第 1頁

Chapter 2 Analytic Function

9. Functions of a complex variable

Let S be a set of complex numbers.

A function defined on S is a rule that assigns to each z in S a complex

number w.

f

value of at , or ( )f z f z

or w f zS is the domain of definition of f

2

1

1

wz

w z

sometimes refer to the function itself, for simplicity.f

Both a domain of definition and a rule are needed in order for a function to be well defined.

第 2頁

Suppose is the value of a function atfw u i z x iy

, ,

u iv f x iy

or f z u x y iv x y

real-valued functions of real variables x, y

or , ,f z u r iv r

Ex.

2

2 2

2 2

2 2

2 2

2

( , ) , ( , ) 2

cos 2 sin 2

, cos 2 , sin 2

i

f z z

f x iy x y i xy

u x y x y v x y xy

f re r ir

u r r v r r

when v=0 is a real-valued function of a complex variable. f Z

第 3頁

is a polynomial of degree n. 20 1 2 ... n

nf z P z a a z a z a z

( )

( )

P z

Q z : rational function, defined when ( ) 0Q z

For multiple-valued functions : usually assign one to get single-valued function Ex.

12 2

2

, 0

, th root

If we choose 0,

2 2 2 and (0) 0,

then is well defined on the entrie complex plane except

the ray = .

i

i

i

z re z

z re n

f z re r

f

f

第 4頁

10. Mappings

w=f (z) is not easy to graph as real functions are.

One can display some information about the function by indicating pairs

of corresponding points z=(x,y) and w=(u,v). (draw z and w planes

separately).

When a function f is thought of in this way. it is often refried to as a mapping, or transformation.

z winverse image

of w

image of z

image of T

T

第 5頁

Mapping can be translation, rotation, reflection. In such cases

it is convenient to consider z and w planes to be the same.

w=z+1 translation +1

w=iz rotation

w= reflection in real axis.

Ex. image of curves

z2

y

x

real number 2y x

2

2 2

2

w z

u x y

v xy

a hyperbola is mapped in a one to one manner onto the line 2 21x y c 1u c

第 6頁

2xy=C2Y

X

2 21x y C

u=C1>1V

U

V=C2>0

right hand branch x>0, u=C1,

left hand branch x<0 u=C1,

image

21

21

2

2

V y y c y

V y y c y

第 7頁

x

y

D

B

A

1

L1 L2

Cx1

Ex 2.

0 1

0

x

y

u

vA’

L2’L1’

B’D’ C’

When , point moves up a vertical half line, L1, as y increases from y = 0.

10 1x 1,x y

2 21

1

2

2 2 2 21 1 1

1

, 2

2

, 42

u x y v x y

vy

x

vu x v x u x

x

a parabola with vertex at 2

1 ,0x

half line CD is mapped of half line C’D’

2 ,0y 0, y

第 8頁

Ex 3.

2 2 2

2 , 2 2 0, 1, 2,...

i

i

w z r e

let w e

r n n

x

y

u

v

20, 0r one to one 0, 0

第 9頁

11. Limits

Let a function f be defined at all points z in some deleted neighborhood of z0

00lim ( ) (1)

( )

z zf z w

f z

means: the limit of as z approaches z0 is w0

can be made arbitrarily close to w0 if we choose the point z close ( )w f z

enough to z0 but distinct from it.

(1) means that, for each positive number ,there is a positive number

such that

0 0( ) 0f z w whenver z z

x

y

z

z0

u

v

ww

w0

0

(2)

第 10頁

Note:

(2) requires that f be defined at all points in some deleted neighborhood of z0

such a deleted neighborhood always exists when z0 is an interior point of a region on which is defined. We can extend the definition of limit to the case in which z0 is a boundary point of the region by agreeing that left of (2) be satisfied by only those points z that lie in both the region and the domain

Example 1. show if

f

00 z z

1

( ) 1,2

lim ( )2z

izf z in z then

if z

第 11頁

1

1

2 2 2 2

when z in z

zi iz if z

For any such z and any positive number

2

if z whenever 0 1 2z

z

y

x

2

v

u

2

i

第 12頁

Then

Let

But

Hence

0 0 0

1 0 1

0

0

f z w whenever z z

f z w whenever z z

0 1

0

min ,

0if z z

0 1 0 1

1 0

( ) ( ) ( ) 2

2

f z w f z w f z w f z w

w w

1 0w w is a nonnegative constant, and can be chosen arbitrarily small.

1 0 1 00,w w or w w

When a limit of a function exists at a point , it is unique.

If not, suppose , and

f z 0z

00lim

z zf z w

01lim

z zf z w

第 13頁

Ex 2. If (4)

then does not exist.

zf z

z

0

limz

f z

show:

0,0 1

00

0, 10

x iwhen z x f z

x iiy

when z y f ziy

since a limit is unique, limit of (4) does not exist.

(2) provides a means of testing whether a given point w0 is a limit, it does

not directly provide a method for determining that limit.

第 14頁

12. Theorems on limits

Thm 1. Suppose that

0 0 0

0 0 0

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

and w u iv

Then iff0

0 0 0 0

0

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

lim ( )

lim ( , ) lim ( , )

z z

x y x y x y x y

f z w

u x y u and v x y v

pf : ” ” 2 20 0 0 1

2 20 0 0 2

1 2

0 ( ) ( )2

0 ( ) ( )2min( , )

u u whenever x x y y

v v whenever x x y y

let

第 15頁

since

and

“ ”

But

and

0 0 0 0 0 0

2 2

0 0 0 0 0 0

0 0

0 0

( ) ( )

2 2

0

u iv u iv u u i v v u u v v

x x y y x x i y y x iy x iy

u iv u iv

whenever x iy x iy

0 0 0 00u iv u iv whenever x iy x iy

0 0 0 0 0

0 0 0 0 0

u u u u i v v u iv u iv

v v u u i v v u iv u iv

2 2

0 0 0 0

0 0

2 20 0

( )

0 ( ) ( )

x iy x iy x x y y

u u and v v

whenever x x y y

第 16頁

Thm 2. suppose that

0 0

0

0

0

0 0

0 0

0 0

0

0

0

lim ( ) lim ( ) (7)

lim ( ) ( )

lim ( ) ( ) (9)

0

( )lim

( )

z z z z

z z

z z

z z

f z w and F z W

Then f z F z w W

f z F z w W

and if W

wf z

F z W

0 0 0 0 0 0 0 0 0

( ) ( , ) ( , )

( ) ( , ) ( , )

, ,

f z u x y iv x y

F z U x y iV x y

z x iy w u iv W U iV

pf: utilize Thm 1.

for (9).

use Thm 1. and (7)

第 17頁

0 0 0 0 0 0 0 0

0 0

( ) ( ) ( )f z F z uU vV i vU uV

u U v V v U u V

w W

have the limits

An immediate consequence of Thm. 1:

0

0

0

0

0 0

0

0

20 1 2

0

0 0

lim

lim

lim ( 1,2,...)

( ) ...

lim ( ) ( )

lim ( ) , lim ( )

z z

z z

n n

z z

nn

z z

z z z z

c c

z z

z z n

P z a a z a z a z

P z P z

if f z w then f z w

by property (9) and math induction.

(11)

利用 whenever0 0( ) ( )f z w f z w 00 z z

第 18頁

13. Limits involving the point at Infinity

It is sometime convenient to include with the complex plane the point at infinity,

denoted by , and to use limits involving it.

Complex plane + infinity = extended complex plane.

NP

zO

complex plane passing thru the equator of a unit sphere.

To each point z in the plane there corresponds exactly one point

P on the surface of the sphere.

intersection of the line z-N with the surface.

north pole

To each point P on the surface of the sphere, other than the north pole N, there corresponds exactly one point z in the plane.

第 19頁

By letting the point N of the sphere correspond to the point at infinity, we obtain a one-to-one correspondence between the points of the sphere and the points of the extended complex plane.

upper sphere exterior of unit circle

points on the sphere close to N

neighborhood of

1z

0

0

0

lim ( )

1( ) 0

10 0

( )

z zf z

f z whenever z z

whenever z zf z

0 0

1 1

1lim ( ) lim 0

( )

3 11. lim since lim 0

1 3

z z z z

z z

f z ifff z

iz zEx

z iz

第 20頁

0

0

0

0 00

lim ( )

1( )

1( ) 0 0

1lim ( ) lim ( )

z

z z

f z w

f z w whenever z

f w whenever zz

f z w iff f wz

Ex 2.0 0

2( )2 2

lim 2 since lim lim 211 1( ) 1

z z z

iz i izzz z

z

第 21頁

10

lim ( )

1 1( )

1 1 1 1( )

10 0 0

(1/ )

1lim ( ) lim 0

( )

z

z zz

f z

f z whenever z

f wheneverz z

whenever zf z

f z ifff

2

3

3

2

1 3

320 0

2 1lim

11

since lim lim 01 2

z

z

z zz

z

z

z z

z

Ex 3.

第 22頁

14. Continuity

A function f is continuous at a point z0 if

0

0

0

0

0 0

lim ( ) exists,

( ) exists,

lim ( ) ( )

( ( ) ( ) )

z z

z z

f z

f z

f z f z

f z f z whenever z z

(1)

(2)

(3)

• if continuous at z0, then are

also continuous at z0.

1 2,f f 1 2 1 2,f f f f

((3) implies (1)(2))

12 0

2

( ) 0f

if f zf

So is

第 23頁

• A polynomial is continuous in the entire plane because of (11), section 12. p.37

• A composition of continuous function is continuous.

z w

f g

0 0

0

( ) ( ) , ( ) ( ) ,g f z g f z whenver f z f z r

whenver z z

• If a function f(z) is continuous and non zero at a point z0, then

throughout some neighborhood of that point.

when let

( ) 0f z

0( ) 0f z 0( )

2

f z

00 0

( )( ) ( )

2

f zf z f z whenever z z

if there is a point z in the at which then

, a contradiction.

0z z 0( ) 0f z

00

( )( )

2

f zf z

第 24頁

Ex. The function

is continuous everywhere in the complex plane since

(i) are continuous (polynomial)

(ii) cos, sin, cosh, sinh are continuous

(iii) real and imaginary component are continuous

complex function is continuous.

2 2 2 2( ) cos( )cosh 2 sin( )sinh 2f z x y xy i x y xy

2 2

2

x y

xy

From Thm 1., sec12.

a function f of a complex variable is continuous at a point

iff its component functions u and v are continuous there.

0 0 0( , )z x y

sinh sin2 2

cosh cos2 2

x x ix ix

x x ix ix

e e e ex x

i

e e e ex x

第 25頁

15. Derivatives

Let f be a function whose domain of definition contain a neighborhood of a point z0. The derivative of f at z0, written , is 0'f z

0

00

0

' limz z

f z f zf z

z z

provided this limit exists.

f is said to be differentiable at z0.

0

0 00 0

0

( ) ( )'( ) lim

( ) ( ).

'( ) lim

z

z

let z z z

f z z f zf z

zlet w f z z f z

dw wf z

dz z

z

0z

0z z

第 26頁

Ex1. Suppose at any point z2( )f z z2 2

0 0 0

( )lim lim lim(2 ) 2 0 2z z z

w z z zz z z z

z z

since a polynomial in .2z z z

'( ) 2dw

f z zdz

Ex2.

2

2 2

( )f z z

z z z z zzz z zw zz z z

z z z z

when thru on the real axis ,

Hence if the limit of exists, its value =

when thru on the imaginary axis.

, limit = if it exists.

0z

0z

,0x

0, y

w

z

z z

z z

z z z z

第 27頁

since limits are unique,

, if is to exist.

observe that when

exists only at , its value = 0

, 0z z z z or z dw

dzw

zz

0z

dw

dz 0z

• Example 2 shows that

a function can be differentiable at a certain point but nowhere else in any

neighborhood of that point.

• Re are continuous, partially

Im differentiable at a point.

but may not be differentiable there.

2 2 2

20

z x y

z

2

z

第 28頁

existence of derivative continuity.

0 0 0

0

00 0

0

0

0

( ) ( )lim ( ) ( ) lim lim( )

'( ) 0 0

lim ( ) ( )

z z z z z z

z z

f z f zf z f z z z

z z

f z

f z f z

continuity derivative exists.not necessarily

• is continuous at each point in the plane since its components are continuous at each point. ( 前一節 )

2( )f z z

第 29頁

16. Differentiation Formulas

1

0 : complex constant

1

( ) '( )

n n

dC C

dzd

zdzd

cf z cf zdzd

z nzdz

n a positive integer.

2

( ) '( ) '( )

( ) ( ) '( ) '( ) ( ) (4)

( ) 0

( ) '( ) ( ) '( )

( ) ( )

df z F z f z F z

dzd

f z F z f z F z f z F zdzwhen F z

f zd F z f z f z F z

dz F z F z

第 30頁

: (4)pf

( ) ( ) ( ) ( )

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

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

f z z F z z f z F z

f z F z z F z f z z f z F z z

f z z F z z f z F z F z z F z f z z f zf z F z z

z z z

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

( ) '( ) '( ) ( )

das z fF f z F z f z F z z

dzf z F z f z F z

(F continuous at z)

第 31頁

f has a derivative at z0

g has a derivative at f(z0)

F(z)=g[f (z)] has a derivative at z0

and chain rule (6)0 0 0'( ) '[ ( )] '( )F z g f z f z

dW dW dw

dz dw dz

pf of (6)

choose a z0 at which f’(z0) exists.

let w0 = f (z0) and assume g’(w0) exists.

Then, there is of w0 such that

we can define a function , with and

0w w

0( ) 0w 0

0 00

( ) ( )( ) '( )

g w g ww g w when w w

w w

(7)

0

lim ( ) 0,w w

w

Hence is continuous at w0

第 32頁

0 0 0 0(7) ( ) ( ) '( ) ( ) ( ) ( ) (9)g w g w g w w w w w w

0w wvalid even when

since exists and therefore f is continuous at z0, then we can

have f (z) lies in

0'( )f z

0 0 0w w of w if z z

substitute w by f (z) in (9) when z in

(9) becomes

0z z

0 00

0 0

0

[ ( )] [ ( )] ( ) ( ){ '[ ( )] [ ( )]} (10)

(0 )

g f z g f z f z f zg f z f z

z z z z

z z

since f is continuous at z0 , is continuous at

is continuous at z0 , and since

0 0( )w f z

[ ( )]f z 0( ) 0w

0

lim ( ) 0z z

f z

0 0 0 0'( ) ' ( ) '( ) as F z g f z f z z z so (10) becomes

第 33頁

17. Cauchy-Riemann Equations

Suppose that 0 00

0

0 0 0,

( ) ( )'( ) lim .

z

f z z f zf z exists

zz x iy z x i y

writing

Then by Thm. 1 in sec 12

0 00 ( , ) (0,0)

0 00 ( , ) (0,0)

( ) ( )Re '( ) lim Re[ ] (3)

( ) ( )Im '( ) lim Im[ ] (4)

x y

x y

f z z f zf z

zf z z f z

f zz

where

0 0 0 0 0 0

0 0 0 0

( ) ( ) ( , ) ( , )(5)

[ ( , ) ( , )]

f z z f z u x x y y u x y

z x i y

i v x x y y v x y

x i y

第 34頁

Let tend to (0,0) horizontally through . i.e.,( , )x y ( ,0)x 0y

0 0 0 00

0

0 0 0 00

0

0 0 0 0 0

( , ) ( , )Re[ '( )] lim

( , ) ( , )Im[ '( )] lim

'( ) ( , ) ( , ) (6)

x

x

x x

u x x y u x yf z

xv x x y v x y

f zx

f z u x y iv x y

Let tend to (0,0) vertically thru , i.e. , then( , )x y (0, )y 0x

0 0 0 00 0 0 00

0 0 0 0

( , ) ( , )( , ) ( , )'( ) ( )

( , ) ( , ) (7)y y

y y

i v x y y v x yu x y y u x yf z

i y i y

v x y iu x y

iu v

(6)= (7)

0 0 0 0

0 0 0 0

( , ) ( , ) (8)

( , ) ( , )

x y

y x

u x y v x y

u x y v x y

Cauchy-Riemann Equations.

第 35頁

Thm : suppose

exists at a point

Then

0 00

( ) ( , ) ( , )

'( )

f z u x y iv x y

f z z x iy

0 0, , , exist at ( , )

and , ; also '( )

x y x y

x y y x x x

u u v v x y

u v u v f z u iv

2 2 2( ) 2

2 2

2 2

,

'( ) 2 2 2( ) 2

x x

y y

x y y x

f z z x y i xy

u x v y

u y v x

u v u v

f z x i y x iy z

Ex 1.

Cauchy-Riemann equations are Necessary conditions for the existence of

the derivative of a function f at z0.

Can be used to locate points at which f does not have a derivative.

第 36頁

2

2 2

( ) ,

( , ) ( , ) 0

2 0 , '( )

2 0

x x x y

y y

f z z

u x y x y v x y

u x v u v f z

u y v

Ex 2.

does not exist

at any nonzero point.

The above Thm does not ensure the existence of f ’(z0)

(say)

第 37頁

18. Sufficient Conditions For Differentiability

but not0'( ) exist ,

" "

x y y xf z u v u v

Let be defined throughout some

neighborhood of a point

suppose exist everywhere in the neighborhood and

are continuous at .

, , ,x y x yu u v v

0 0( , )x y

( ) ( , ) ( , )f z u x y iv x y Thm.

0 0 0z x iy

0 0

0

, at ( , )

'( ) .

x y y xu v u v x y

f z exists

Then, if

第 38頁

0 0

0 0 0 0

: , 0

( ) ( )

( ) ( ) [ ( ) ( )]

pf let z x i y where z

w f z z f z

w u i v u z z u z i v z z v z

Thus

where0 0 0 0

0 0 0 0

( , ) ( , )

( , ) ( , )

u u x x y y u x y

v v x x y y v x y

Now in view of the continuity of the first-order partial derivatives of u and v at the point

0 0 0 0 0 0 0 0( , ) ( , ) ( , ) ( , )x y xyu u x y u x y x u x y y u x y x y 2

0 0( , )2!xx

xu x y

2

0 0( , )2!

...

yy

yu x y

0 0

2 20 0 0 0 1

( , )

( , ) ( , ) ( ) ( )x y

u x y

u x y x u x y y x y

0 0( , )x y

第 39頁

2 20 0 0 0 2

1 2

( , ) ( , ) ( ) ( )

, 0, ( , ) (0,0)x yv v x y x v x y y x y

as x y

w u i v

above

where and tend to 0 as in the -plane.

(3)

1 2 ( , ) (0,0)x y z

assuming that the Cauchy-Riemann equations are satisfied at , we can

replace in (3), and divide thru by

to get

0 0( , )x y,y x y xu by v and v by u z

2 2

0 0 0 0 1 2

2 2

2 2

( , ) ( , ) (4)

1

x x

x ywu x y iv x y i

z z

but x y z

x yso

z

also tends to 0, as

The last term in(4) tends to 0 as

1 2i , 0,0x y

0z

0 0 0 0 0 The limit of exists, and '( ) ( , ) ( , ).x x

wf z u x y iv x y

z

第 40頁

Ex 1. ( ) (cos sin )

( , ) cos

( , ) sin

,

'( )

'( ) (cos sin )

x

x

x

x y y x

xx x

f z e y i y

u x y e y

v x y e y

u v u v

f z

f z u iv e y i y

everywhere, and continuous.

exists everywhere, and

Ex 2.2

2 2

( )

( , ) 2 2

( , ) 0 0 0

'(0) 0 0

x y

x y

f z z

u x y x y u x u y

v x y v v

f i

has a derivative at z=0.

can not have derivative at any nonzero point.

第 41頁

19. Polar Coordinates

cos sin

( 0)i

x r y r

z x iy re z

Suppose that exist everywhere in some neighborhood of a

given non-zero point z0 and are continuous at that point.

also have these properties, and ( by chain rule )

, , ,x y x yu u v v

, , ,r ru u v v

cos sin (2)

sin cosr x y

x y

u u x u y

r x r y r

u u x u y

x y

u u u

u u r u r

Similarly,cos sin (3)

sin cosr x y

x y

v v v

v v r v r

第 42頁

If ,

cos sin (5)

sin cos

x y y x

r y x

y x

u v u v

v u u

v u r u r

from (2) (5),

Thm. p53…

0

1at (6)

1

r

r

u v zr

u vr

第 43頁

2

cos sin

sin cos sin sin

cos sin

r y x

r y x

r r x

u v v

u v v

v u v

2

cos sin

cos cos sin cos

r x y

r x y

v v v

v v v

0'( ) cos sin cos sin

(cos sin )( )

( ) (7)

r r r r

r r

ir r

f z u v i v u

i u iv

e u iv

0

2

'( )

?

cos sin

cos cos sin cos

x x

r x y

r x y

f z u iv

u u u

u u u

2

cos sin

cos sin

sin cos sin sin

r x y

y x

r y x

v v v

u u

v u u

cos sinr r xu v u

第 44頁

1 1,

'

r ru v u vr r

f

at any non-zero point iz re

exists

2 2

22 2 2

1' ( cos sin )

1 1 1( )

i

i i i

if e

r r

e e er r z

2 2

1 1( )

1 1( , ) cos ( , ) sin

1 1cos sin

1 1sin cos

i

r r

f zz re

u r v rr r

u vr r

u vr r

Ex : Consider

第 45頁

A function f of the complex variable z is analytic in an open set if it has a derivative at each point in that set.

f is analytic at a point z0 if it is analytic in a neighborhood of z0.

20. Analytic Functions

Note:

• is analytic at each non-zero point in the finite plane

• is not analytic at any point since its derivative exists only at z = 0 and

not throughout any neighborhood.

- An entire function is a function that is analytic at each point in the entire finite

plane. Ex: every polynomial.

- If a function f fails to be analytic at a point z0 but is analytic at some point in

every neighborhood of z0 , then z0 is called a singular point, or singularity, of f .

2

1( )

( )

f zz

f z z

第 46頁

Ex: • z=0 is a singular point of

• has no singular point since it is no where analytic.

1( )f z

z

2( )f z z

Sufficient conditions for analyticity

• continuity of f throughout D.

• satisfication of Cauchy-Riemann equation.

• Sum, Product of analytic functions is/are analytic.

• Quotient of analytic functions is/are analytic . if denominator 0

A composition of two analytic function is analytic

( ) ' ( ) '( )d

g f z g f z f zdz

第 47頁

Thm : If everywhere in a domain D, then must be constant thr

oughout D.

pf : write

'( ) 0f z ( )f z

( ) ( , ) ( , )

'( ) 0

0by cauchy-Riemann

0

0

x x

y y

x y x y

f z u x y iv x y

Assume f z

u iv

v iu

u u v v

in D

at each point in D.

,x yu u

are x and y components of vector grad u.

grad u is always the zero vector.

( , )

( , )

( )

u x y a

similarly v x y b

f z a bi

u is constant along any line segment lying entirely in D.

第 48頁

21. Reflection Principle

predicting when the reflection of f (z) in the real axis corresponds to the

reflection of z. ( )f z

( )f z

Thm: f analytic in domain D which contains a segment of the x axis and is symmetric to that axis.

Then for each point z in the domain iff f (x) is real for each point x on the segment.

( ) ( )f z f z

pf: “ ” f (x) real on the segment

Let ( ) ( )F z f z

( ) ( , ) ( , ), ( ) ( , ) ( , )

(z) ( , ) ( , ) (3)

( , ) ( , ) ( , ) (4)

( , ) ( , ) ( , )

f z u x y iv x y F z U x y iV x y

f u x y iv x y

U x y u x y u x t where t y

V x y v x y v x t

write

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Now, since f (x+i t) is an analytic function of x + i t. , (5)x t t xu v u v

From (4),

x x

y t t

U u

dtV v v

dy

From (5), x yU V

Similarly, y t x x

y x

dtU u ut V v

dy

U V

From (5),

since , continuous

F(z) is analytic in D.

,x y y xU V U V

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Since f (x) is real on the segment of the real axis lying in D,

on the segment

,0 0v x

( ) ( ,0) ( ,0)

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

( ) ( )

F x U x iV x

u x iv x u x

F z f z

This is at each point z = x on the segment. (6)From Chap 6 (sec.58)A function f that is analytic in D is uniquely determined by its value along any line segment lying in D. (6) holds throughout D.

( ) ( ) ( ) ( ) ( ) (7)F z f z f z or f z f z

“ ” ( ) ( )

(3) ( ) ( , ) ( , )

f z f z

from f z u x y iv x y

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expand (7):

for (x,0) on real axis

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

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

( ,0) 0

( )

u x iv x u x iv x

v x

f x

is real on the segment of real axis lying in D.

Ex:

22

2

1 1

1,

z z

z z

x x

for all z.

since are real when x is real.

do not have reflection property since are not real when x is real.

2

,z i

iz

2

x i

ix

第 52頁

22. Haronic Functions

A real-valued function H of two real variables x and y is said to be harmonic in a given domain of the xy plane if, throughput that domain, it has continuous partial derivatives of the first and second order and satisfies the partial differential equation

( , ) ( , ) 0xx yyH x y H x y

known as Laplace's equation.

Applications: Temperatures T(x,y) in thin plate.

Electrostatic potential in the interior of a region of 3-D space.

Ex 1. is harmonic in any domain of the xy plane.( , ) sinyT x y e x

sin

sin

yxx

yyy

T e x

T e x

0xx yyT T

Thm 1. If a function is analytic in a domain D, then its component functions u and v are harmonic in D.

( ) ( , ) ( , )f z u x y iv x y

第 53頁

Pf: Assume f analytic in D. (then its real and imaginary components have

continuous partial derivatives of all orders in D)

“proved in chap 4”

,

,:

,:

,

0 0

x y y x

xx yx yx xxx

xy yy yy xyy

yx xy xy yx

xx yy yy xx

then u v u v

u v u v

u v u v

u u v v

u u v v

the continuity of partial derivatives of u and v ensures this (theorem in calculus)

Ex 2. ( ) sin cos

( , ) sin

y y

y

f z e x ie x

T x y e x

is entire.

in Ex 1. must be harmonic.Hence

cos sin

sin cos

y yx y

y yx y

u e x u e x

v e x v e x

第 54頁

is harmonic.

• If u , v are harmonic in a domain D, and their first-order partial derivatives satisfy Cauchy-Riemann equation throughout D, v is said to be harmonic conjugate of u. (u is not necessary harmonic conjugate of v )

• Thm 2( ) ( , ) ( , ) analyic ifff z u x y iv x y � v is a harmonic conjugate of u.

Ex 4: and are real & imag. parts of 2 2( , )u x y x y ( , ) 2v x y xy 2( )f z z

v is a harmonic conjugate of .

But is not analytic.2 2( ) 2 ( )

2 2

2 2

x y

x y

g z xy i x y

u y u x

v x v y

u

2 2Re ( ) ( ) ( )sin 2 cosyf z g z e x y x xy x

Ex 3. 2 2 2 2( ) ( ) 2g z z x iy x y i xy is entire.

so is f (z)g(z). ( f (z) in Ex 2 )

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• It can be shown that [Ex 11(b).] if two functions u and v are to be harmonic conjugates of each other, then both u and v must be constant functions.

• In chap 9, shall show that a function u which is harmonic in a domain of certain type always has a harmonic conjugate.

─ Thus, in such domains, every harmonic function is the real part of an analytic function.

─ A harmonic conjugate, when it exists, is unique except for an additive constant.

• If v is a harmonic conjugate of u in a domain D, then -u is a harmonic

conjugate of v in D.

( ) analytic

( ) ( , ) analytic

f z u iv

if z v iu x y

第 56頁

2 2 2

2

3

2 3

3 '( ) 3 3

'( ) 3

( )

( , ) 3 is a harmonic conjugate of .

y x y x

x x

x x c

v x y xy x c u

3 2

2

( , ) 3

6 6

6

since

6

3 ( )

xx yy

x

x y

y

x y

u x y y x y

u u

u xy

u v

v xy

v xy x

v u

harmonic

Ex 5. To obtain a harmonic conjugate of a given harmonic function