Functions of a Complex Variable

By

Dr. Pragya Mishra

Maharana Pratap Govt. P. G.

College,Hardoi

1

Complex number:

Complex algebra:

Complex conjugation:

2

.)1 real, are and (both iyxiyx z

Cauchy-Riemann conditions

Complex algebra

2112212121221121

2121221121

vectors.)2d to(Anologous

zzicycxiyxcczyxyxiyyxxiyxiyxzz

yyixxiyxiyxzz

iyxiyx z ** )(22* ))(( yxiyxiyx zz

Modulus (magnitude): 212122* zzzz yxrzzz

Polar representation: ireiriyx z )sin(cos

Argument (phase):

)arg()arg()arg(

)quadrants. 3rdor 2nd in the is if ( arctan)arg(

2121 zzzz

zx

yz

3

Functions of a complex variable:

All elementary functions of real variables may be extended into the complex plane.

0

2

0

2

!!2!11

!!2!11 :Example

n

nz

n

nx

n

zzze

n

xxxe

A complex function can be resolved into its real part and imaginary part:

2222

2222

11

2)()( :Examples

),(),()(

yx

yi

yx

x

iyxz

xyiyxiyxz

yxivyxuzf

Multi-valued functions and branch cuts:

ivunirrereznii )2(ln]ln[)ln(ln :1 Example )2(

To remove the ambiguity, we can limit all phases to (-,). = - is the branch cut. lnz with n = 0 is the principle value.

2/)2(2121)2(2121 )( :2 Example niniierrerez

We can let z move on 2 Riemann sheets so that is single valued everywhere. 21)()( irezf

4

Analytic functions: If f (z) is differentiable at z = z0 and within the neighborhood of z=z0, f (z) is said to be analytic at z = z0. A function that is analytic in the whole complex plane is called an entire function.

Cauchy-Riemann conditions for differentiability

Cauchy-Riemann conditions

z

zf

z

zfzzf

dz

dfzf

zz

)(lim

)()(lim)('

00

In order to let f be differentiable, f '(z) must be the same in any direction of z.

Particularly , it is necessary that

x

v

y

u

y

v

x

u

y

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ui

yi

viuzfyiz

x

vi

x

u

x

viuzfxz

y

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,

have we themEquating

.lim)(' ,For

.lim)(' ,For

0

0

Cauchy-Riemann conditions x

v

y

u

y

v

x

u

,

5

Conversely, if the Cauchy-Riemann conditions are satisfied, f (z) is differentiable:

.

1 and ,lim

limlim)(

lim

0

000

y

vi

y

u

ix

vi

x

u

yix

yixx

vi

x

u

yix

yx

ui

x

vx

x

vi

x

u

yix

yy

vi

y

ux

x

vi

x

u

z

zf

dz

df

z

zzz

More about Cauchy-Riemann conditions:

1) It is a very strong restraint to functions of a complex variable. 2) 3)

4)

2100 cvcuvuvuy

v

y

u

x

v

x

u

.)()( iy

vi

iy

u

y

ui

y

v

x

vi

x

u

dz

df

analytic.not but continuous everywhere is ,e.g.

0002

1

2

1

:on dependsonly )( that so ,0 toEquivalent

***

*

iyxf

y

vi

y

ui

x

vi

x

u

y

fi

x

f

iy

f

x

f

z

y

y

f

z

x

x

f

z

f

zz,zfz

f *

6

Reading: General search for Cauchy-Riemann conditions: Our Cauchy-Riemann conditions were derived by requiring f '(z) be the same when z changes along x or y directions. How about other directions? Here I do a general search for the conditions of differentiability.

x

v

y

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dp

d

dp

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pzf

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dy

dx

dyi

dx

dy

y

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vi

dx

dy

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idydx

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dz

dfzf

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7

Cauchy’s theorem

Cauchy’s integral theorem

Contour integral: Cauchy’s integral theorem: If f (z) is analytic in a simply connected region R, [and f ′(z) is continuous throughout this region, ] then for any closed path C in R, the contour

integral of f (z) around C is zero:

CCC

z

zudyvdxivdyudxidydxivudzzf )()())(()(

2

1

C

dzzf 0)(

Proof using Stokes’ theorem: σVλV ddSC

dxdyy

V

x

VdyVdxV

S

xy

Cyx

0

)()()(

dxdy

y

v

x

uidxdy

y

u

x

v

udyvdxivdyudxdzzf

SS

CCC

8

Contour deformation theorem: A contour of a complex integral can be arbitrarily deformed through an analytic region without changing the integral.

1) It applies to both open and closed contours. 2) One can even split closed contours. Proof: Deform the contour bit by bit.

Examples: 1) Cauchy’s integral theorem. (Let the contour shrink to a point.) 2) Cauchy’s integral formula. (Let the contour shrink to a small circle.)

Cauchy-Goursat proof: The continuity of f '(z) is not necessary.

Corollary: An open contour integral for an analytic function is independent of the path, if there is no singular points between the paths.

1

2

2

1

)()()()( 12

z

z

z

zdzzfzFzFdzzf

z1

z2

z1

z2 C2

C1

C1

C2

C1

C2 C3

“nails”

“rubber bands”

9

Cauchy’s integral formula

Cauchy’s integral formula: If f (z) is analytic within and on a closed contour C, then for any point z0 within C,

)(2

)0(Let )()()(

0)()()()(

:Proof

)(

2

1)(

0

0

2

0

00

0000

0

0

0

201

zif

rdriere

rezfdz

zz

zfdz

zz

zf

dzzz

zfdz

zz

zfdz

zz

zfdz

zz

zf

dzzz

zf

izf

i

i

i

CC

LCLC

C

C C0

L1

L2

z0

Can directly use the contour deformation theorem.

10

Derivatives of f (z):

C n

ndz

zz

zf

i

nzf

1

0

0)( )(

2

!)(

Corollary: If a function is analytic, then its derivatives of all orders exist. Corollary: If a function is analytic, then it can be expanded in Taylor series.

Cauchy’s inequality: If is analytic and bounded,

then

n

nzazf )( ,)(||

Mzfrz

bounded.) is is,(That . n

n

n aMra

Mrar

Mdz

z

zfadz

z

zf

i

nanf

n

nnrz nnrz nn

n || 1|| 1

)( )(

2

1)(

2

!!)0( :Proof

Liouville’s theorem: If a function is analytic and bounded in the entire complex plane, then this function is a constant.

.)( 0.for 0 then ,let , :Proof 0azfnarr

Ma nnn

Fundamental theorem of algebra: has n roots. Suppose P(z) has no roots, then 1/P(z) is analytic and bounded as Then P(z) is constant. That is nonsense. Therefore P(z) has at least one root we can divide out.

)0,0( )(0

n

n

i

i

i anzazP

.z

11

Morera’s theorem: If f (z) is continuous and for every closed contour within a simply connected region, then f (z) is analytic in this region.

C

dzzf 0)(

analytic is )()('

analytic is )(

)()(')()()(0)(

:Proof

12

2

1

zfzF

zF

zfzFzFzFdzzfdzzfz

zC

)()(),0(),0(

),0()0,(),(

then),,()(Let

?)()()(Why

2121

2121

21

12

2

1

2

1

zFzFzGzG

zGzGzzG

zzGdzzf

zFzFdzzf

z

z

z

z

12

Analytic continuation

Laurent expansion

Taylor expansion for functions of a complex variable: Expanding an analytic function f (z) about z = z0, where z1 is the nearest singular point.

00

0)(

01

0

00

10

0

0

0 0

0

0

00

00

!

)(

')'(

)'(

2

1'

)'(

)'(

2

1

')'(

)'('

2

1'

'1)'(

)'(

2

1

')()'(

)'(

2

1'

'

)'(

2

1)(

n

nn

nC n

n

Cn

n

n

C

n

n

C

CC

zzn

zf

dzzz

zfzz

idz

zz

zfzz

i

dzzz

zfzz

zz

idz

zz

zzzz

zf

i

dzzzzz

zf

idz

zz

zf

izf

z

13

Schwarz’s reflection principle: If f (z) is 1) analytic over a region including the real axis, and 2) real when z is real, then

).()( **zfzf

)()(

!

)()( :Proof

**

00

0)(

zfzf

xzn

xfzf

n

nn

Examples: most of the elementary functions.

Analytic continuation: Suppose f (z) is analytic around z = z0, we can expand it about z = z0 in a Taylor series: This series converges inside a circle with a radius of convergence , where a0 is the nearest singularity from z = z0. We can also expand f (z) about another point z = z1 within the circle R0: . In general, the new circle has a radius of convergence and contains points not within the first circle.

14

0

00

)(

!

)()(

m

mm

zzm

zfzf

000 zR a

0

11

)(

!

)()(

n

nn

zzn

zfzf

z0

a0

z1

R0

R1 a1

nmn

nnmm

nm

nmm

n

zznmn

zzzfzf

zznm

zfzf

,01

010)(

010

)(

1)(

)!(!

)()( expansion, second theinto Plug

)!(

)()( expansion,first theFrom

111 zR a

Consequences: 1) f (z) can be analytically continued over the complex plane, excluding singularities. 2) If f (z) is analytic, its values at one region determines its values everywhere.

15

Laurent expansion

Laurent expansion

Problem: Expanding a function f (z) that is analytic in an annular region (between r and R).

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n

nC n

n

nC n

n

mC

m

mn

C n

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1

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~

)'(

')'()(

2

1

)'(

')'()(

2

1

)'(

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2

1

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1

2

1

)'(

')'()(

2

1

')'()'()(

1

2

1

)'(

')'()(

2

1

'1)(

')'(

2

1

'1)'(

')'(

2

1

)()'(

')'(

2

1

)()'(

')'(

2

1

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')'(

2

1

'

')'(

2

1

'

')'(

2

1)(

21

21

21

21

21

21

2211

L1

L2

C is any contour that encloses z0 and lies between r and R (deformation theorem).

16

C nn

n

n

nzz

dzzf

iazzazf

10

0)'(

')'(

2

1 ,)()(

Laurent expansion:

1) Singular points of the integrand. For n < 0, the singular points are determined by f (z). For n ≥0, the singular points are determined by both f (z) and 1/(z'-z0)

n+1. 2) If f (z) is analytic inside C, then the Laurent series reduces to a Taylor series:

3) Although an has a general contour integral form, In most times we need to use straight forward complex algebra to find an.

.0 ,0

,0 ,!

)( 0)(

n

nn

zf

a

n

n

17

Laurent expansion: Examples

Example 1: Expand about z0=1. 2

3

1)(

z

zzf

)1(3

1

3

1

1

1

1)1(3)1(3)1(

1

]1)1[(

122

23

2

3

2

3

zzzz

zzz

z

z

z

z

Example 2: Expand about z0=i.

)(84

11

2

)(2

1

2

11

2

1

21

1

2

11

2

1

2

11

2

111

2

1

1

1)(

02

2

izi

iz

i

iziiizi

i

iziizi

iziiziizizizzf

n

n

n

1

1)(

2

zzf

18

Branch points and branch cuts

Singularities

Poles: In a Laurent expansion

then z0 is said to be a pole of order n. A pole of order 1 is called a simple pole. A pole of infinite order (when expanded about z0) is called an essential singularity.

The behavior of a function f (z) at infinity is defined using the behavior of f (1/t) at t = 0.

Examples:

,0 and 0for 0 if ,)()( 0

nm

m

m

m anmazzazf

1200

12 1

)!12(

)1(1sin ,

)!12(

)1(sin )2

nn

n

n

nn

tntn

zz

.at pole single a has 22

14

11

2

1

2/)(1

1

4

11

2

1

)(2

11

2

111

2

1

))((

1

1

1)1

2

2

izi

iz

i

iz

izi

iiziziiziiziiziziizizz

sinz thus has an essential singularity at infinity.

infinity.at 2order of pole a has 1 )3 2 z

19

Branch points and branch cuts:

Branch point: A point z0 around which a function f (z) is discontinuous after going a small circuit. E.g.,

Branch cut: A curve drawn in the complex plane such that if a path is not allowed to cross this curve, a multi-valued function along the path will be single valued. Branch cuts are usually taken between pairs of branch points. E.g., for , the curve connects z=1 and z = can serve as a branch cut.

.lnfor 0 ,1for 1 00 zzz-z

1 z-

Examples of branch points and branch cuts:

)sin(cos)( .1 aiarzzfaa

If a is a rational number, then circling the branch point z = 0 q times will bring f (z) back to its original value. This branch point is said to be algebraic, and q is called the order of the branch point. If a is an irrational number, there will be no number of turns that can bring f (z) back to its original value. The branch point is said to be logarithmic.

,/ qpa

20

)1)(1()( .2 zzzf

We can choose a branch cut from z = -1 to z = 1 (or any curve connecting these two points). The function will be single-valued, because both points will be circled.

Alternatively, we can choose a branch cut which connects each branch point to infinity. The function will be single-valued, because neither points will be circled.

It is notable that these two choices result in different functions. E.g., if , then iif 2)(

choice. second for the

2)( and choicefirst for the 2)( iifiif

A B

A B

21

Mapping

Mapping

Mapping: A complex function can be thought of as describing a mapping from the complex z-plane into the complex w-plane. In general, a point in the z-plane is mapped into a point in the w-plane. A curve in the z-plane is mapped into a curve in the w-plane. An area in the z-plane is mapped into an area in the w-plane.

),(),()( yxivyxuzw

z0

z1

z

y

x

w0

w1

w

v

u

Examples of mapping:

Translation:

0 zzw

Rotation:

0

0

0

0

0

or ,

rrerree

zzw

iii

22

Inversion:

r

ree

zw

i

i

11

or ,1

In Cartesian coordinates:

.,11

22

22

22

22

vu

vy

vu

ux

yx

yv

yx

xu

iyxivu

zw

A straight line is mapped into a circle:

.0)( 22

2222

vauvub

bvu

au

vu

vbax y

23

Conformal mapping

Conformal mapping: The function w(z) is said to be conformal at z0 if it preserves the angle between any two curves through z0.

If w(z) is analytic and w'(z0)0, then w(z) is conformal at z0.

Proof: Since w(z) is analytic and w'(z0)0, we can expand w(z) around z = z0 in a Taylor series:

.)(

).)(('or ),('lim

))((''2

1))((')(

121200

0000

0

0

0

200000

0

aa

zzAeww

zzzwwwzwzz

ww

zzzwzzzwzww

i

zz

1) At any point where w(z) is conformal, the mapping consists of a rotation and a dilation.

2) The local amount of rotation and dilation varies from point to point. Therefore a straight line is usually mapped into a curve.

3) A curvilinear orthogonal coordinate system is mapped to another curvilinear orthogonal coordinate system .

24

What happens if w'(z0) = 0?

Suppose w (n)(z0) is the first non-vanishing derivative at z0.

n

n

Br

reBen

ezzn

zwww

n

niiinn

!)(!

1)(

!

)(0

0)(

0

This means that at z = z0 the angle between any two curves is magnified by a factor n and then rotated by .