unit-iii complex variables - SNS Courseware

UNIT-III COMPLEX VARIABLES

19MAB102- INTEGRAL CALCULUS AND LAPLACE TRANSFORM SATHYA.S /AP/MATHS Page 1 of 13

2 MARKS QUESTION BANK

1. For what values of a, b and c the function f z x 2ay i bx cy is analytic.

f z x 2ay i bx cy

u+iv =(x-2ay)+i(bx-cy)

u=x-2ay v=bx-cy

u1

x

u2a

y

vb

y

vc

y

Given f(z) is analytic

x yu v ,

1 c, 2a b

c 1

y xu v

2a b

2a b

2. Define analytic function of a complex variable.

A function is said to be analytic at a point if its derivative exists not only

at that point but also in some neighborhood of that point.

3. If f(z) = u+iv be an analytic function P.T. u satisfies the Laplace equation.

Let f(z) = u+iv

Given f(z) is an analytic function

x yu v and y xu v

u v

x y

u v

y x

To prove: 2 2

2 2

u u0

x y

u v

x y

--------(1)

u v

y x

-----------(2)

Diff (1) partially w.r.t ‘x’ , we get

2 2

2

u v

x x x y

u v(3)

x yx

Diff (2) partially w.r.t ‘y’ , we get

2 2 2

2

u v

y y y x

u v v(4)

y x x yy



(3) + (4) implies 2 2 2 2

2 2

u u v v0

x y x yx y

Hence u satisfies the Laplace equation.

4. Show that 2 3u 3x y y is a harmonic function.

Given, 2 3

2

2

2 3

2

2

2 2

2 2

u 3x y y

u6xy

x

u6y

x

u3x y y

y

u6y

y

u u6y 6y 0

x y

Hence u is harmonic.

5. Find a function w such that w = u + iv is analytic, if xu e siny

Given x

u e siny

x1

z1

ux,y e sin y

x

z,0 e (0) 0

x2

z2

ux,y e cosy

y

z,0 e

z z1 2f(z) z,o dz i z,o dz 0 i e dz ie c

zf(z) ie c where c is the complex constant

6. Find the analytic function where real part is 2xe sin2y

Given xu e siny

2x1

2z1

2x2

2z2

u(x,y) 2e sin2y

x

(z,0) 2e (0) 0

u(x,y) 2e cos2y

y

(z,0) 2e

By Milne’s Thomson method,



z

2z1 2

z

ef (z) z,o dz i z,o dz 0 i 2e dz i2 c

2

f (z) ie c

7. Construct the analytic function f(z) for which the real part is 2xe cosy

Given 2xu e sin2y

x1

z z1

x2

z2

u(x,y) e cosy

x

(z,0) e cos0 e

u(x,y) e siny

y

(z,0) e (0) 0

By Milne-Thomson method

z1 2

z

f (z) z,o dz i z,o dz e dz 0

f (z) e c

8. Determine the analytic function where real part is 3 2 2 2u x 3xy 3x 3y 1

Given 3 2 2 2u x 3xy 3x 3y 1

2 21

21

2

2

u(x,y) 3x 3y 6x

x

(z,0) 3z 6z

u(x,y) 6xy 6y

y

(z,0) 0


1 2

2

3 2

3 2

f (z) (z,0)dz i (z,0)dz

(3z 6z)dz 0

z z3 6 c

3 2

f (z) z 3z c

9. . Find the analytic function f z u(x,y) iv x,y where real part is xy e cosy

Given xu y e cosy

x x1 2

z1 2

u u(x,y) e cosy ; (x,y) 1 e siny ;

x y

(z,0) e (z,0) 1




1 2

z

f (z) (z,0)dz i (z,0)dz

e iz c

10. .Find the image of the circle z 2 under the transformation w 3z .

Given w 3z u iv 3(x iy)

u 3x; v 3y

u vx ;......(1) y ......(2)

3 3

Also given 2 2

2 2

z 2

x iy 2 x y 2

x y 4.....(3)

Sub (1) & (2) in (3) 2 2

2 2 2

u v4

3 3

u v (6)

In the z-plane the circle z 2 transformed into 2 2 2u v (6) .

11. Write down the formula for finding an analytic function f (z) u iv , whenever

the real part is given by using Milne Thomson method.

1 2

1

2

v (z,0)dz i (z,0)dz c

u(x,y)(z,0)

x

v(x,y)(z,0)

x

12. Define bilinear transformation or [Mobius Transformation]

The transformation az b

wcz d

, ad bc 0 where a, b, c, d are complex

numbers is called a bilinear transformation.

13. Verify the function xx,y e sin y is harmonic or not.

Given xe siny

To prove 2 2

2 20

x y

x xe siny e cosyx x

2 2x x

2 2e siny e siny

x y

2 2x x

2 2e siny e siny

x y



= 0

satisfies the Laplace equation.

is harmonic.

14. Find the critical points for the transformation 2w z z .

Given 2w z z ............. 1

Critical points occur at dw

0dz

Diff. (1) w.r.to z,

dw

2w z zdz

dw

2w 2z .......... 2dz

dw0

dz

0 2z

z2

from (2),

dz 2w

dw 2z

dz0

dw

2w

02z

2w=0

w=0

from (1) z z 0

z z 0

z ,

The critical points are , &2

.

15. Find the constant a so that 2 2u x,y x y xy is harmonic.

Given u is harmonic.

i.e u satisfies Laplace equation 2 2

2 2

u u0

x y

u u2ax y 2y x

x y



2 2

2 2

u u2a 2

x y

2 2

2 2

u u0

x y

2a-2=0

2(1-a)=0

1-a=0

a=1.

16. Find the invariant of bilinear transformation 1 z

w1 z

.

1 zw

1 z

Replace w by z, 1 z

z1 z

2

2

2

z z 1 z

1 z z z 0

z 1 0

z i

The invariant points are i.

17. Conformal Mapping

A mapping w f z is said to be conformal at 0z z if it preserves the angle

between any two curves through 0z in z plane both in magnitude &

direction.

18. Find the image of x=1 under the transformation of 1

wz

.

1z

w

2 2

1x iy

u iv

1 u iv

u iv u iv

u iv

u v

2 2 2 2

u vx , y

u v u v

Given x=1

2 2

u1

u v

2 2u v u 0



This is the equation of the circle with centre at 1

,02

and radius 1

r2

.

19. Find the critical points of the transformation 1

w zz

.

Let 1

w zz

2

2 2

dw 1 z 11

dz z z

2

2

dz z

dw z 1

Now the critical points are 1,0 .

20. Find the critical points of the transformation w f z sin z .

Given w sinz

dwcos z

dz

dw0 cosz 0

dz

1z cos 0

2n 1 , n 1,2, ......2

dz 1

dw cos z

dz 10 0

dw cosz

1=0 which is impossible.

The critical points are 2n 1 , n 1,2,.......2

21. Find the image of the region 1y under the transformation z)i1(w .

Given z)i1(w

)iyx)(i1(ivu

)xy(i)yx(

yixiyx

2

vuy

y2vu

,yxu

2

vux

x2vu

,xyv

Hence image region 1y is 12

vu

is the w-plane.



22. Find the image of the circle z under the transformation z5w .

z5w

5

vyy5v

5

uxx5u

)iyx(5ivu

Given

2222

222

222

)5(25vu

5

v

5

u

yxiyx

z

Hence the image of z is 2222)5(25vu

23. Find the image 03yx2 under the transformation i2zw .

Given i2zw

)2y(ixivu

i2)iyx(ivu

ux

,xu

2vy

2yv

05vu2

03)2v(u2

03yx2Given

Hence the image of 03yx2 is 05vu2

24. Obtain the invariant point of the transformation z

22w

The invariant points are given by

i1z

i12

i22

2

842z

z

2z2z;

z

22z

2



25. Find the fixed point of the transformation z

9z6w

The fixed points are given by replacing zw

3,3z0)3z(

9z6zz

9z6z

z

9z6w,.,ie

2

2

The fixed points are 3,3.

26. Find the bilinear transformation that maps the points 0,i, onto ,i,0

Respectively.

Given

321

321

w,iw,0andw

0z,iz,z

Let the transformation be,

z

1w

2

iw

z

i

i

w

)10)(z(

)i)(10(

)i)(i0(

)1)(w(

1i

)0z(

)0i(z

)0i(1w

1i

)0w(

1z

zz)zz(

)zz(1z

zz

)ww(1w

ww

1w

ww)ww(

)zz)(zz(

)zz)(zz(

)ww)(ww(

)ww)(ww(

2

1

213

321

1

`123

3

3

231

`123

321

`123

321

27. Define a critical point of the bilinear transformation.

The point at which the mapping )z(fw critical point of the mapping. The

transformation )z(fw is conformal, )w(fz1 is also conformal its critical

points occur at .0dw

dz

Hence the critical paths of )z(fw occur at 0dz

dw and .0

dw

dz



28. Test the analyticity of the function zsinw

Let zsin)z(fw

ysinhxcosiycoshxsinivu

iysinxcosiycosxsinivu

)iyxsin(ivu

ysinhxsinu

ycoshxcosu

ycoshxsinu

y

x

ycoshxcosv

ysinhxsinv

ysinhxcosv

y

x

xyyx vuandvu

C.R equations are satisfied.

Hence the function is analytic.

29. Show that 2

z)z(f is differentiable at z=0 but not analytic.

Let iyxzandiyxz

0v;y2u

0v;x2u

0v;yxu

0i)yx(z)z(f.,ie

yxzzz

yy

xx

22

222

222

So f(z) may be differentiable only at z=0. Now ,y2u,x2u yx and

0v,0v yx are continuous everywhere and in particular at

(0,0).

Hence the sufficient conditions for differentiability are satisfied by f(z) at

z=0.So f(z) is differentiable at z=0 only and not analytic there.

30. Write the necessary condition for f(z) to be analytic.

The necessary condition for a complex function )y,x(iv)y,x(u)z(f to be

analytic in a region R are

yxyx uv,vu.,ie

y

u

x

vand

y

v

x

u



31. Write the sufficient condition for f(z) to be analytic.

If the partial derivatives yxyx uandv,u,u are all continuous in D and

yxyx uvandvu . Then the function f(z) is analytic in domain D.

32. Show that an analytic function with constant real part is constant.

Given u=constant

.ttanconsais)z(f

c)z(f,0)z(f

0i0x

vi

x

u)z(f

0v0u

0v0u

,REquationsByC

0y

uand0

x

u

;cuie.,

constant. a is f(z) :prove To

'

xy

yx

33. Determine whether the function )yx(ixy222 is analytic or not.

Let )yx(ixy2)z(f22

y2y

vx2

y

u

x2x

vy2

x

u

yxv,x2u22

xyyx vuandvu

C-R equations are not satisfied.

Hence f(z) is not an analytic function.

34. If ivu is analytic, show that iuv is also analytic.

Given ivu is analytuic. Ie., C-R equations are satisfied.

)2.........(vu

)1.(..........vu

xy

yx

Since the derivatives of u& v exist.therefore it is continuous.

To prove iuv is analytic.

)4.....(..........uv

)3.(..........uv.,ie

xy

yx



Results (3) & (4) follows fom (1) & (2).Since the derivatives of u & v exists

from (1) & (2) the derivatives of u & v should be continuous.

35. Find the image of the transformation z

1w .

Given w

1z

z

1w

2222

22

vu

vy;

vu

ux

vu

ivuiyx

ivu

1iyx

Given

0u610vu

u61

0vu

u6

)vu(

v

)vu(

u

0y4yx

0x6yx3iy)3x(

33iyx

33z

22

22222

2

222

2

22

22

Which is a straight line.

6

1u which is the required image.

36. Under the transformation z

1w find the image of 2i2z

Given w

1z

z

1w

2222

22

vu

vy;

vu

ux

vu

ivuiyx

ivu

1iyx



4

1v0v410

vu

v41

0vu

v4

)vu(

v

)vu(

u

0y4yx

4)2y(x2)2y(ix

2i2iyx

2i2z

22

22222

2

222

2

22

22

Which is a straight line in w-plane.

37. Find whether zlog is analytic or not.

1u0u

0vr

1u

vrloguHere

)relog(ivu

zlogw

rr

i

C-R Equations are ,

u

r

1

r

vand

r

v

r

1

r

u

We know that

z

1

r

e)z(f

)0(ir

1e

r

vi

r

ue)z(f

i'

ii'

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unit-iii complex variables - SNS Courseware