

f(z

+!;.z) -

f(z)

= lim

f(z)f(!;.z)

f(z)

lim

f(!;.z)

- I

..6.z~O

.6.z

..6.z--+0

..6.z-+0

ilz

Properly (b) now implies that

lim f(!;.z) - I = I

1'(0)

= I

..6.z--+0

..6.z

and f(O) =

Although

the

derivative

Izl

existsatz= 0, itis not

analytic

there(or

anywhere

else). Tobe

analytic

atapoint,a

function

musthave

derivatives

atall points in some

neighborhood

of thegivenpoint.

9.2

ANALYTIC

FUNCTIONS

235

The first implicationfollows from the definition

derivative, and the second from the fact

thattheonlyotherchoice,namely

1(0)

= 0, wouldyield

-00

for the limit.

Now,we write

I(z)

=u(x, y) +

iv(x,

y},for whichproperty(b) becomes

au av au av

-+i-=u+iv;;;}

-=u,

-=v.

ax ax ax ax

Theseequatioushavethe mostgeneralsolutionu(x, y) =

a(y)e'

and v(x, y) = b(y)eX,

where

a(y)

and bey) are the "constants" of integration. The Cauchy-Riemann conditions

now yield

aryl

dbldy

and

daldy

-b(y),

whose most general solutionis aryl =

Acosy+Bsiny,b(y)

Asiny-Bcosy.Ontheotherhand,/(O)

=lyieldsu(O,O) =I

and

v(O,0) =0,implyingthata(O)=I, b(O)=0or A =I, B =

Wethereforeconclude

that

I(z)

=a(y)e

ib(y)e'

=eX(cosy +i siny) =

exi

•

Both

and e

are well-defined in the entire complex plane. Hence, e

is defined and

differentiableover all C; therefore, it is entire. II

Example 9.2.7 shows that any polynomial in z is entire. Example 9.2.8 shows

that the exponential function

is also entire. Therefore, any product and/or sum

polynomials and e

will also he entire. We

can

bnildother entire functions. For

instance, e

and

are entire functions; therefore. the trigonometric functions,

defined as

eiz _

e-iz

eiz +

sinz=

and

cosz=

2 (9.4)

are also entire functions. Problem 9.5 shows that sin

z and cos z have only real

zeros.

The

hyperholic functions

can

be defined similarly:

sinhz = and cosh z = (9.5)

2 2

Although the sum and the product

entire functions are entire, the ratio, in

general, is not.

For

instance,

f(z)

and

g(z)

are polynontials

degrees m and

n, respectively, then for n > 0, the ratio f (z)I

g(z)

is not entire, because at the

zeros

g(z)-which

always existand we assume thatit is not a zero

f(z)-the

derivative is not defined.

The

functions u(x, y) and v(x, y)'

an analytic function have an interesting

property that the following example investigates.

9.2.9.

Example.

Thefamilyof curvesu(x, y) = constant is perpendicularto thefamily

curves vex, y) = constant at each point of the complex plane where fez) = u +iv is

analytic.

Thiscaneasilybe seenby lookingat thenormalto thecurves.Thenormalto thecurve

u(x, y) = constant is simplyVu =

(aulax,

aulay). Similarly, the normal to the curve

v(x, y) = constant is Vv =

(avlax,

avlay).

Takingthedotproductofthesetwononna1s,

we obtain

au av au av au

(au)

(Vu) . (Vv) = ax ax + ay ay = ax - ay + ay ax =0

by the

C-R

conditions.

III

236 9.

COMPLEX

CALCULUS

9.3 Conformal Maps

The real and imaginary parts

an analytic function separately satisfy the two-

dimensional Laplace's equation:

u a

v a

-+--0

-+-=0

(9.6)

This can easily be verified from the C-R conditions. Laplace's equation in three

dimensions,

<1>

-+-+--0

- ,

describes the electrostatic potential

in a charge-free region of space. In a typ-

ical electrostatic problem the potential

is given at certain bonndaries (usually

conducting surfaces), and its value at every point in space is sought. There are

numerous techniques for solving such problems, and some of them will be dis-

cussed later in the book. However, some of these problems have a certain degree

symmetry that reduces them to two-dimensional problems. In such cases, the

theory

analytic functions can be extremely helpful.

The symmetrymentionedabove is cylindricalsymmetry, where the potentialis

known a priori to be independent of the z-coordinate (the axis

symmetry). This

situation occurs when conductors are cylinders

and-if

there are charge distribu-

tions in certain regions of

space-the

densities are z-independent. In such cases,

a<l>jaz

= 0, and the problem reduces to a two-dimensional one.

harmonic

functions

Functions satisfyingLaplace'sequation are called

harmonic

functions. Thus,

the electrostatic potential is a three-dimensioual harmonic function, and the po-

tential for a cylindrically symmetric charge distribution and boundary conditionis

a two-dimensional harmonic function. Since the real and the imaginary parts of a

complex analytic function are also harmonic, techniques

of complex analysis are

sometimes useful in solving electrostatic problems with cylindrical symmetry.'

To illustrate the connection between electrostatics and complex analysis, con-

sider a long straight filament with a constantlinearcharge density

is shown in

introdnctoryelectromagnetismthat the potential

(disregardingthe arbitrary con-

stant that determines the reference potential) is given, in cylindrical coordinates,

2Alnp

= 2AIn[(X

y2)1/2]

= 2AIn [z]. Since

satisfies Laplace's

equation, we conclude that

could be the real part of an analytic function w(z),

complex

potential

which we call the complex potential. Example 9.2.9, plus the fact that the curves

u =

constant

are

circles, imply

that

the

constant-v

curves

are

rays,

i.e.,

()(

rp.

Choosing the constant of proportionality as 2A,we obtain

w(z) =

2Alnp

+ i2Arp =

2AIn(pei~).

= 2AlnZ

5Weuse

electrostatics

because

itis more

familiar

tophysics

students.

Engineering

students

are

familiar

withsteady

state

heat

transfer

aswell,whichalso

involves

Laplace's

equation,

and

therefore

amenable

tothis

technique.

9.3

CONFORMAL

MAPS

237

It is useful to know the complex potential of more than one filament of charge.

Tofind such a potentialwe must first find

w(z)

for

aline

charge when it is displaced

from the origin.

the line is located at zo = xo +iyo, then it is easy to show that

w(z)

= 2Aln(Z - zo).

there

aren

line charges located at

zjv

...

, Zn, then

w(z) = 2

I>k

In(z -

Zk)·

k=l

(9.7)

The function w(z) can be used directly to solve a number

electrostaticproblems

involving simple charge distributions and conductor arrangements. Some of these

are illustrated in problems at the

end

this chapter. Iostead

treating

w(z)

as a

complex potential, let us look at it as a map from the z-plane (or

xy-plane) to the

w-plane (or uv-plane).

particular, the equipotential curves (circles) are mapped

onto

lines parallel to the v-axis in the w-plane. This is so because equipotential

curves are defined by

u = constant. Similarly, the constant-v curves are mapped

onto horizontal lines in the w-plane.

This is an enormous simplification

the geometry. Straight lines, especially

when they are parallelto axes, are by far simplergeometricalobjects than circles,"

especially if the circles are not centered at the origin. So let us consider two

complex "worlds." One is represented by the

xy-plane and denoted by z. The

other, the "prime world," is represented? by z', and its real and imaginary parts

x' and y'. We start

z, where we need to find a physical quantity such as the

electrostatic potential

(x,

y).

the problem is too complicated in the z-world,

we transfer it to the z'-world, in which it may be easily solvable; we solve the

problem there (in terms

and

y')

and then transfer back to the z-world (x and

y). The mapping that relates z and z'must be cleverly chosen. Otherwise, there is

no guarantee that the problem will simplify.

Two conditions are necessary for the above strategy to work. First, the dif-

ferential equation describing the physics must

not

get more complicated with the

transferto

z'.Since Laplace's equationis already of the simplesttype, the z'-world

must also respect Laplace's equation. Second, and more importantly, the map-

ping mustpreserve the angles between curves. This is necessary because we want

the equipotential curves and the field lines to be perpendicular in both worlds. A

mapping that preserves the angle between two curves at a given point is called

conformal

mapping

conformal

mapping. We already have such mappings at our disposal, as the

following proposition shows.

9.3.1. Proposition.

Let

and

be curvesin the complexz-plane that intersectat

apoint

at an angle

a.Let

f :

--+

Cbe a mappinggiven by

f(z)

=z' =

x'+iy'

that is analytic at

zoo

Let

and

be the images

and

underthis mapping,

which intersect at an angle a', Then,

6This statement is valid only in Cartesian coordinates. But these are precisely the coordinates we are using in this discussion.

7We are using Zl instead of w, and

(x',

y') instead

Cu,v).

for i =

1,2.

238 9.

COMPLEX

CALCULUS

(a) a' = a, that is, the mapping f is conformal, if (dz'ldzlz

(b)

Iff

is harmonic in (x, y), it is also harmonic in (x', y').

Proof

We sketch the

proof

the first part.

The

details, as well as the

proof

the

second part, involve partial differentiation and the chain rule and are left for the

reader.

The

angle between the two curves is obtained by taking the inner product

the two unit vectors tangent to the curves at

zoo

A small displacement along Yi

canbe

written

asexAxi +

eyD..Yi

fori = 1,2,

and

theunit

vectors

.6.

xi +

eylJ.Yi

J(!;'Xi)2 +(!;.Yi)2

Therefore,

, , !;.XI

!;.X2

+!;.YI

!;.Y2

el . ez = . .

J (!;.XIP +

(!;'YI)2J

(!;.X2)2

+(!;.Y2P

Sintilarly, in the prime plane, we have

!;.x;

!;.x~

+!;.Y;

!;.y~

,,===='==~r=~==~===

J(!;.x;)2

(!;.y;pJ(!;.x~)2

(!;.y~)2'

(9.8) .

translation

dilation

inversion

where

u(x,

andy'

= vex, y)

andu

and v are the real

andimaginaryPj

the analytic function

Using the relations

, au au , av av

!;.x,

-!;.x,

-a

!;.Yi,

!;.Yi

-a

!;.Xi

-!;.Yi,

i =

1,2,

Y x ay

and the Cauchy-Riemannconditions, the readermay verify that

1';

.e~

1'1

·1'2.

The

following are some examples

conformal mappings.

(a)

z' = z +a,where ais an arbitrary complex constant. This is simply a trans-

lation

the z-plane,

(b) z' = bz; where b is an arbitrary complex constant. This is a dilation whereby

distances are dilated by a factor

Ibl.

A graph in the z-plane is mapped onto a sim-

ilar

(congruent) graph in the

z'·plane

that will be reduced

(Ibl

< 1) or enlarged

(!bl

> 1) by a factor

Ibl.

a mapping, circles are mapped onto circles or straight lines.

(d) Combining the preceding three transformations yields the general mapping

az+b

Z=--,

cz+d

which is conformal

cz +d

dz'ldz.

The

latter conditions are eqnivalent

toad

- be

9.3

CONFORMAL

MAPS

239

9.3.2.

Example,

A circle of radius r whose center is at a in the a-plane is described by

the equation lz - aI = r.

When

transforming to the z'-plane

under

inversion, this equation

becomes II/z' - a] = r, or 11-

all

rill.

Squaringboth sidesand simplifying yields

lal

2)lz'1

Re(al)

- I =

termsof Cartesiancoordinates, thisbecomes

(9.9)

where a

a- +ia;. We now consider two cases:

1. r

14Divideby r

-lal

and comptetethe squaresto get

(

' ar

ai )2 a;+af 1

x + + y - - -

-0

lal

- lal

(r2 _

la12)2

- lal

- ,

or definiug a; sa

-ar/(r

-laI

a; ea

ai!(r

-laI

andr'

r/lr

-lal

2l,

have (x' - a;)2 +

(y'

- aj)2 = r

ll,

which can also be written as

a'l

= r',

I ,

a = a

la;

= 2

lal

-r

homographic

transformations

This is a circle in the z'-plane with center at a' and radius of r',

2. r = a:

Then

Equation (9.9) reduces to

a-x'

- aj y' = l.which is the equation

liue.

we use the transformation z' = 1/(z - c) instead

z' = lIz, then lz - at = r

becomes 11/z' - (a - c)I = r, and all the above analysis willgo throughexactly as before,

exceptthata is replacedby a - c.

III

Mappiugs

the form given in Equation (9.8) are called

homographic

trans-

formations. A nseful property

such transformations is that they can map an

iufinite region

the z-plane onto a finite region

the z'-plane. In fact, points

with very large values

z are mappedonto a neighborhood

the pointz'= a/c.

course, this argmnent goes both ways: Eqnation (9.8) also maps a neighbor-

hood

-d

/ c in the z-plane onto large regions

the z'-plane.

The

nsefulness

homographic transformations is illustrated in the followiug example.

9.3.3.

Example.

Cousidertwocyliudricalconductorsofequal

rar1ius

r, beldatpotentials

Ul and U2. respectively, whose centers are D units

length apart. Choose the

x-and

the

y-axes such that the centers of the cylinders are located on the x-axis at distances

and

«a from the origin, as shown in Figure 9.3.

Let

us find the electrostatic potential produced

by sucha configuration iu thexy-plane.

We know from elementary electrostatics that the problem becomes very simple

the

two cylinders are concentric (and,

course,

different radii). Thus, we try to map the

two circles onto two concentric circles in the z'-plane such that the infinite region outside

the two circles in the z-plane gets mapped onto the finite annular region between the two

concentric circles in the zl-plane. We then (easily) find the potential in the z'-plane, and

transfer it back to the z-plane.

The most general mapping that may be able to do the

job

is that given by Equation

(9.8). However, it turns out that we do not have to be this general. In fact, the special case

240 9.

COMPLEX

CALCULUS

Figure 9.3 In thez-plane, we see twoequal

cylinders

whose

centers

are

separated.

z' =

I/(z

- c) in which c is a real constant will be snfficient. So, z =

(I/z')

+ c, and the

circles

-akl

=I"

fork

1,2

will be mapped onto the circles [z' -

a,(1

1"'(,

where (by

Example 9.3.2)

a,( = (ak -

c)/[(ak

- c)2 - r

and r,( =

r/I(ak

- c)2 _ r

Canwe

arrange

the

parameters

so thatthe circlesin thez'

-plane

are

concentric,

i.e.,

thata~

?Theansweris yes.Wesetal = a

andsolveforaz in

terms

al'

The

result

either

the

trivial

solution az =

at,

orcz = c - r

2/(al

- c).

we placetheoriginof

the

z-plane atthecenter

the

first

cylinder,

thenat = 0 andaz = D = c +r

[c. Wecan

alsofindat and

:at = a

ee a' =

-c/(c

- r

), andthe

geometry

the

problem

is as

shown

Figure

9.4.

Forsucha

geometry

the

potential

at a pointin the

annular

regionis givenby

4>'

A Inp + B = A In

Iz'

- a'i + B, where A and B are real constants determined by the

conditions

<1>'

(r;)

= "1 and

<1>'

(r~)

"2,

which yields

and

Thepotential

is thereal

part

of thecomplexfuncttonf

F(Z') =

Aln(z'

- a') +B,

whichis

analytic

exceptatz' = a', apointlying

outside

theregionof

interest.

Wecannow

go back to the z-plane by substituting z' =

I/(z

- c) to obtain

G(z)

Ain

(_1

a')

+B,

z-c

8Writing

Z =

Izle

wenotethatlnz = In[z] +ie, so

that

thereal

part

of acomplexlog

function

is thelog of the

absolute

value.

9.4

INTEGRATION

COMPLEX

FUNCTIONS

241

Figure 9.4 In

-plane,

we see two

concentric

unequal

cylinders.

whosereal

part

is the

potential

inthez-plane:

I - a'z+a'cl

(x, y) =Re[G(z)] =

Aln

+ B

z-c

Alnl(1

+a'c-a'x)

-ia'yl

(x-c)+iy

[(I

+a'c

-a'x)2

+a'2

y2]

=-In

2 2

+B.

2 (x - c) + y

Thisis thepotential we

want.

9.4 Integrationof ComplexFunctions

iii

The derivative of a complex function is an important concept and, as the previous

section demonstrated, provides a powerful tool in physical applications. The con-

cept

integration is even more important.

fact, we will see in the next section

that derivatives can be written in terms of integrals. We will study integrals

complex functions in detail in this section.

The definite integral of a complex function is defined in analogy to that of a

real function:

' 2 N

!(z)

dz = lim L !(Zi)Ll.Zi,

N....:,-oo.

..6.zr+

242

COMPLEX

CALCULUS

where I1Zi is a small segment, situated at zs,

the curve that conoects the complex

number

to the complex numbera2 in the z-plane, Since there are infinitelymany

ways

connecting

to a2, it is possible to obtaindifferentvalues for the integral

for different paths.

One

encounters

similar

situation

whenone triesto

evaluate

theline

integral

of a vector field. In fact, we can

tum

the integral

a complex function into a line

integral as follows. We substitute

f(z)

= u

+iv

anddz

= dx

+idy

in the integral

to obtain

c c c

f(z)dz

(udx

vdy)

(vdx

udy).

we define the two-dimensional vectors

sa (u,

-v)

andA2

(v, u), then we

get

f(z)

dz = r

Al .

ira,

A2 . dr. It follows from Stokes' theorem

JUI

jUt

(or Green's theorem, since the vectors lie in a plane) that the integral of f is path-

independent only if both A

I and A2 have vanishing curls. This in

tum

follows if

and only

u and v satisfy the CoRconditions, and this is exactly what is needed

for

f (z) to be analytic.

Path-independenceof a line integral

avectorA is equivalentto the vanishing

of the integral along a closed path, and the latter is equivalent to the vanishing of

V x A

= 0 at every point of the regionborderedby the closedpath. The preceding

discussion is encapsnlated in an important theorem, which we shall state shortly.

First, however, it is worthwhile to become familiar with some terminology used

frequently in complex analysis.

curve

defined

1. A

curve

is a map Y : [a, b]

from the real interval into the complex

planegiven by

y(t)

= Yr(t)

+iYi(t),

where a

:s:

b, and Yrand Yi are the

real and imaginary parts

Y; Y(a) is called the

initial

point

the curve

and

y(b)

its final point.

2. A simple arc, or a

Jordan

arc,is a curve

that

doesnotcrossitself,i.e., y is

injective (or one to one), so that

y(tl)

"" y(t2) when tl ""

ti.

3. A path is a finite collection

In,

... ,

Yn}

of simple arcs such that the

initial point of

JIk+1

coincides with the final point

JIk.

4. A

smooth

arc is a curve for which

ldt

dYr/dt

idYi/dt

exists and

is nonzero for

t E [a, b].

contour

defined

5. A

contour

is a path whose arcs are smooth.

When

the initial point

coincides with the final point

Yn,the contour is said to be a simpleclosed

contour.

Cauchy-Goursat

9.4,1.

Theorem.

(Cauchy-Goursat theorem) Let f : C

be analytic on a

theorem

simple closed contour C and at all points inside C. Then

f(z)dz