Shroff S.R. Rotary Institute Of Chemical Technology

Sub:-COMPLEX VARIABLES AND NUMERICAL METHODSSUBJECT CODE:-2141905

Topic:-ANALYTIC FUNCTION, Cauchy-riemann equations, Harmonic Conjugate functions.

Branch:-Mechanical Engineering( 4th Sem.)

Prepared by:-Malaysinh Borasiya(130990119004)

Chintan Charola(130990119005) Mayurrajsinh

Chauhan(130990119006)

ANALYTIC FUNCTIONA single valued function which is defined and

differentiable at each point of a domain D is said to be analytic in that domain.

A function is said to be analytic at a point if its derivative exists not only at that point but in some neighbourhood of that point.

The point where the function is not analytic is called a singular point of the function.

Notes Let f(z) and g(z) be analytic functions in some

domain D;

(i) and f(z).g(z) are analytic functions in D.

(ii) is analytic in D except where g(z)=0.

(iii) f[g(z)] and g[f(z)] are analytic functions in D.

)()( zgzf

)(

)(

zg

zf

Example -1 : State where the function is not analytic?

Solution:- f(z) is not analytic at as the function is not defined at these points. Hence, the points are singular points.

Example-2 : State where the function is not analytic?

Solution:- The function is not analytic at the points and and

1

2)(

2 z

zzf

iz iz

)22)(2(

1)(

2

2

zzz

zzf

02z 0222 zz

2z iz

12

842

CAUCHY-RIEMANN PARTIAL DIFFERENTIAL EQUATIONS

Theorem : If a function f(z) = u(x,y) + iv(x,y) is analytic at any point z = x+iy, then the partial derivatives ux , uy , vx , vy should exist and satisfy the equations ux =vy , uy = -vx or

Proof : Let f(z) be an analytic function at any point z.

exist

Now

…..(1)

x

v

y

u

y

v

x

u

,

z

zfzzf

zzf

)()(

0

lim)('

z

ivuvviuu

zzf

)()()(

0

lim)('

z

viu

z

0

lim

CAUCHY-RIEMANN PARTIAL DIFFERENTIAL EQUATIONS For equation (1) limit exist along any path as Consider the path along real axis(z=x).

from (1),

….(2) Now consider the path along y axis i.e. z=iy

from (1), ….(3)

.0z

00 xz

x

viu

xzf

0

lim)('

x

vi

x

u

00 yz

y

v

y

ui

yi

viu

yzf

0

lim)('

CAUCHY-RIEMANN PARTIAL DIFFERENTIAL EQUATIONS

It is obvious that the limits (2) and (3) are same.

and

y

v

y

ui

x

vi

x

u

yx vu xy vu

Example 3 : Use C-R equation concept to find derivative of

Solution:- We have

and So, C-R equations are satisfied everywhere in z-plane. exists everywhere. Thus we get

Example 4 : Show that neither nor is an analytic function. Solution:- We have

So, C-R equation is not satisfied. is not analytic.

.)( 2zzf

xyiyxzzf 2)( 222 xyyxvyxyxu 2),(,),( 22

yx vxu 2xy vyu 2

)(' zf

zyixiuvivuzf yyxx 222)('

zzf )( zzf )(

iyxzzf )(

yyxvxyxu ),(,),(

1,0,0,1 yxyx vvuu yx vu

zzf )(

Continue… Now

So, C-R equation is not satisfied. is not analytic.

Example 5 : Show that is an analytic function, find

Solution:- We have,

and . So, w is analytic.

22)( yxzzf

0),(,),( 22 yxvyxyxu

0,22

yx vyx

xu yx vu

zzf )(

2222 yx

iy

yx

xw

.

dz

dw

2222,

yx

yv

yx

xu

,)( 222

22

yx

xyux

222 )(

2

yx

xyvx

222

22

)( yx

xyvy

,

)(

2222 yx

xyu y

yx vu xy vu

Continue… Now,

Example 6 : Check whether the following functions are analytic or not at any point:

Solution:- (a) we have

is not analytic anywhere. (b)We have

is not analytic anywhere.

xx ivudz

dw

222222

22

)(

2

)( yx

xyi

yx

xy

.2)()()()( 2ixyxzfbezfa z

)sin(cos)( yiyeeezf xiyxz

,cos yeu x ,sin yev x,cos yeu x

x yev xy cos

yx vu zezf )(

22)( ixyxzf

,2

,2

xu

xuxyv

xyv

y 2

2

yx vu 22)( ixyxzf

Example 7 : Examine the analyticity of sinh z.

Solution:- Let z0 be any point in the domain.

exist at any point z. is analytic.

0

0

00

0

0

sinhsinhlim)()(lim

zz

zz

zzzz

zfzf

zz

0

00

0

2sinh

2cosh2

lim

zz

zzzz

zz

0

0

0

0

0

0

cosh

2

2sinh

lim

2cosh

lim

z

zz

zz

zz

zz

zz

)(' zf)(zf

Continue…. OR We have

and So, C-R equation is satisfied for any point. So, f(z) is analytic function.

)sinh(sinh)( iyxzzf xyixy coshsinsinhcos

,sinhcos),( xyyxu xyyxv coshsin),( ,coshcos xyux xyvy coshcos,sinhsin xyu y ,sinhsin xyvx

yx vu xy vu

POLAR FORM OF C.R. EQUATIONSWe have

are C-R equations in polar form.

and

sin,cos ryrx

x

yyxr 122 tan,

r

vu

r

v

rr

u

1,

1

r

vi

r

uezf i)('

Harmonic Function & Conjugate harmonic functionHarmonic Function:- A function is said to be harmonic in a domain D if (1) Satisfy Laplace’s equation and (2) are continuous functions of x and y in D.

Conjugate harmonic function:- If f(z)=u+iv is an analytic function of z, then v is called a conjugate harmonic function of u and u in its turn is termed a conjugate harmonic function of v. Or u and v are called conjugate harmonic functions.

),( yx

),( yx 0 yyxx

yyxyxx ,,

Example 8 : Is the function u=x sin x cosh y - cos x sinh y harmonic?

Solution:- We have

And

also are continuous functions. So, u is a harmonic function.

yxyyxxyxyxu

yxyyxxyxu

yxyyxxu

xx

x

sinhcoscoshsincoshcoscoshcos

sinhsincoshcoscoshsin

sinhcoscoshsin

yxyyxxyx sinhcoscoshsincoshcos2

yxyyxyxyxxu

yxyyxyxxu

yy

y

sinhcoscoshcoscoshcoscoshsin

coshcossinhcossinhsin

yxyyxyxx sinhcoscoshcos2coshsin 0 yyxx uu

yyxyxx uuu ,,

Example 9 : Show that is harmonic.

Solution:- We have

….(1)

22 yx

xu

22 yx

xu

222

22

222

22

)()(

)2()1)((

yx

xy

yx

xxyxux

422

2222222

)(

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

yx

xyxxyxyxuxx

322

23

322

3223

422

222222

)(

62

)(

4422

)(

)](4)2)()[((

yx

xyx

yx

xxyxyx

yx

xyxxyxyx

Continue…

…(2)

So, From equation (1) & (2),

Also are continuous functions.

So, u is a harmonic function.

422

22222

222

)(

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

)(

2

yx

yyxxyxyxu

yx

xyu

yy

y

322

23

322

223

422

22222222

)(

62

)(

]822[

)(

)](8)2)([()(

yx

xyx

yx

xyxyx

yx

yxxyxyxyx

0 yyxx uu

yyxyxx uuu ,,

Example 10 : Determine a and b such that is harmonic and find its conjugate harmonic.

Solution:- We have

since u is harmonic function,

and b assumes any value

Now,

bxyaxu 3

bxyaxu 3

0

63 2

yyy

xxx

ubxu

axubyaxu

0 yyxx uu

006 aax

bxubyubxyu yx ,

dyvdxvdv yx

bydybxdx

dyudxu xy

Continue…

cy

bxbv

22

22

c

bybxibxyivuzf

22)(

22

Method of constructing a regular functionIf only the real part of an analytic function f(z) is given then

where c is a real constant.

Replace x by and y by to find and put x=y=0 to find u(0,0)

in u(x,y)

ciui

zzuzf

)0,0(

2,2

2)(

2

z

i

z

2

i

zzu

2,2

Example 11 : Find an analytic function if

Solution:- We have

and u(0,0)=0

.33 xyxu ivuzf )(

xyxu 33

izz

i

zzz

i

zzu 2

33

4

3

8223

22,2

ciui

zzuzf

)0,0(

2,2

2)(

ciizz

zf 23

2

3

4)(

Example 12 : Show that the function is harmonic and find the corresponding analytic function.

Solution:- We have

So, u is harmonic.

Now,

xyxu 22

xyxu 22

22

212

yyy

xxx

uyu

uxu

0 yyxx uu

0)0,0(

222442,2

222

u

zzzzz

i

zzu

cizzciui

zzuzf

2)0,0(

2,2

2)(

Thank you