physics department yarmouk university - yuctaps.yu.edu.jo/physics/.../pdf/1_phys601_chapter1.pdf ·...

© Dr. Nidal M. Ershaidat

Phys. 601: Mathematical Physics

Physics Department

Yarmouk University

Chapter 1: Functions of a Complex

Variable

© Dr. Nidal M. Ershaidat - Mathematical Physics - Phys. 601 - Chapter 1: Complex Functions

2

Overview

1.Complex Algebra2.Calculus in the Complex space3.Cauchy-Riemann Conditions4.Anayltic Functions (Elements of Riemann’s Theory)

5.Cauchy’s Integral Theorem6.Taylor Expansion7.Laurent Expansion8.Singularities9.Mapping


3

� Some operators in Quantum mechanics are complex entities.

Complex Analysis

Functions of a complex variable provide us some powerful and widely useful tools in theoretical physics.

The study of functions of a complex variable is known as complex analysiscomplex analysis and has enormous practical use in applied mathematics as well as in physics.

Often, the most natural proofs for statements in real analysis or even number theory employ techniques from complex analysis

� Some important physical quantities are complex variables (the wave-function ψ)


4

Many physical quantities that were originally real become complex as a simple physical theory is made more general.

Generalizing Physics

Another example: the real index of refraction of light becomes a complex quantity when absorption is included.

The most famous example is real energy associated with an energy level which becomes complex when the finite lifetime of the level is considered.

The energy has the form En = En + i Γ Γ Γ Γ (where ��/ΓΓΓΓ is the finite lifetime of this state).

Importance in Importance in

PhysicsPhysics


6

Mapping

1. For many pairs of functions u and v, both uand v satisfy Laplace’s equation

Either u or v may be used to describe a two-dimensional electrostatic potential.

2222 =,0=),(=)( yxuru ++++ρρρρθθθθρρρρ∇∇∇∇∇∇∇∇�

The other function, which gives a family of

curves orthogonal to those of the first

function, may then be used to describe the

electric field .E�

12222 =,0=),(=)( yxvrv ++++ρρρρθθθθρρρρ∇∇∇∇∇∇∇∇

�


7

Mapping2

In many cases in which the functions u and vare unknown, mappingmapping permits us to create a coordinate system tailored to the particular problem.

Transforming such a real problem in the complex plane, we also say domain, is

called mappingmapping.

A similar situation holds for the hydrodynamics of an ideal fluid in irrotational motion. The

function u might describe the velocity

potential, whereas the function v would then be

the stream function.


8

Analytic Continuation

2. Second-order differential equations of interest in physics may be solved by power series,

The same power series may be used in the

complex plane to replace x by the complex

variable z.

The dependence of the solution f(z) at a given z0

on the behavior of f(z) elsewhere gives us

greater insight into the behavior of our solution and a powerful tool (analytic continuation) for extending the region in which the solution is valid.


9

Real to Imaginary Change

The same change transforms the Helmholtz equation solutions (Bessel and spherical Bessel functions) into the diffusion equation solutions (modified Bessel and modified spherical Bessel functions).

3. The change of a parameter k from real to

imaginary, k→→→→ik, transforms the Helmholtz equation into the diffusion equation.


10

Helmholtz – Diffusion

It is the "space part" of the wave equation-type:

The Helmholtz equation appears in many fields in physics: electromagnetism, seismology, …

(((( )))) 0,1

2

2

2

2 ====

∂∂∂∂

∂∂∂∂−−−−∇∇∇∇ tru

tc

�2

Such a 2nd order "partial" differential equation is solved by separation of variables which gives, for the space part, Helmholtz equation:

022 ====ψψψψ++++ψψψψ∇∇∇∇ k 3


11

Diffusion Equation

The diffusion equation is a partial differential equation which describes density dynamics in a material undergoing diffusion.

(((( )))) 0,1

2

2

2

2 ====

∂∂∂∂

∂∂∂∂−−−−∇∇∇∇ tru

tc

�

The diffusion equation can be derived in a straightforward way from the continuity equation, which states that a change in density in any part of the system is due to inflow and outflow of material into and out of that part of the system. Effectively, no material is created or destroyed:


12

1D Diffusion EquationFor 1D systems the space part of the diffusion equation, also obtained by separation of variables, has the simple form:

which is a Helmholtz equation type if k is

replaced by ik.

022 ====ψψψψ−−−−ψψψψ∇∇∇∇ k


13

Integrals in the Complex Plane

o Evaluating definite integrals;

o Inverting power series;

o Forming infinite products;

o Obtaining solutions of differential equations

for large values of the variable (asymptotic

solutions);

o Investigating the stability of potentially

oscillatory systems;

o Inverting integral transforms.

4. These integrals have a wide variety of useful applications:

Complex AlgebraComplex Algebra

Analytic Properties, Analytic Properties,

MappingMapping


15

Complex Algebra Review

Please review:

1. Arfken: Paragraph 6.1

2. Complex numbers in Introduction to Mathematical Physics, 2nd Edition 2004, Nabil M. Laham and Nabil Y. Ayoub

3. See also:http://ctaps.yu.edu.jo/Physics/Phys201

4. Exercises 6.1.1-6.1.26

Function of a Function of a

complex variablecomplex variable


17

Definitions� A function of a complex variable f(z) has a

real part and an imaginary part as complex

numbers, i.e.:

where x and y are real variables and u(x,y) and

v(x,y) are real-valued functions.

4

The basic concepts of complex analysis are often introduced by extending the elementary real functions (e.g., exponentials, logarithms, and trigonometric functions) into the complex domain.

f(z)= u(x,y) + i v(x,y)


DefinitionA complex function is a one in which the independent variable and the dependent variable are both complex numbers. More precisely, a complex function is a function whose domain and range are subsets of the complex plane.

For any complex function, both the independent variable and the dependent variable may be separated into real and imaginary parts:

yixz +=independent variable

( ) ( ) ( )zvizuzf +=

dependent variable


19

Domain and Range

� As for real functions, a domaindomain of a

function of a complex variable should be

defined. In general we say that it is a subset

A of C → → → → C

We also define the rangerange of a function of a

complex variable as being the set of all

values that the function takes in the

complex plane for all values of z defined in

the domain.A complex function is a function whose

domain and range are subsets of the

complex plane. © Dr. Nidal M. Ershaidat - Mathematical Physics - Phys. 601 - Chapter 1: Complex Functions

20

Domain and Range - Real

The quadratic parent function is y = -x2 + 5

Function of a Complex Function of a Complex

VariableVariable


22

u(x,y) and v(x,y) ∈∈∈∈ R

x and y ∈∈∈∈ R

Where

f(z)= u(x,y) + i v(x,y)

Functions of a complex number z are

defined in the same manner as the

functions of real variables. In a rectangular

form a function f(z) is written in the form:

Complex Functions Space

Re[f(z)]= u(x,y)

Im[f(z)]= v(x,y)

Trigonometry & the Trigonometry & the

Complex SpaceComplex Space


24

We define:

The Exponential

where

(((( )))) (((( ))))yi+ye=e=e xyi+xz sincos

Re[f(z)]= u(x,y) = ye x cos

yex sin

When z is a purely complex number, i.e. z = i y

then:

yi+y=e=e yiz sincos

Im[f(z)]= v(x,y) =


25

Let’s write e+ix and its complex conjugate e-ix .

Complex Representation of cos x & sin x

xi+x=e xi sincos

It’s easy to see that:

2cos

xixi e+e=x

−−−−

Since i2 = i . i = -1 then i

=i1

−−−−

xix=e xi sincos −−−−−−−−

i

ee=x

xixi

2sin

−−−−−−−−


26

The previous definition of the cosine and sine

functions of a real variable is used to define

the cosine and sine functions, and all the

derived trigonometric functions, of a complex

variable z.

cos z & sin z

,2

sini

ee=z

zizi −−−−−−−−

++++

−−−−−−−−

++++

−−−−−−−−

−−−−

−−−−

−−−−

zizi

zizi

zizi

zizi

ee

eei=

ee

ee

i=

z

z=z

1

cos

sintan

2cos

zizie+e

=z−−−−

⇒⇒⇒⇒


27

The hyperbolic functions of a complex variable

z are defined as:

The Hyperbolic Functions

2cosh

zz e+e=z

−−−−

⇒⇒⇒⇒zz

zz

e+e

ee=

z

z=ztanh

−−−−

−−−−−−−−

cosh

sinh

2sinh

zz ee=z

−−−−−−−−


28

The logarithmic function of a complex variable

z is defined as w = ln z which leads to z = ew.

The Logarithmic Function

Let’s calculate the logarithm of the product of

by , i.e.22

we=z1

1w

e=z

(((( ))))212121

w+e=ee=zz

www⋅⋅⋅⋅⋅⋅⋅⋅

(((( )))) 212121 lnlnln z+z=w+w=zz ⋅⋅⋅⋅

which is the same as for the real variables.


29

Writing z in its (general) polar form

Infinite logarithms of a Complex Number

We have

(((( ))))ππππθθθθ m+i+r=z=w 2lnln

(((( ))))ππππθθθθ m+ier=z

2

Where m is an integer number (m=0, ± ± ± ±1, ±±±±2, …)

This means that a complex number has an

infinite number of logarithms. Conventionally

we take m = 0 and limit the phase to its

principal value defined by an interval of 2ππππ

(normally the interval [-ππππ,+ππππ])


30

In the real numbers space, a positive real

variable x raised to a (positive or negative) real

number y is denoted xy

Raising a Complex Number to a complex power

We have (((( )))) xy=xy

lnln

Taking the exponential of this equation, we can

write: (((( )))) (((( ))))yxxyye=e=x

lnln

This definition is taken to define the operation

of raising a complex number to a complex

power. Thus we have

(((( ))))wzzwwe=e=z

lnln

Examples and Examples and

ApplicationsApplications


32

(((( ))))tωxkieA=Y −−−−

The function (which we call the wave function)

Waves

ννννππππ====ωωωω 2Where A is the amplitude,

represents a plane wave of wavenumber k and

angular frequency ωωωω in one dimension traveling

in the positive x direction.

λλλλ

ππππ2=k and

The real amplitude (or displacement) is

y = Re[Y] = A cos (k x – ωωωω t)

λ λ λ λ is the wavelength and νννν is the frequency of

the wave.


33

The amplitude A of a wave function

Amplitude of a Wave

(((( ))))tωxkieA=Y −−−−

is, generally, obtained as follows

(((( )))) (((( ))))

2*

**

A=AA

eAeA=YY tωxkitωxki

⇒⇒⇒⇒

−−−−−−−−−−−−

In the case of a plane wave, A is real and

*YY=A

Calculus in the Calculus in the

Complex SpaceComplex Space


35

Differentiating Complex Functions1

�The derivative of a complex function f(z) of a complex variable is defined, as for a real function, by:

An important condition is that the limit be independent of the particular approach to the point z

(((( )))) (((( ))))(((( ))))

(((( ))))zfdz

df

z

f

zzz

zfzzf

zz′′′′========

δδδδ

δδδδ====

−−−−δδδδ++++

−−−−δδδδ++++

→→→→δδδδ→→→→δδδδ 00limlim

This requirement is more restrictive than for a

real function. Simply because the point z at

which we want to calculate the derivative is in a plane!

5


36

Differentiating Complex Functions2Consider a differential change in x (δδδδx) and y(δδδδy) in a complex plane where we represent the

function f(z).

In this case we have:

yixz δδδδ++++δδδδ====δδδδ

viuf δδδδ++++δδδδ====δδδδ

yix

viu

z

f

δδδδ++++δδδδ

δδδδ++++δδδδ====

δδδδ

δδδδ⇒⇒⇒⇒

and

Figure 1

37

Differentiating Complex Functions3

Taking the limit in the x “direction” (δδδδy=0) we have*:

x

vi

x

u

x

vi

x

u

x

viu

z

f

xxz ∂∂∂∂

∂∂∂∂++++

∂∂∂∂

∂∂∂∂====

δδδδ

δδδδ++++

δδδδ

δδδδ====

δδδδ


δδδδ

δδδδ

→→→→δδδδ→→→→δδδδ→→→→δδδδ 000limlimlim

y

v

y

ui

y

v

y

ui

yi

viu

z

f

yyz ∂∂∂∂

∂∂∂∂++++

∂∂∂∂

∂∂∂∂−−−−====

δδδδ

δδδδ++++

δδδδ

δδδδ−−−−====

δδδδ


δδδδ

δδδδ

→→→→δδδδ→→→→δδδδ→→→→δδδδ 000limlimlim

Taking the limit in the y “direction” (δδδδx=0) we have*:

Assuming the partial derivatives exist

6a

6bCauchyCauchy--Riemann Riemann

ConditionsConditions


39

Cauchy-Riemann Conditions

f(z) has a derivative df/dz if and only if equations 6a and 6b are equal, i.e.

y

v

y

ui

x

vi

x

u

∂∂∂∂

∂∂∂∂++++

∂∂∂∂

∂∂∂∂−−−−====

∂∂∂∂

∂∂∂∂++++

∂∂∂∂

∂∂∂∂

x

v

y

uand

y

v

x

u

∂∂∂∂

∂∂∂∂−−−−====

∂∂∂∂

∂∂∂∂

∂∂∂∂

∂∂∂∂====

∂∂∂∂

∂∂∂∂

Which means that we must have:

Cauchy-Riemann Conditions

Inversely, if df/dz exists at some point z then Cauchy-Riemann conditions hold.

Complex partsReal parts

7


40

Baron Augustin-Louis Cauchy (21 August 1789 –23 May 1857); a French mathematician who was an early pioneer of analysis.

Baron Augustin-Louis Cauchy

His writings cover the entire range of mathematics and

mathematical physics.

He started the project of formulating and proving the theorems

of infinitesimal calculus in a rigorous manner, rejecting the

heuristic principle of the generality of algebra exploited by

earlier authors. Cauchy defined continuity in terms of infinitesimals and gave

several important theorems in complex analysis and initiated

the study of permutation groups in abstract algebra.

Cauchy was a prolific writer; he wrote approximately eight

hundred research articles and five complete textbooks.


41

Other NotationSome authors use the following notations

yx vu ====

Using this simple notation, the Cauchy-Riemann conditions are simply written as:

9

y

uu

x

uu yx

∂∂∂∂

∂∂∂∂====

∂∂∂∂

∂∂∂∂==== ,

y

vv

x

vv yx

∂∂∂∂

∂∂∂∂====

∂∂∂∂

∂∂∂∂==== ,

xy vu −−−−====

8and

and


42

Other Representations

Other representations of the Cauchy–Riemann equations occasionally arise in other coordinate systems.

n

v

s

u

s

v

n

u

∂

∂−=

∂

∂

∂

∂=

∂

∂,

rr

vu

r

v

rr

u

∂

∂−=

θ∂

∂

θ∂

∂=

∂

∂ 1,

1

Combining these into one equation for ƒ givesθ∂

∂=

∂

∂ f

rir

f 1

If equations 9 and 10 hold for a continuously differentiable pair of functions u and v, then so do

for any coordinate system (n(x,y), s(x,y)) such that the

pair ( ) is orthonormal and positively oriented. sn ∇∇��

,

As a consequence, in particular, in the system of coordinates given by the polar representation z = r eiθ, the equations then take the form:


43

Other Representations

Conversely

If the partial derivatives

exist then we have:

y

uu

x

uu yx

∂∂∂∂

∂∂∂∂====

∂∂∂∂

∂∂∂∂==== ,

y

vv

x

vv yx

∂∂∂∂

∂∂∂∂====

∂∂∂∂

∂∂∂∂==== ,

(((( ))))x

vi

x

u

z

fzf

z ∂∂∂∂

∂∂∂∂++++

∂∂∂∂

∂∂∂∂====

δδδδ

δδδδ====′′′′

→→→→δδδδ 0lim 6c

and

© Dr. Nidal M. Ershaidat - Mathematical Physics - Phys. 601 - Chapter 1: Complex Functions© Dr. Nidal M. Ershaidat - Mathematical Physics - Phys. 601 - Chapter 1: Complex Functions

44

Example 1.1Find the derivative of f(z) = z3

Solution

(((( ))))

(((( )))) (((( ))))(((( ))))

(((( ))))(((( )))) 222

0

33223

0

3

0

333lim

33lim

lim

zzzzz

z

zzzzzzz

z

zzzf

dz

df

z

z

z

====∆∆∆∆++++∆∆∆∆++++====

∆∆∆∆

−−−−∆∆∆∆++++∆∆∆∆++++∆∆∆∆++++====

∆∆∆∆

−−−−∆∆∆∆++++====

→→→→∆∆∆∆

→→→→∆∆∆∆

→→→→∆∆∆∆


45

Example 1.2Check Cauchy-Riemann conditions for f(z) = z3.

Solution: (((( )))) (((( ))))

(((( ))))(((( ))))

(((( ))))(((( ))))��yxvyxu

yyxiyxx

yiyxyxix

yixzzf

,

32

,

23

3223

33

33

33

−−−−++++−−−−====

−−−−−−−−++++====

++++========

,33 22yx

x

u−−−−====

∂∂∂∂

∂∂∂∂

,6 yxy

u−−−−====

∂∂∂∂

∂∂∂∂

,33 22 yxy

v−−−−====

∂∂∂∂

∂∂∂∂

yxx

v6and ====

∂∂∂∂

∂∂∂∂

x

v

y

uand

y

v

x

u

∂∂∂∂

∂∂∂∂−−−−====

∂∂∂∂

∂∂∂∂

∂∂∂∂

∂∂∂∂====

∂∂∂∂

∂∂∂∂We do have:

CauchyCauchy--Riemann Conditions Riemann Conditions

in Polar Coordinatesin Polar Coordinates

See See Appendix 1Appendix 1--11

Analytic FunctionsAnalytic Functions(Elements of Riemann(Elements of Riemann’’s Theory)s Theory)


48

Analyticity Concept

If f(z) has a derivative df/dz at z=z0 and in some

small region around z0, we say that f(z) is

analytic (holomorphic) at z=z0.

In this course we shall only consider analytic functions.In this course we shall only consider analytic functions.

If f(z) is analytic everywhere in the (finite)

complex plane, the function is said to be entireentire.

If f’(z) does not exist at z=z0 then we say that z0

is a singularity point.


49

Examples1�f(z) = z2 is analytic while f(z) = z* is not. See

examples 6.2.1 and 6.2.1 in Arfken.

�f(z) = z3 is analytic.

( ) ( ) ( ) ( )yxyiyxxiyxzzf 23233333 +−+−=+==

2233 yx

x

u−=

∂

∂yx

y

u6−=

∂

∂

yxx

v6=

∂

∂ 2233 yx

y

v−=

∂

∂

We see that Cauchy-Riemann are verified.3. The function is not analytic at any

point (though it is differentiable at z=0).

( ) 2zzf =

50

Examples21. For n ∈∈∈∈ N and complex numbers a0, . . . , an the

polynomial is an analytic function

for all z ∈∈∈∈ C.

2. The function is analytic for all z≠≠≠≠0.( )z

zf1

=

(((( )))) ∑∑∑∑====

====n

k

kk zazf

0

In fact, any rational function (functions of the

form where p and q are polynomial

functions) is analytic in their domain of

definition.

( )( )zq

zp


51

The derivative of a function of a real variable is

essentially a local characteristic, in that it provides

information about the function only in a local

neighborhood—for instance, as a truncated Taylor

expansion.

Real and Complex Derivatives

The existence of a derivative of a function of a

complex variable has much more far-reaching

implications. The real and imaginary parts of our

analytic function must separately satisfy Laplace’s

equation. See Exercise 6.2.1.

In this sense the derivative not only governs the

local behavior of the complex function, but controls

the distant behavior as well.© Dr. Nidal M. Ershaidat - Mathematical Physics - Phys. 601 - Chapter 1: Complex Functions

52

A large class of examples of analytic functions

can be built out of polynomials,

Constructing Analytic Functions

Further reading:

http://home.iitk.ac.in/~psraj/mth102/lecture_notes/comp3.pdf

That is, they are actually given by power series

(this is another difference with the reals: if f : R

→→→→ R is differentiable everywhere on R then it is

not necessary that f is given by a power

series). So we need to develop the notion of

power series.

CauchyCauchy’’s Integral s Integral

TheoremTheorem


54

The integral of a function of a complex variable over a contour in the complex plane may be defined in close analogy to the (Riemann) integral of a real function integrated along the

real x-axis.

Contour Integrals

This means that the integral of a function of a complex variable between two points is simply the area enclosed by this function between these 2 points.

The integral of f(z) along a specified contour C

(from z=z0 to z=z’0) is called the contour integral

of f(z).


55

We divide the contour from z0 to z′0 into n

intervals by picking n−1 intermediate points z1,

z2, . . . on the contour (Fig. 2).

Calculating the area

Figure 2

56

Trapezoidal SumA good approximation of the integral is the sum (trapezoidal sum):

(((( ))))(((( )))) ,11

−−−−====

−−−−ζζζζ==== ∑∑∑∑ jj

n

jjn zzfS

12 zz −−−−

(((( ))))1ζζζζf

(((( ))))0ζζζζf

01 zz −−−−

where ζζζζj is a

point on the curve between

zj-1 and zj


57

Calculating the areaSn is nothing else but the sum of (rectangles)

surfaces between the 2 limits of the integral.

(((( ))))(((( )))) ,11

−−−−====

−−−−ζζζζ==== ∑∑∑∑ jj

n

jjn zzfS


58

If we now let n→ ∞→ ∞→ ∞→ ∞ with for all j and

if exists and is independent of the details

of choosing the points zj and ζζζζj then:

Contour Integrals

(((( ))))(((( )))) (((( ))))dzzfzzf

z

z

jj

n

jj

n∫∫∫∫∑∑∑∑′′′′

−−−−====

∞∞∞∞→→→→====−−−−ζζζζ

0

0

11

lim

01 →→→→−−−− −−−−jj zz

nn

S∞∞∞∞→→→→

lim

The preceding development of the contour integral is closely analogous to the Riemann integral of a real function of a real variable.

10


59

Alternative MethodAlternatively, the contour integral may be

defined by:

(((( )))) (((( )))) (((( ))))[[[[ ]]]][[[[ ]]]]

(((( )))) (((( ))))[[[[ ]]]] (((( )))) (((( ))))[[[[ ]]]]∫∫∫∫∫∫∫∫

∫∫∫∫∫∫∫∫

++++++++−−−−====

++++++++====

22

11

22

11

22

11

2

1

,

,

,

,

,

,

,,,,

,,

yx

yx

yx

yx

yx

yx

z

z

dyyxudxyxvidyyxvdxyxu

idydxyxviyxudzzf

The complex contour integral is thus reduced to

the complex sum of real (line) integrals.

This is analogous to the replacement of a vector

integral by the vector sum of scalar integrals.

11


60

Compute the integral:

Example 1-2

In this case dz = i r eiθθθθ dθθθθ and we have:

∫∫∫∫Cn dzz

All points z ∈∈∈∈ C are defined by z = r eiθθθθ.

(((( ))))

(((( ))))(((( ))))[[[[ ]]]] 0

1

2

01

12

0

11 ====++++

====θθθθ====ππππθθθθ++++

++++ππππθθθθ++++++++

∫∫∫∫∫∫∫∫ni

nnin

C

n en

rderidzz

where C is a circle of radius r > 0 around the

origin z = 0 in the mathematical sense

(counterclockwise).

This result is valid for n ≠≠≠≠ -1


61

Example 1-3

Compute the integral: ∫∫∫∫Cn dzz

We can write:

(((( )))) (((( )))) (((( )))) (((( ))))0

1111

1

4

4

3

3

2

2

1

1111

====++++

++++++++

++++++++

++++++++

====++++++++++++++++

∫∫∫∫z

z

nz

z

nz

z

nz

z

n

C

n

n

z

n

z

n

z

n

zdzz

where C is a rectangle of corners z1, z2, z3 and z4.

We leave the case n=-1 as an exercise.

(((( )))) 01

1 14

11

13

14

12

13

11

12 ====

−−−−++++

−−−−++++

−−−−++++

−−−−

++++====

++++++++++++++++++++++++++++++++ nnnnnnnnzzzzzzzz

n

StokesStokes’’ Theorem Theorem

ProofProof

See Appendix 1See Appendix 1--22

CauchyCauchy’’s Integral s Integral

FormulaFormula


64

For an analytic function f(z) on a closed contour

C and within the interior region bounded by C.

We have

Second Basic Theorem

z0 is any point in the interior region bounded by C.

(((( )))) (((( ))))002

1zfdz

zz

zf

i C====

−−−−ππππ ∫∫∫∫ 12

Note: since z is on the contour C while z0 is in the

interior, z − z0 ≠ ≠ ≠ ≠ 0 and the integral Eq. 12 is well

defined.

Although f(z) is assumed analytic, the integrand

f(z)/(z - z0) is not analytic at z = z0 unless f(z0) = 0.


65

Conditions on Boundary

where C is the original outer

contour and C2 is the circle

surrounding the point z0traversed in a counterclockwise direction (see the arrow).

(((( )))) (((( ))))0

200

====−−−−

−−−−−−−−

∫∫∫∫∫∫∫∫ dzzz

zfdz

zz

zf

CC13

If the contour is deformed as shown in Fig. 3, Cauchy’s

integral theorem applies. We have (Eq. 12)

It’s practical to write z0 = + r eiθ, using the polar

representation thanks to the circular shape of the

path around z0.

Figure 3

66

Proof of the Cauchy Integral Formula

Thus we have z = z0 + r eiθ. Here r is small and will

eventually be made to approach zero. We have

(with dz = i r eiθ dθ)

(((( )))) (((( ))))θθθθ

++++====

−−−−θθθθ

θθθθ

θθθθ

∫∫∫∫∫∫∫∫ deirer

erzfdz

zz

zf i

C i

i

C 22

0

0

14

Taking the limit as r → → → → 0, we obtain:

(((( )))) (((( )))) (((( ))))000

222

zfidzfidzzz

zf

CCππππ====θθθθ====

−−−−∫∫∫∫∫∫∫∫ 15

since f(z) is analytic and therefore continuous at z=z0.

Here is a remarkable result. The value of an

analytic function f(z) is given at an interior point z =

z0 once the values on the boundary C are specified.


67

Gauss and KirchhoffIn other words, once the values on the boundary Care specified, we can get the value of an analytic

function f(z) at any interior point z = z0.

This is closely analogous to a two-dimensional form of Gauss’ law in which the magnitude of an interior line charge would be given in terms of the

cylindrical surface integral of the electric field E.

A further analogy is the determination of a function in real space by an integral of the function and the corresponding Green’s function (and their derivatives) over the bounding surface. Kirchhoff diffraction theory is an example of this.


68

z0 exterior to CWhat happens if z0 is exterior to C? In this case

the entire integrand is analytic on and within C. Cauchy’s integral theorem applies and the integral vanishes. We have

(((( )))) (((( ))))

====−−−−ππππ ∫∫∫∫ ,0

,

2

1 0

0

zfdz

zz

zf

i C

z0 interior

z0 exterior16


69

We return back to the expression of f’(z)

Cauchy’s Integral formula and derivatives

(((( )))) (((( )))) (((( )))) (((( ))))

−−−−−−−−

δδδδ−−−−−−−−δδδδππππ====

δδδδ

−−−−δδδδ++++∫∫∫∫∫∫∫∫ dz

zz

zfdz

zzz

zf

ziz

zfzzf

CC00000

000

2

1

From Eq. 16 f(z) we have:

(((( ))))(((( )))) (((( ))))(((( )))) zzz

zfzzfzf

z −−−−δδδδ++++

−−−−δδδδ++++====′′′′

→→→→δδδδ 0lim

(((( ))))(((( ))))

(((( ))))(((( ))))(((( ))))

(((( ))))dz

zz

zf

i

dzzzzzz

zfz

zizf

C

Cz

∫∫∫∫

∫∫∫∫

−−−−ππππ====

−−−−δδδδ−−−−−−−−

δδδδ

δδδδππππ====′′′′

→→→→δδδδ

20

000

0

00

0

2

1

2

1lim0

The 1st derivative is then:

17

18


70

The 2nd derivative is the first derivative of the 1st

derivative we have just calculated, i.e.

Second Derivatives

(((( )))) (((( )))) (((( ))))(((( ))))

(((( ))))(((( ))))

−−−−−−−−

δδδδ−−−−−−−−δδδδππππ====

δδδδ

′′′′−−−−δδδδ++++′′′′∫∫∫∫∫∫∫∫ dz

zz

zfdz

zzz

zf

ziz

zfzzf

CC 20

20000

000

2

1

Again, from Eq. 16, we have:

(((( ))))(((( )))) (((( ))))(((( )))) zzz

zfzzfzf

z −−−−δδδδ++++

′′′′−−−−δδδδ++++′′′′====

→→→→δδδδ 0

)2( lim

(((( ))))(((( ))))(((( ))))

(((( )))) (((( ))))(((( ))))

(((( ))))(((( ))))∫∫∫∫

∫∫∫∫

−−−−ππππ====

−−−−δδδδ−−−−−−−−

δδδδδδδδ−−−−−−−−

δδδδππππ====

→→→→δδδδ

C

Cz

dzzz

zf

i

dzzfzzzzz

zzzz

zizf

30

20

200

000

00

0)2(

2

2

2

2

1lim0

The 2nd derivative is then:

19

20

21


71

We need to calculate:

Second Derivatives - Details

(((( )))) (((( ))))(((( )))) (((( ))))(((( )))) (((( ))))

(((( ))))(((( ))))(((( )))) (((( ))))

(((( ))))(((( ))))(((( )))) (((( ))))2

02

00

000

20

200

000000

20

200

200

20

20

200

2

11

zzzzz

zzzz

zzzzz

zzzzzzzzzz

zzzzz

zzzzz

zzzzz

−−−−δδδδ−−−−−−−−

δδδδδδδδ−−−−−−−−====

−−−−δδδδ−−−−−−−−

δδδδ++++++++−−−−−−−−δδδδ−−−−−−−−++++−−−−====

−−−−δδδδ−−−−−−−−

δδδδ−−−−−−−−−−−−−−−−====

−−−−−−−−

δδδδ−−−−−−−−

where we used the difference of squares

a2-b2 = (a-b) (a+b)


72

The 1st and 2nd derivatives are independent of

the direction of δδδδz0 as it must be. The nth

derivative is obtained in the same manner and

finally we get:

nth Derivatives

The analyticity of f(z) guarantees not only a

first derivative but derivatives of any order!

( ) ( )( )

dzzz

zf

i

nzf

Cn

n ∫ +−π

=1

0

)(

2

!

The derivatives of f(z) are automatically

analytic. This assumes the Goursat version of

the Cauchy integral theorem.

22

MoreraMorera’’s Theorems Theorem


74

This is the converse of Cauchy’s

integral formula.

Morera’s Theorem

It states that:

We leave the proof of this theorem as an

exercise!

If a function f(z) is continuous in a simply

connected region R and for every

closed contour C within R, then f(z) is analytic

throughout R.

(((( )))) 0====∫∫∫∫C dzzf

Laurent Laurent

ExpansionExpansion


77

Taylor Expansion of Functions of

a Complex Variable

Suppose we are trying to expand f(z) about

z = z0 and we have z = z1 as the nearest point on

the Argand diagram for which f(z) is not

analytic.

We construct a circle C centered at z = z0 with

radius less than |z1 − z0| (Fig. 5). Since z1 was

assumed to be the nearest point at which f(z)

was not analytic, f(z) is necessarily analytic on

and within C.


78

Step 1

Figure 5: z’ is a point on the contour C and z is any

point interior to C.


79

Step 1We start from Cauchy Integral Formula

Here z’ is a point on the contour C and z is any

point interior to C (Fig. 3).

(((( ))))(((( ))))

(((( ))))(((( )))) (((( ))))

(((( ))))(((( )))) (((( )))) (((( ))))[[[[ ]]]]∫∫∫∫

∫∫∫∫

∫∫∫∫

−−−−′′′′−−−−−−−−−−−−′′′′

′′′′′′′′

ππππ====

−−−−−−−−−−−−′′′′

′′′′′′′′

ππππ====

′′′′−−−−′′′′

′′′′

ππππ====

C

C

C

zzzzzz

zdzf

i

zzzz

zdzf

i

zdzz

zf

izf

000

00

12

1

2

1

2

1

23


80

Step 2

Using the identity*:

*Multiply both sides by (1-t) in order to verify this identity.

∑∑∑∑∞∞∞∞

====

====++++++++++++++++====−−−− 0

3211

1

n

nttttt

�

We have: (((( ))))(((( )))) (((( ))))

(((( ))))∫∫∫∫ ∑∑∑∑∞∞∞∞

====++++−−−−′′′′

′′′′′′′′−−−−

ππππ====

Cn

n

n

zz

zdzfzz

izf

01

0

0

2

1

Interchanging the order of integration and summation (valid because the infinite series is uniformly

convergent for |t | < 1), we obtain

(((( )))) (((( ))))(((( ))))

(((( ))))∫∫∫∫∑∑∑∑ ++++

∞∞∞∞

==== −−−−′′′′

′′′′′′′′−−−−

ππππ====

C nn

n

zz

zdzfzz

izf

100

02

126

25

24


81

27

28

Taylor Expansion

The integral in the previous equation is nothing

else but the nth derivative of f(z) and finally we get the Taylor expansion of the

function f(z) about z=z0.

(((( )))) (((( ))))(((( ))))

(((( ))))∫∫∫∫∑∑∑∑ ++++

∞∞∞∞

==== −−−−′′′′

′′′′′′′′−−−−

ππππ====

C nn

n

zz

zdzfzz

izf

100

02

1

(((( )))) (((( ))))(((( ))))(((( ))))

!n

zfzzzf

n

n

n 0

00∑∑∑∑

∞∞∞∞

====

−−−−====


82

Circle of Convergence

The expansion (Eq. 28) converges when

010 zzzz −−−−<<<<−−−−

The circle defined by is called the

circle of convergence of Taylor series.

010 zzzz −−−−====−−−−

The distance is called the radius of

convergence of Taylor series.

01 zz −−−−


83

Example – See Exercise 6.5.1

Taylor expansion of ln(1+z)

( ) ( )( )( )

!

0

0

0n

zfzzzf

n

n

n∑∞

=

−=

Solution: First we define the contour C.

We take a circle centered on z=0 of radius <1

Secondly we rewrite the Taylor expansion as:

( ) ( )( )( )

( )( )( )

( )( )( )

+−+−+−= �

!2!1!0

0

22

0

0

1

0

0

00

0

zfzz

zfzz

zfzzzf


84

Example: Taylor Expansion(((( ))))(((( )))) (((( )))) (((( ))))

(((( )))) (((( )))) (((( )))) (((( ))))

(((( )))) (((( )))) (((( )))) (((( ))))

(((( )))) (((( )))) (((( )))) (((( ))))�etc

fzzf

fzzf

fzzf

ff

2012012

10101

10101

001ln00

33

22

11

0

====++++====′′′′′′′′′′′′++++====′′′′′′′′′′′′

−−−−====++++−−−−====′′′′′′′′++++−−−−====′′′′′′′′

====++++====′′′′++++====′′′′

====++++========

−−−−−−−−

−−−−−−−−

−−−−−−−−

which gives

(((( ))))(((( ))))

11

1

321ln 1

0

22

<<<<++++

−−−−====++++++++−−−−====++++ ++++

∞∞∞∞

====∑∑∑∑ zz

n

zzzz n

n

n

�

or (((( )))) (((( )))) 111ln1

1<<<<−−−−====++++ ∑∑∑∑

∞∞∞∞

====

−−−−z

n

zz

n

nn

Binomial Theorem in Binomial Theorem in

the Complex Spacethe Complex Space

See Arfken - Exercise 6.5.2


86

Deriving the Binomial theorem

using Taylor Expansion (Exercise 11.5.2)

(((( )))) nm

n

nmm zzn

mzz 2

0121 ∑∑∑∑

====

−−−−

====++++

The binomial theorem in the complex space is

written as follows:

where is the nth binomial coefficient

n

m

It gives the number of ways, disregarding order, that n

objects can be chosen from among m objects; more

formally, the number of n-element subsets (or n-

combinations) of an m-element set.

29


87

Binomial CoefficientThe binomial coefficient can be written in one of the

following formats:

Factorial format( )!!

!

nmn

m

n

m

−=

Multiplicative format(((( ))))

∏∏∏∏====

−−−−−−−−====

n

i i

nm

n

m

1

1

Recursive format

−−−−++++

−−−−

−−−−====

n

m

n

m

n

m 1

1

1

With initial values

000

1

>>>>====

∈∈∈∈====

nintegersallforn

Nmallforn

m


88

Binomial Theorem using Taylor Expansion

(((( ))))(((( )))) (((( ))))(((( ))))

∑∑∑∑∞∞∞∞

====

====++++

⋅⋅⋅⋅⋅⋅⋅⋅

−−−−−−−−++++

⋅⋅⋅⋅

−−−−++++++++====++++

0

32

321

21

21

111

n

nmz

n

mz

mmmz

mmzmz �

Derive the binomial expression

for m any real number. The expansion is

convergent for |z| < 1. Why?

We leave this problem as an exercise


89

Binomial Theorem using Taylor Expansion

Solution

(((( ))))(((( ))))

(((( ))))(((( )))) (((( ))))

(((( ))))(((( )))) (((( )))) (((( ))))[[[[ ]]]]∫∫∫∫

∫∫∫∫

∫∫∫∫

−−−−′′′′−−−−−−−−−−−−′′′′

′′′′′′′′

ππππ====

−−−−−−−−−−−−′′′′

′′′′′′′′

ππππ====

′′′′−−−−′′′′

′′′′++++

ππππ====++++

−−−−

C

C

C

mm

zzzzzz

zdzf

i

zzzz

zdzf

i

zdzz

zm

iz

000

00

1

12

1

2

1

1

2

11

Schwarz Reflection Schwarz Reflection

PrinciplePrinciple


91

Consider the binomial expansion of g(z) = (z − x0)n for

integral n. Let' s compute the complex conjugate of the

function g is the function of the complex conjugate for

real x0:

Definition

(((( )))) (((( ))))[[[[ ]]]] (((( )))) (((( ))))*** 0

*

0 zgxzxzzgnn

====−−−−====−−−−==== 30

This leads us to the Schwarz reflection principle:

(((( )))) (((( ))))** zfzf ==== 31

If a function f(z) is (1) analytic over some region including the real axis

and

(2) real when z is real, then


92

Fig. 6 illustrates SRP

SRP - Illustration

Figure 6


• Since f(z) is real when z is real, must be real

for all n. Then when we use Eq. 26 and Eq. 27, the

Schwarz reflection principle, follows immediately.


93

Analyticity and SRP

See Exercise 6.5.6 for another form of this principle. Analytic continuation permits extending this result to the entire region of analyticity.

• Since f(z) is analytic at z = x0, this Taylor expansion

exists. ( )( )

0xfn

Expanding f(z) about some (nonsingular) point x0

on the real axis, we have:

(((( )))) (((( ))))(((( ))))(((( ))))

∑∑∑∑∞∞∞∞

====

−−−−====0

00

!n

n

n

xfxzzf 32

Analytic Analytic

ContinuationContinuation


95

It is natural to think of the

values f(z) of an analytic

function f as a single entity,

which is usually defined in

some restricted region S1 of the

complex plane, for example, by a Taylor series (see Fig. 7).

Analyticity →→→→ Analytic Continuation

Then f is analytic inside the circle of convergence C1,

whose radius is given by the distance r1 from the center

of C1 to the nearest singularity of f at z1.

Figure 7


96

DefinitionA singularity is any point where f is not analytic.

If we choose a point inside C1 that is farther than r1

from the singularity z1 and make a Taylor expansion of f

about it (z2 in Fig. 7), then the circle of convergence, C2

will usually extend beyond the first circle, C1.

In the overlap region of both circles, C1,C2, the function

f is uniquely defined. In the region of the circle C2 that

extends beyond C1, f(z) is uniquely defined by the

Taylor series about the center of C2 and is analytic

there, although the Taylor series about the center of C1

is no longer convergent there.


97

Analytic Continuation

After Weierstrass this process is called analytic continuation. It defines the analytic functions in terms of its original

definition (in C1, say) and all its

continuations.

Laurent SeriesLaurent Series

99

Simply Connected Region1Functions that are analytic and single-valued in an annular region, say, of inner radius r and outer radius

R, as shown in Fig. 8 are frequent in Physics.

Figure 8


100

Simply Connected Region2Drawing an imaginary contour line to convert our region into a simply connected regionsimply connected region, we apply

Cauchy’s integral formula, and for two circles C2 and C1

centered at z = z0 and with radii r2 and r1, respectively,

where r < r2 < r1 < R, we have

33(((( ))))(((( )))) (((( ))))

∫∫∫∫∫∫∫∫ −−−−′′′′

′′′′′′′′

ππππ−−−−

−−−−′′′′

′′′′′′′′

ππππ====

21 2

1

2

1

CC zz

zdzf

izz

zdzf

izf

The explicit minus sign in Eq. 33 is introduced so that

the contour C2 (like C1) is to be traversed in the

positive (counterclockwise) sense.


101

Simply Connected Region3The treatment of Eq. 33 now proceeds exactly as we did (Eq. 23) in the development of the Taylor series.

Each denominator is written as (z′ − z0) − (z − z0) and expanded by the binomial theorem, which now follows

from the Taylor series, i.e.

34

(((( )))) (((( ))))(((( ))))

(((( ))))

(((( ))))(((( ))))

(((( ))))∫∫∫∫∑∑∑∑

∫∫∫∫∑∑∑∑

−−−−

∞∞∞∞

====

−−−−

++++

∞∞∞∞

====

−−−−′′′′

′′′′′′′′−−−−

ππππ++++

−−−−′′′′

′′′′′′′′−−−−

ππππ====

2

1

101

0

100

0

2

1

2

1

C nn

n

C nn

n

zz

zdzfzz

i

zz

zdzfzz

izf

for C1, |z′ −z0| > |z −z0| and

for C2, |z′ −z0| < |z −z0|.

102

SeriesThe minus sign of Eq. 33 has been absorbed by the

binomial expansion. Defining the series S1 and the

second S s as

35(((( ))))(((( ))))

(((( ))))∫∫∫∫∑∑∑∑ ++++

∞∞∞∞

==== −−−−′′′′

′′′′′′′′−−−−

ππππ====

C nn

n

zz

zdzfzz

iS

100

012

1

36

where S1 is the regular Taylor expansion, convergent

for ||zz −− zz00| < || < |zz′′ −− zz00| = | = rr11, that is, for all z interior to the

larger circle, C1.

(((( )))) (((( )))) (((( ))))∫∫∫∫∑∑∑∑ ′′′′′′′′−−−−′′′′−−−−ππππ

====−−−−

∞∞∞∞

====

−−−−

C

n

n

nzdzfzzzz

iS

10

102

2

1

and S2 is convergent for ||z z −− zz00| > || > |zz′′ −− zz00| = | = rr22, that is, for

all z exterior to the smaller circle, C2, which goes counterclockwise. © Dr. Nidal M. Ershaidat - Mathematical Physics - Phys. 601 - Chapter 1: Complex Functions

103

Laurent Series*

37

These two series are combined into one series - (a Laurent series) by

(((( )))) (((( ))))∑∑∑∑∞∞∞∞

∞∞∞∞−−−−====

−−−−====n

nn zzazf 0

where

38(((( ))))

(((( )))).

2

11

0

∫∫∫∫ ++++−−−−′′′′

′′′′′′′′

ππππ====

C nnzz

zdzf

ia

*named after Pierre Alphonse Laurent in 1843


104

Laurent Series - DefinitionSince, in Eq. 38, convergence of a binomial

expansion is no longer a problem, C may be any

contour within the annular region r < |z −z0| < R

encircling z0 once in a counterclockwise sense.

If we assume that such an annular region of convergence does exist, then Eq. 37 is the Laurent

series, or Laurent expansion, of f(z).

The use of the contour line (Fig. 8) is convenient in converting the annular region into a simply connected region. Since our function is analytic in this annular region (and single-valued), the contour line is not essential and, indeed, does not appear in the finalresult, Eq. 38.


105

Use of Laurent Series

Laurent series coefficients need not come from evaluation of contour integrals (which may be very intractable). Other techniques, such as ordinary series expansions, may provide the coefficients.

Numerous examples of Laurent series appear in Chapter 7. We limit ourselves here to one simple example to illustrate the application of Eq. 37.


106

Redefining AnalyticityAnother definition of analytic function is given in terms of Taylor expansion.

An analytic function is a function that is locally An analytic function is a function that is locally

given by a convergent power series. given by a convergent power series.

A function is analytic if and only if it is equal to A function is analytic if and only if it is equal to

its Taylor series in some neighborhood of every its Taylor series in some neighborhood of every

point.point.


Laurent Series – Summary1W

107

Laurent series of a complex function f(z) is a

representation of that function as a power series which includes terms of negative degree.

It is defined by It is defined by Eq. 37Eq. 37. .

It may be used to express complex functions in cases where a Taylor series expansion cannot be applied.

37(((( )))) (((( ))))∑∑∑∑∞∞∞∞

∞∞∞∞−−−−====

−−−−====n

nn zzazf 0

The series coefficients are defined by Eq. 38:

38(((( ))))

(((( )))).

2

11

0

∫∫∫∫ ++++−−−−′′′′

′′′′′′′′

ππππ====

C nnzz

zdzf

ia


Laurent Series – Summary2W

108

The path of integration C is counterclockwise around a

closed, rectifiable path containing no self-

intersections, enclosing z0 and lying in an annulus A (in

red) in which f(z) is analytic (holomorphic). The

expansion for f(z) will then be valid anywhere inside

the annulus.


Laurent Series – ExamplesIn order to calculate the Laurent series of a function, it is always worth it to try to find first the Taylor series and define the radius of convergence.

109

We proceed following three steps.

First step: Define the contour C. For this one

should have an idea about the radius of

convergence

Second step: write the Taylor expansion as:

( ) ( )( )( )

( )( )( )

( )( )( )

+−+−+−= �

!2!1!0

0

22

0

0

1

0

0

00

0

zfzz

zfzz

zfzzzf

Third step: which means that one needs the 1st,

2nd, … derivatives.

SingularitiesSingularities


111

Isolated Singular PointThe Laurent expansion represents a generalization of the Taylor series in the presence of singularities.

if f(z) is not analytic at z = z0 but is

analytic at all neighboring points, z0 is an

isolated singular point of the function

f(z).

PolesPoles


114

Pole of order n

In the Laurent expansion of f(z) about the isolated singular point z0,

(((( )))) (((( )))) ,zzazfm

mm∑∑∑∑

∞∞∞∞

∞∞∞∞−−−−====

−−−−==== 0 28

if am = 0 for m<−n<0 and a−n = 0, we say

that z0 is a pole of order n.

As an example, if n = 1,i.e. if a−1/(z − z0) is the first nonvanishing term in the Laurent

series, we have a pole of order 1, often

called a simple polesimple pole.


115

Essential Singularity

If, on the other hand, the summation continues to m=−∞, then z0 is a pole of infinite orderpole of infinite order and

is called an essential singularity.

These essential singularities have many pathological features.

See Mapping


117

The sin z ExampleThe behavior of f(z) as z→→→→∞ is defined in terms

of the behavior of f (1/t) as t →0. Consider the

function

(((( ))))∑∑∑∑

∞∞∞∞

====

++++

++++

−−−−====

0

12

12

1sin

n

nn

n

zz 39

As z→∞, we replace the z by 1/t to obtain

(((( ))))(((( ))))∑∑∑∑

∞∞∞∞

====++++++++

−−−−====

012!12

11sin

nn

n

tnt40


118

Essential Singularity for sin zFrom the definition, sin z has an essential

singularity at infinity. See Exercise 6.1.9.

(((( ))))∑∑∑∑

∞∞∞∞

====

++++

++++

−−−−====

0

12

12

1sin

n

nn

n

zz 41

sin z = sin iy = i sinh y

which approaches infinity exponentially as y→→→→∞.

Thus, although the absolute value of sin x for

real x is equal to or less than unity, the

absolute value of sin z is not bounded.


119

Meromorphic FunctionsA function that is analytic throughout the finite complex plane except for isolated poles is called meromorphic.

Ratios of two polynomials or tan z, cot z are

examples of meromorphic functions.

Entire functions that are analytic in the finite

complex space, i.e. they have no singularities in

this plane, such as exp(z), sin z, cos z (see Sections 5.9, 5.11) are meromorphic functions.

MappingMapping

Visualizing a Function of Visualizing a Function of

a Complex Variablea Complex Variable

http://www.pacifict.com/ComplexFunctions.htmlwww.youtube.com/watch?v=s1DFa1dCss0 http://my.fit.edu/~gabdo/introduc.html


121

Geometric Aspects In the preceding sections we have defined

analytic functions and developed some of their

main features.

Here we introduce some of the more geometric

aspects of functions of complex variables,

aspects that will be useful in visualizing the

integral operations (See Arfken 6th Edition Chapter

7) and that are valuable in their own right in

solving Laplace’s equation in two dimensional two dimensional

systemssystems.


122

z-Plane and w-Plane - Mapping

Then for a point in the z-plane (specific values

for x and y) there may correspond specific

values for u(x,y) and v(x,y) that then yield a

point in the w-plane. This is what we call

mapping

In ordinary analytic geometry we may take

y = f(x) and then plot y versus x. Our problem

here is more complicated, for z is a function of

two variables, x and y. We use the notation

(((( )))) (((( )))) (((( ))))yxviyxuzfw ,, ++++======== 42


123

Mapping lines and Areas

As points in the z-plane transform, or are

mapped into points in the w-plane, lines or

areas in the z-plane will be mapped into lines

or areas in the w-plane.

We shall see how lines and areas map from the

z-plane to the w-plane for a number of simple

functions.

For this three important transformations in the

complex plane will be discussed

TranslationTranslation


125

Transformations (in Physics)Fundamental transformations are necessary for the definition of the symmetry of a physical system.

A symmetry of the system is a physical or mathematical property of the system that remains unchanged under some change which we call transformations

A Transformation can be continuous as in translation, rotation or the so-called time reversal.

A Transformations can be discontinuous as the parity (which is simply the inversion of coordinates in space).


126

Invariance and Conservation Laws

Invariance under translation leads to the law of conservation of momentum.

Invariance under rotation leads to the law of conservation of angular momentum.

Three of the four fundamental interactions are invariant under the parity transformation, namely the strong, electromagnetic and gravitational interactions. Parity is not conserved in weak interactions. (But this is another course!)

Invariance under time reversal leads to the law of conservation of energy.

127

Translation in the Real SpaceTranslation in Euclidean geometry means simply moving without rotation or resizing.Translation of an object is equivalent to moving every point of the object the same distance in the same direction.In general, it is represented by the addition of a constant vector representing the "translation distance".

A mechanical system defined by the position

vector will have the position vector after a

translation defined by the vector and we write:

r�

r ′′′′�

0r�

0rrr��

++++====′′′′

(((( )))) rrTorrrT ′′′′====′′′′→→→→��

:

An operator is defined and we also write:

42


128

The function w is equal to the variable z plus a

constant, z0 = x0 + iy0. Using the definition of z (x

+ iy) and w (w=u(x,y)+i v(x,y); Eq. 43 ) we get:

Translation in the complex space is defined similarly to translation in the real space.

Translation in the Complex Plane

u = x + x0, v = y + y0

representing two pure real translations of the coordinate axes, as shown in Fig. 9.

Under a given translation, to each point corresponds another point defined by:

w = z + z0


129

Translation – Graphical Representation

Figure 9: Translation - Definition

RotationRotation

131

, and 000

θθθθ====i

erz

Polar Representation

Here it is more convenient to return to the polar representation, using

then

ϕϕϕϕρρρρ==== iew 45

46(((( )))),0

0θθθθ++++θθθθϕϕϕϕ ====ρρρρ

iierre

θθθθ==== ierz

or 00 , θθθθ++++θθθθ====ϕϕϕϕ====ρρρρ rr 47

• The modulus r has been modified, either expanded or

contracted, by the factor r0.

• The argument θθθθ has been increased by the additive

constant θθθθ (Fig. 10).

0zzw ==== 44Rotation is defined by:

This represents a rotation of the complex variable

through an angle θ0. For the special case of z0 = i = ei ππππ/2,

the transformation is a pure rotation through π π π π/2

radians.

132

Rotation in the Complex Plane

Figure 10: Rotation - Definition

InversionInversion


134

• ρρρρ = 1/r shows clearly the inversion.

zw

1====

Inversion – Definition - Mapping

Using the polar form, we have

48

which shows that

θθθθ−−−−

θθθθ

ϕϕϕϕ ========ρρρρ i

i

i erer

e11

θθθθ−−−−====ϕϕϕϕ====ρρρρ ,1

r

Inversion is defined by:

49

•The interior of the unit circle is mapped onto the exterior and vice versa (Fig. 11).

• In addition, the second part of Eq. 49 shows that the polar angle is reversed in sign. Eq. 49 therefore also

involves a reflection of the y-axis, exactly like the complex conjugate equationcomplex conjugate equation.


135

Inversion – Graphical Representation

Figure 11: Inversion - Definition


136

yixviu

++++====++++

1

Back to Cartesian FormTo see how curves in the z-plane transform into the w-

plane, we return to the Cartesian form:

50

Rationalizing the right-hand side by multiplying

numerator and denominator by z∗∗∗∗, we have2222

1

yx

yi

yx

x

yix

yix

yixviu

++++−−−−

++++====

−−−−

−−−−××××

++++====++++

2222,

vu

vy

yx

yv

++++−−−−====

++++−−−−====

51

Equating the real parts and the imaginary parts, we have:

andandandand2222 vu

ux,

yx

xu

++++====

++++====

Circle Circle -- LineLine


138

Mapping a CircleA circle centered at the origin in the z-plane has the form

52

and by Eqs. 51 transforms into

222ryx ====++++

(((( )))) (((( ))))2

222

2

222

2

rvu

v

vu

u====

++++++++

++++

54

Simplifying Eq. 53, we obtain

2

2

22 1ρ==+

rvu

A circle centered at the origin in the z-plane transforms

into a circle in the w-plane also centered at the origin.

53


139

Mapping a Horizontal Line

The horizontal line y = c1 transforms into

(((( )))) 1222c

vu

v====

++++

−−−−

56or(((( ))))2

1

2

1

2

2

1

2

1

ccvu ====

++++++++

which describes a circle in the w-plane of radius

(½ c1) and centered at u = 0, v = − ½ c1

55


140

Transforming a Horizontal Line


A straight line or a circle in the z-plane

transforms into a circle in the w-plane.

141

Horizontal Line Parallel to the x-axis

The horizontal lines: y=-c1 x=±±±±c1 also transform

into a circle in the w-plane. (ExerciseExercise)

The previous result is generalized as follows

Generalization

Branch Points and Branch Points and

Multivalent FunctionsMultivalent FunctionsBranch PointsBranch Points


144

Branch Points - DefinitionThere is another sort of important singularity (See Arfken Chapter 7).

(((( )))) 102 ====≠≠≠≠→→→→ ⋅⋅⋅⋅ππππ aia eezf

for nonintegral a.

57

Consider f(z) = za, in which a is not an integer.

As z moves around the unit circle from e0 to e2πi,


145

Branch Points: z = 0 and z = ∞∞∞∞We have a branch point at the origin and another

at infinity. If we set z = 1/t, a similar analysis of f(z)

for t →→→→ 0 shows that t = 0; that is, z = ∞ is also a

branch point.

The points e0i and e2πi in the z-plane coincide, but

these coincident points lead to different values of

f(z); that is, f(z) is a multivalued function.

The problem is resolved by constructing a cut line

joining both branch points so that f(z) will be

uniquely specified for a given point in the z-plane.

For za, the cut line can go out at any angle.


146

Branch PointsNote that the point at infinity must be included here; that is, the cut line may join finite branch points via the point at infinity. The next example is

a case in point. If a = p/q is a rational number, then

q is called the order of the branch point, because

one needs to go around the branch point q times

before coming back to the starting point.

If a is irrational, then the order of the branch point

is infinite, just as for the logarithm.

Note that a function with a branch point and a required cut line will not be continuous across the cut line. Often there will be a phase difference on opposite sides of this cut line.


147

Branch PointsHence line integrals on opposite sides of this branch point cut line will not generally cancel each other.

Numerous examples of this case appear in the exercises. The contour line used to convert a multiply connected region into a simply connected region (Arfken Section 6.3) is completely different. Our function is continuous across that contour line, and no phase difference exists.

TwoTwo--toto--One One

CorrespondenceCorrespondence


149

The three transformations just discussed have all

involved one-to-one correspondence of points in the z plane to points in the w-plane. Now to illustrate the

variety of transformations that are possible and the problems that can arise, we introduce first a two-to-one

correspondence and then a many-to-one correspondence. Finally, we take up the inverses of these two

transformations.

Two-to-One Correspondence


150

Two-to-One CorrespondenceLet’s consider the transformation (function)

which leads to

2zw ====

58θθθθ====ϕϕϕϕ====ρρρρ 22 ,r

This transformation is nonlinear, for the modulus is squared, but the significant feature of Eq. 57 is that the phase angle or argument is

doubled. This means that:

57


151

� The first quadrant of z, transforms in the

upper half-plane of w

Mapping

20 ππππ<<<<θθθθ≤≤≤≤

ππππ<<<<ϕϕϕϕ≤≤≤≤0

� The upper half-plane of z, transforms in the

whole plane of w, ππππ<<<<ϕϕϕϕ≤≤≤≤ 20

ππππ<<<<θθθθ≤≤≤≤0

The lower half-plane of z maps into the already covered

entire plane of w, thus covering the w-plane a second time.

This is an example of two-to-one correspondence, i.e. two

distinct points in the z-plane, z0 and z0eiπ = −z0,

corresponding to the single point w = z02.

Hence the lines u = c1, v = c2 in the w-plane correspond to x2 − y2 = c1, 2xy = c2, rectangular (and orthogonal) hyperbolas in the z-plane (Fig. 6.21). To every point on the hyperbola x2 −

y2 = c1 in the right half-plane, x >0, one point on the line u = c1 corresponds, and vice versa. However, every point on the line u = c1 also corresponds to a point on the hyperbola x2 −

y2 = c1 in the left half-plane, x <0, as already explained.

152

The lower half-plane of z maps into the already covered

entire plane of w, thus covering the w-plane a second

time. This is our two-to-one correspondence, that is, two

distinct points in the z-plane, z0 and z0eiπ = −z0,

corresponding to the single point w = z02.

Two-to-One Correspondence

57(((( )))) yixyxyixviu 2222++++−−−−====++++====++++

which leads to yxvyxu 2,22 ====++++====


153

It will be shown in Section 6.8 that if lines in the w-

plane are orthogonal, the corresponding lines in the

z- plane are also orthogonal, as long as the

transformation is analytic.

Physics

Since u = c1 and v = c2 are constructed perpendicular to

each other, the corresponding hyperbolas in the z-planeare orthogonal. We have constructed a new orthogonal system of hyperbolic lines (or surfaces if we add an

axis perpendicular to x and y).

It might be noted that if the hyperbolic lines are electric or magnetic lines of force, then we have a quadrupole lens useful in focusing beams of high energy particles (see Arfken: Exercise 2.1.3).


154

Study the inverse transformation of w = z2 is:

Application

21zw ====


Next Lecture

Chapter 2: THE GAMMA FUNCTION(FACTORIAL FUNCTION)

1



Physics Department

Yarmouk University

Appendix 1-1: Cauchy-Riemann in polar

coordinates

© Dr. Nidal M. Ershaidat - Mathematical Physics - Phys. 601 - Chapter 1 Appendices

2

Hint. Set up the derivative first with δδδδz radial

and then with δδδδz tangential.

Cauchy-Riemann in polar coordinates.

Using , in which and

are differentiable real functions of r and θθθθ, show that

the Cauchy–Riemann conditions in polar coordinates

become

(((( )))) (((( )))) (((( ))))θθθθΘΘΘΘθθθθ θθθθ==== ,, rii erRerf (((( ))))θθθθΘΘΘΘ ,r

θθθθ∂∂∂∂

ΘΘΘΘ∂∂∂∂====

∂∂∂∂

∂∂∂∂

r

R

r

R

(((( ))))θθθθ,rR

rR

R

r ∂∂∂∂

ΘΘΘΘ∂∂∂∂−−−−====


∂∂∂∂1

Solution: We take z0≠≠≠≠0, z = r ei and and we

express the real and imaginary parts of f as functions

of r and θθθθ, i.e.

θθθθ==== ierz 00

θθθθ==== ierz

(((( )))) (((( )))) (((( )))) (((( ))))θθθθ++++θθθθ======== θθθθ ,, rviruerfzf i


3

Radial δδδδzStep I: In the definition of “differentiable at z0" let z →→→→z0 along the ray θθθθ = θθθθ0 (draw a picture to illustrate this!).

Then the following limit exists:

(((( ))))(((( )))) (((( ))))

(((( )))) (((( )))) (((( )))) (((( ))))[[[[ ]]]]

(((( )))) (((( )))) (((( )))) (((( ))))

−−−−

θθθθ−−−−θθθθ++++

−−−−

θθθθ−−−−θθθθ====

−−−−

θθθθ−−−−θθθθ++++θθθθ−−−−θθθθ====

−−−−

−−−−====′′′′

→→→→→→→→


→→→→θθθθ

θθθθθθθθ

θθθθθθθθ

→→→→

0

000

0

000

0

000000

0

00

,,lim

,,lim

,,,,lim

1

lim

00

0

00

00

00

0

rr

rvrvi

rr

rurue

rr

rvrviruru

e

erer

erferfzf

rrrr

i

rri

ii

ii

rr

1

Both limits in the last line exist because the limit in the first line does exit (and a complex function has a limit at a point if and only if its real and imaginary parts do).


4

Radial δδδδzNow these limits equal the respective partial

derivatives of u and v with respect to r, at the

polar coordinates (r0,θθθθ0). The result is:

(((( )))) (((( )))) (((( ))))

θθθθ

∂∂∂∂

∂∂∂∂++++θθθθ

∂∂∂∂

∂∂∂∂====′′′′ θθθθ−−−−

00000 ,,0 rr

vir

r

uezf i 2

5

Tangentiel δδδδzStep II: In the definition of “differentiable at z0" let z →→→→z0 along the circle r = r0 (draw a picture to illustrate

this!). Then the following limit is true:

(((( ))))(((( )))) (((( ))))

(((( )))) (((( )))) (((( )))) (((( ))))[[[[ ]]]]

(((( )))) (((( )))) (((( )))) (((( ))))0

00

0

00

0

0

0

0

0

000

0

000

000000

0

00

000

,,lim

,,lim

,,,,lim

1

lim

θθθθθθθθθθθθ→→→→θθθθθθθθ→→→→θθθθ


θθθθθθθθθθθθ→→→→θθθθ

θθθθθθθθ

θθθθθθθθ

θθθθ→→→→θθθθ

−−−−

θθθθ−−−−θθθθ





−−−−

θθθθ−−−−θθθθ++++θθθθ−−−−θθθθ====

−−−−

−−−−====′′′′

iii

ii

ii

ii

ee

rvrvi

rurue

ee

rvrviruru

r

erer

erferfzf

3

As θ→θθ→θθ→θθ→θ0 the difference quotients in the square brackets

converge - if they converge at all - to the partial

derivatives of u and v with respect to θθθθ, evaluated at the polar coordinates (r0,θθθθ0).


6

Tangentiel δδδδz

This convergence will happen if the last fraction, i.e.

has a limit at θ→θθ→θθ→θθ→θ0 .0

0θθθθθθθθ −−−−

θθθθ−−−−θθθθii ee

Let’s rather study the fraction

,sinsincoscos

0

0

0

0

0

0






−−−− θθθθθθθθ

iee ii

which, as θ→θθ→θθ→θθ→θ0 , tends to

0

00

00 cossinsincos θθθθ

θθθθ====θθθθθθθθ====θθθθ

====θθθθ++++θθθθ−−−−====

θθθθ

θθθθ++++

θθθθ

θθθθieii

d

di

d

d

which is a finite limit! Thus we can write:

0

000

1lim 0 θθθθ−−−−

θθθθθθθθθθθθθθθθ→→→→θθθθ−−−−========

−−−−

θθθθ−−−−θθθθ iiii

eieiee

4

2


7

Tangentiel δδδδz

And

Finally, the two expressions of the derivative are the same and we have:

(((( )))) (((( )))) (((( ))))

(((( )))) (((( ))))

θθθθ


∂∂∂∂++++θθθθ


∂∂∂∂−−−−====

θθθθ


∂∂∂∂++++θθθθ


∂∂∂∂====′′′′


θθθθ

0000

0

0000

0

0

,,

,,1

0

0

rv

ru

ir

e

rv

iru

erizf

i

i

5

r

vu

r

v

rr

u

∂∂∂∂

∂∂∂∂====


∂∂∂∂−−−−


∂∂∂∂====

∂∂∂∂

∂∂∂∂ 1&

1

Everything was supposed to be evaluated at the polar

coordinates (r0, θθθθ0).

6


8

Summary

If f = u + i v is differentiable at , then the polar

Cauchy-Riemann (Eq. 6) hold at (r0,θθθθ0). In addition, we

have these formulas for the derivative of f:

(((( )))) (((( )))) (((( )))) (((( )))) (((( ))))

θθθθ


∂∂∂∂−−−−θθθθ


∂∂∂∂====

θθθθ

∂∂∂∂

∂∂∂∂++++θθθθ

∂∂∂∂

∂∂∂∂====′′′′

θθθθ−−−−θθθθ−−−−

0000

0

00000 ,,,,0

0 ru

irv

r

er

r

vir

r

uezf

ii

00

00 ≠= θierz

7



Physics Department

Yarmouk University

Appendix 1-2: Stokes’ Theorem

Stokes’ Theorem


12

Stokes’ Theorem

→→→→→→→→→→→→→→→→→→→→

⋅⋅⋅⋅××××∇∇∇∇====⋅⋅⋅⋅∫∫∫∫ adFdrF

The line integral of over a closed path

limiting an area element equals the flux

of .→→→→→→→→→→→→

××××∇∇∇∇==== FFcurl

→→→→F

→→→→ad

Evaluating the line integral

when we go around a complete loop gives:

(((( )))) →→→→→→→→

⋅⋅⋅⋅drz,y,xF


13

Stokes’ Theorem

Stokes’ theorem states that:

∫∫→→→→→

⋅=⋅×∇CS

a �dFdF

Where S is a surface bounded (limited) by a

curve C

Proof:

Let’s consider the surface in

the figure. Let’s divide the

area into differential

(infinitesimal) area elements

3


14

For the ith element of area the equation we

obtained in the previous paragraph can be

written here as:

Stokes’ Theorem - Proof

→→→→→→→→→→→→→→→→→→→→⋅⋅⋅⋅××××∇∇∇∇====⋅⋅⋅⋅∫∫∫∫ ii adFdF �

∑∑∑∑∑∑∑∑∫∫∫∫→→→→→→→→→→→→→→→→→→→→

⋅⋅⋅⋅××××∇∇∇∇====⋅⋅⋅⋅i

i

i

i adFdF �

∫∫∫∫∫∫∫∫→→→→→→→→→→→→→→→→→→→→

⋅⋅⋅⋅××××∇∇∇∇====⋅⋅⋅⋅SC

adFdF �

Summation over all the area elements gives:

Which tends toward the limit when the number

of elements become large enough and the

summation becomes simply an integral, i.e.


15

Stokes’ Theorem - Note

Note: When making the sum of the linear

integral of the vector the terms involving

common boundaries between the area elements

will vanish because they cancel in pairs.

∑∑∑∑∑∑∑∑∫∫∫∫→→→→→→→→→→→→→→→→→→→→

⋅⋅⋅⋅××××∇∇∇∇====⋅⋅⋅⋅i

i

i

i adFdF �

∫∫∫∫∑∑∑∑∫∫∫∫→→→→→→→→→→→→→→→→

⋅⋅⋅⋅====⋅⋅⋅⋅C

i

i dFdF ��



Physics Department

Yarmouk University

Appendix 1-3: Kirchhoff Diffraction Theory


17

Kirchhoff’s Diffraction TheoryAlso known as Scalar Diffraction Theory

Diffraction and the limitations it imposes on optical performances should be appreciated. This is essential in optical imaging (instruments) but also in the data processing systems.

As we do in many other domains in physics, we also prefer to deal with scalars rather than with vectors. The amplitude of the em wave is taken as a scalar function (of position and time) and treating diffraction, which requires the solution of Maxwell’s equations, becomes easier .

Disturbance of light (em waves) by an obstacle can be represented by a scalar function which obeys Helmholtz equation.

18

Green + HelmholtzGiven two scalar functions u and v, Green's second identity asserts that:

From: http://scienceworld.wolfram.com/physics/ScalarDiffractionTheory.html

( ) ( )∫∫ ⋅∇−∇=∇−∇SV

aduvvudVuvvu��

221

If u and v are both solutions of the Helmholtz equation, then

0

0

22

22

=+∇

=+∇

vkv

uku 2

3

where k is the wavenumber.

Replacing u and v in Eq. 1, by their values from equations 2 and 3,

we get:

( ) ( ) ( )( )∫∫ =⋅−×−−×=∇−∇SV

dVukvvkudVuvvu 022224


19

Kirchhoff Diffraction Theoryso

Now define:

( ) 0=⋅∇−∇∫S

aduvvu��

5

r

ev

u

kri

=

ψ=6

7

Combining Eq. 1 with Eq. 5 we obtain:

where S’ is a small sphere surrounding the central

singular point.

0=⋅

ψ∇−

∇ψ+⋅

ψ∇−

∇ψ ∫∫ ′S

krikri

S

krikri

adr

e

r

ead

r

e

r

e ��8

physics department yarmouk university - yuctaps.yu.edu.jo/physics/.../pdf/1_phys601_chapter1.pdf ·...

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