David R. Jackson

ECE 6382

Differentiation of Functions of a Complex Variable

Notes are adapted from D. R. Wilton, Dept. of ECE

1

Notes 2

Fall 2017

Functions of a Complex Variable

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

2

2

2

sinh( )n

nn

n

z x iy, w u ivw f z u x, y iv x, y

f z z u x, y x - y , v x, y xy

w a bz cz , w A z

a bzw , w a zc dz ez

=∞

=−∞

= + = +

= = +

= = =

= + + =

+= =

+ +∑

Function of a complex variable :

( e.g., , )

Examples :

2

Differentiation of Functions of a Complex Variable

( ) ( ) ( )0 0

lim limz z

f z z f zdf ff zdz z z∆ → ∆ →

+ ∆ −∆′ = = =∆ ∆

Derivative of a function of a complex variable :

z

z z+ ∆z

x

yz∆

( )f z

( )f z z+ ∆

w

u

vf∆

( )arg

zz

z z z∆ = ∆

To define a unique derivative at a point , the limit must exist at must be independent of the direction of at

3

The Cauchy – Riemann Conditions

( ) ( )

( ) ( ) ( ) ( )

( ) ( )

( )

0 0

0 0

0 0

0

lim lim

lim lim

lim lim

lim

x x

x x

y y

y

z x i yz x

f z x f zdf fdz x x

u x x, y u x, y v x x, y v x, yi

x xdf u vidz x x

z i yf z i y f zdf f

dz i y i yu x, y y

∆ → ∆ →

∆ → ∆ →

∆ → ∆ →

∆ →

∆ = ∆ + ∆∆ = ∆

+ ∆ −∆= =

∆ ∆+ ∆ − + ∆ −

= +∆ ∆

∂ ∂⇒ = +

∂ ∂

∆ = ∆

+ ∆ −∆= =

∆ ∆

+ ∆ −=

Define First, let :

Next let :

( )u x, yi

i y+

∆( ) ( )

0limy

v x, y y v x, yi∆ →

+ ∆ −y

df v u u v v ui i idz y y x x y y

∆

∂ ∂ ∂ ∂ ∂ ∂⇒ = − + = −

∂ ∂ ∂ ∂ ∂ ∂Question: Is ?

4

The Cauchy – Riemann Conditions (cont.)

df u v df v ui idz x x dz y y

∂ ∂ ∂ ∂= + = −

∂ ∂ ∂ ∂

We found

For a unique derivative, these expressions must be equal. That is, a condition

for the existence of a derivative of function of a complex

necessary

u v u vx y y x

dfdz

∂ ∂ ∂ ∂= = −

∂ ∂ ∂ ∂

⇒

(implies)

variable is that

Cauchy-Riemann conditions

We've proved that if exists,

Cauchy-Riemann conditions (necessity).5

dfdz

u u v vx y i x yx y x yf u i v

z z x i y

u v u vi x i yx x y y

x i yux

∂ ∂ ∂ ∂∆ + ∆ + ∆ + ∆ ∂ ∂ ∂ ∂∆ ∆ + ∆ = =

∆ ∆ ∆ + ∆

∂ ∂ ∂ ∂ + ∆ + + ∆ ∂ ∂ ∂ ∂ =∆ + ∆

⇒

∂∂=

Use C.R. conditions

Next, we prove that Cauchy-Riemann conditions exists (sufficiency) :

( )

( )

v ux x

vi x i yx

x i yu v u vi x i i yx x x x

x i yu vi x i yx x

∂ + ∆ + + ∆ ∂ ∆ + ∆

∂ ∂ ∂ ∂ + ∆ + + ∆ ∂ ∂ ∂ ∂

∂ ∂

=∆ + ∆

∂ ∂ + ∆ + ∆ ∂ ∂

∂

=

−∂

x i y∆ + ∆1arg tanu v yi z

x x x−∂ ∂ ∆

= + ∆ =∂ ∂ ∆

, independent of

( ) ( )( )

( ) ( )( )

u x, y u x, yu x, y x yx y

v x, y v x, yv x, y x yx y

∂ ∂∆ = ∆ + ∆

∂ ∂∂ ∂

∆ = ∆ + ∆∂ ∂

Total differentials :

6

The Cauchy – Riemann Conditions (cont.)

dfdz

dfdz

• ⇔

•

•

exists Cauchy - Riemann conditions.

or

exists (iff) the Cauchy - Riemann conditions hold.

or

The Cauchy - Riemann conditions are a

condition for the e

if and only if

necessary and sufficientdf fdz

xistence of the derivative of a complex variable .

7

The Cauchy – Riemann Conditions (cont.)

Hence we have the following equivalent statements:

We say that a function is "analytic" at a point if the derivative exists there(and at all points in some neighborhood of the point).

( )f zD D

is said to be "analytic" in a domain if the derivative exsits at each point in .

8

The Cauchy – Riemann Conditions (cont.)

The theory of complex variables largely exploits the remarkable properties of analytic functions.

The terms " holomorphic", "regular", and "differentiable" are also used instead of "analytic."

D z

x

y

Applying the Cauchy – Riemann Conditions

( ) ( )( )

( )

( )

( ) ( )( )

( )( )

1

0

1 1

u x,y v x,y

v x,yu x,y

f z z x iy x i yu vx y

zu vy x

z

f z z* x iy * x i yu vx y

uy

= = + = +

∂ ∂= =

∂ ∂∂ ∂

= = −∂ ∂

= = + = + −

∂ ∂= ≠ = −

∂ ∂∂

=

⇒

◊

◊

∂

⇒

C.R. conditions hold everywhere for finite

is analytic everywhere

Example 1 :

Example 2

:

0 vx

z*

∂= −

∂

⇒

⇒

C.R. conditions hold

is analytic nowhere

nowhereX

9

Applying the Cauchy – Riemann Conditions (cont.)

( )

( ) ( )

2 2

2 2 2 2

2

1 1

u x,y v x,y

x iyf zz x iy x y

x yix y x y

u xx

−= = =

+ +

−= +

+ +

∂∂

◊

=

Example 3

:

2 2y+ −

( )2 22

22 2

x yx vyx y

− −∂= =

∂+

? 2+

( )

( )

( )

2

22 2

22 2

2 0 ( 0)

0 0

y

x y

u xy v x y z .y xx y

f z z z.

+

∂ − ∂= = − = =

∂⇒ =

∂ +

= =

C.R. conditions hold everywhere except

is analytic everywhere except at The point is called a "singularity."

1 0

: 0

zz

D z

=

⇒ >

is analytic except at

10

D

z

x

y

( ) ( ) ( ) ( ) ( )

( )( ) ( )

( )( ) ( )

( ) ( )( ) ( )

sin sin sin cos sin cos

cos cosh2 2

sin sinh2 2

sin sin sin cosh sinh cos

cos cosh

u x,y v x,y

i iy i iy y y

i iy i iy y y

f z z x iy x iy iy x

e e e eiy y,

e e e eiy i i yi

z x iy x y i y x

u v ux y ,x y y

− −

− −

= = + = +

+ += = =

− −= = − =

= + = +

∂ ∂ ∂= =

◊

∂ ∂⇒

∂

Example

4

ut

so

:

b

( )

( ) ( )( )

sin sinh

cos cosh sin sinh

cos cos sin sin

coscos

vx yx

z

u vf z i x y i x yx x

x iy x iy

x iyz

∂= = −

∂

∂ ∂′ = + = −∂ ∂

= −

= +

=

⇒ C.R. conditions hold for all finite

Now use :

( ) ( )sin cosdf z z zdz

′ = =⇒11

Applying the Cauchy – Riemann Conditions (cont.)

Differentiation Rules

x z Replacing by in the usual derivations for functions of a real variable, we find practically all differentiation rules for functions of a complex

variable turn out to be identical to t

( ) ( )( ) ( ) ( )

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

( )( ) ( )2

d f z g zf z g z

dzd f z g z

f z g z f z g zdz

f zd g f f gdz g z g

±′ ′= ±

′ ′= +

′ ′−=

hose for real variables :

12

Differentiation Rules (cont.)

It is relatively simple to prove on a case -by- case basis that practically all formulas for differentiating functions of real variables also apply to the corresponding function of a complex

( ) ( )

( )( )

1

1 sin coscos sin etc.

n nN

n azn azd z d e d z d znz , ae , z, z,

dz dz dz dz

d z nz N P zdz z

P zz

Q z

−

−= = = = −

⇒

⇒

=

variable :

every polynomial of degree , , in is analytic (differentiable).

every rational function in is analytic

w

except

( )Q zhere vanishes.

13

Differentiation Rules

( ) ( ) ( ) ( )22 22

0 0lim limz z

zz z zd zdz z∆ → ∆ →

+ ∆ −= =

∆

◊ Example

( ) ( )2 22z z z z+ ∆ + ∆ −

02 lim

2z

z

z z

z∆ →

∆

= + ∆

=

14

A Theorem Related to z*

( )

2 2

1 1

u vx yv ux y

z z* z z*z x iy z* x iy x , yi

x y z z*

u z,z* u z u z*x z x z* x

,

∂ ∂=

∂ ∂∂ ∂

= −∂ ∂

+ −= + = − = =

∂ ∂ ∂ ∂ ∂= + =

∂ ∂ ∂ ∂ ∂

⇒

= =

C.R.con

C.R. conditions :

Note that

Make the above substitution for and and treat and as independent variables :

( )

v z,z* v z v z*y z y z* y

u u v viz z* z z*

i i

∂ ∂ ∂ ∂ ∂= +

∂ ∂ ∂ ∂ ∂

∂ ∂ ∂ ∂+ = − ∂ ∂ ∂ ∂

⇒

= = −

ds.

(1)

15

If f = f (z,z*) is analytic, then

0fz*∂

=∂

(The function cannot really vary with z*.)

( )

( )

1 1

v z,z* u z,z*v z v z* u z u z*x z x z* x y z y z* y

v v u uiz z* z z*

f u v vi iz* z* z* z

ii

∂ ∂∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂= + = − =− −

∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂

∂ ∂ ∂ ∂+ = − − ∂ ∂ ∂ ∂

∂ ∂ ∂ ∂= + = −

∂ ∂

⇒

∂ ∂

= = =−=

C.R.conds.

Similarly,

(2)

Next, consider

v uz* z

∂ ∂+ + ∂ ∂

viz∂

+∂

from (2)

uz∂

−∂

0

f

uz*

u v f fiz* z* z* z*

z*

∂+ ∂

∂ ∂ ∂ ∂= − + = − = ∂ ∂ ∂ ∂

⇒

⇒

is independent of

from (1)

16

A Theorem Related to z* (cont.)

( )

( ) ( )

( ) 2

1 0

sin cos 0 2 1 2

0

ff z z* z*z*

ff z z* z* z n /z*

ff z z zz* zz*

π

∂= = ≠

∂∂

= = ≠ = +∂

∂= = = ≠

∂

unless

unless

is analytic nowhere since (not independent of )

is analytic almost nowhere since ( )

is analytic almost nowhere since (

Examples:

0z = )

17

A Theorem Related to z* (cont.)

Analytic Functions

1 22 2

2 3 4 51

sin cos sinh cosh

1 1, tan cot tanh c1

/

n

z, z, z , z , z ,z , ,z ,

e , z, z, z, z

, z , z, z, z,z z −

Typical functions analytic everywhere (in the finite complex plane) :

Typical functions analytic everywhere :almost

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

oth z

f z ,g z ,h z

a f z b g z c h z+ +⇒

If of analytic functions are analytic :

are analyticanalytic

(series) of analytic functions may be : - Analytic everyw

Finite linear combinations

Infinite linear combinationshere

- Analytic nowhere - Analytic almost everywhere

18

Composite functions of analytic functions are also analytic.

Derivatives of analytic functions are also analytic. (This is proven later.)

A function that is analytic everywhere is called “entire”.

Real and Imaginary Parts of Analytic Functions Are Harmonic Functions

( )f z u iv= +Assume an analytic function:

This result is extensively used in conformal mapping to solve

electrostatics and other problems involving the 2D Laplace equation

(discussed later).

19

2 2 0u v∇ =∇ =

u vThe functions and are harmoni (i.e., satisfy Laplace's ec q.)

Real and Imaginary Parts of Analytic Functions Are Harmonic Functions

( ) ( )

( )

( )

df u v v ui i U x, y iV x, ydz x x y y

f z U iV

u vU x, y , ix y

∂ ∂ ∂ ∂= + = − ≡ +

∂ ∂ ∂ ∂′ ≡ +

∂ ∂≡ =

⇒

∂ ∂

Analytic

where

We have : ( )V x, y i≡v ix

∂= −

∂

( )2 2

2 2

2 2

2 2

2

2

0

0

uy

f z

U V u ux y x y

V U v vx y x y

u

v

∂∂

′

∂ ∂ ∂ ∂= = −

∂ ∂ ∂ ∂

∂ ∂

⇒ ⇒

⇒ ⇒∂ ∂

= − = −∂ ∂ ∂ ∂

∇ =

∇ =

Apply the C.R. conditions to :

Assume df/dz is also analytic (see note on previous slide)!

20

Proof