SRM UNIVERSITY Department of Mathematics

Prepared by Mr R.Manimaran,Assistant Professor,Department Of Mathematics,SRM UNIVERSITY, City Campus,Vadapalani,Chennai-26 Page 1

SRM UNIVERSITY

RAMAPURAM PART- VADAPALANI CAMPUS, CHENNAI – 600 026

Department of Mathematics

Sub Title: ADVANCED CALCULUS AND COMPLEX ANALYSIS

Sub Code:15 MA102

Unit -IV - ANALYTIC FUNCTIONS

Part – A

1. Cauchy-Riemann equations are

(a) yx vu and xy vu (b) yx vu and xy vu (c) xx vu and yy vu

(d) yx vu and xy vu Ans : (a)

2. If ivuzf )( in polar form is analytic then r

u

is

(a)

v (b)

vr (c)

v

r

1 (d)

v Ans : (c)

3. If ivuzf )( in polar form is analytic then

uis

(a) r

v

(b)

r

v

r

1 (c)

r

v

(d)

r

vr

Ans : (d)

4. A function u is said to be harmonic if and only if

(a) 0 yyxx uu (b) 0 yxxy uu (c) 0 yx uu (d) 022 yx uu Ans : (a)

5. A function )(zf is analytic function if

(a) Real part of )(zf is analytic (b) Imaginary part of )(zf is analytic

(c) Both real and imaginary part of )(zf is analytic (d) none of the above Ans : (c)

6. If u and v are harmonic functions then ivuzf )( is

(a) Analytic function (b) need not be analytic function

(c) Analytic function only at 0z (d) none of the above Ans : (a)

7. If )()( cybxiayxzf is analytic then a,b,c equals to

(a) 1c and ba (b) 1a and bc (c) 1b and ca (d) 1 cba

Ans : (a)

8. A point at which a function ceases to be analytic is called a

(a) Singular point (b) Non-Singular point (c) Regular point (d) Non-regular point

Ans : (a)

9. The function ||)( zzf is a non-constant

(a) analytic function (b) nowhere analytic function (c) non-analytic function (d) entire function

Ans : (b)


10. A function v is called a conjugate harmonic function for a harmonic function u in whenever

(a) ivuf is analytic (b) u is analytic (c) v is analytic (d) ivuf is analytic

Ans : (a)

11. The function 3223)( cybxyyaxxiyxf is analytic only if

(a) 3,3 bia and ic (b) 3,3 bia and ic (c) 3,3 bia and ic

(d) 3,3 bia and ic Ans : (c)

12. There exist no analytic functions f such that

(a) Re xyzf 2)( (b) Re xyzf 2)( 2 (c) Re22)( xyzf (d) Re xyzf )(

Ans : (b)

13. If yeax cos is harmonic, then a is

(a) i (b) 0 (c) -1 (d) 2 Ans : (a)

14. The harmonic conjugate of 23 32 xyxx is

(a) 323 yyxx (b)

3232 yyxy (c) 323 yyxy (d)

3232 yyxy Ans : (b)

15. The harmonic conjugate of )1(2),( yxyxu is

(a) Cxyx 222 (b) Cyyx 222

(c) Cyyx 222 (d) Cyyx 222

Ans : (d)

16. harmonic conjugate of xeyxu y cos),( is

(a) Cyex cos (b) Cyex sin (c) Cxe y sin (d) Cxe y sin Ans : (d)

17. If the real part of an analytic function )(zf is ,22 yyx then the imaginary part is

(a) xy2 (b) xyx 22 (c) yxy2 (d) xxy2 Ans : (d)

18. If the imaginary part of an analytic function )(zf is ,2 yxy then the real part is

(a) yyx 22 (b) xyx 22

(c) xyx 22 (d) yyx 22

Ans : (c)

19. zzf )( is differentiable

(a) nowhere (b) only at 0z (c) everywhere (d) only at 1z Ans : (a)

20. 2

)( zzf is differentiable

(a) nowhere (b) only at 0z (c) everywhere (d) only at 1z Ans : (b)

21. 2

)( zzf is

(a) differentiable and analytic everywhere

(b) not differentiable at 0z but analytic at 0z

(c) differentiable at 1z and not analytic at 1z only

(d) differentiable at 0z but not analytic at 0z Ans : (d)

22. If

,0,0

;0,)()(

22

zif

zifyx

xy

zf then )(zf is

(a) continuous but not differentiable at 0z (b) differentiable at 0z

(c) analytic everywhere except at 0z (d) not differentiable at 0z Ans : (d)

23. zezf )( is analytic


(a) only at 0z (b) only at iz (c) nowhere (d) everywhere Ans : (d)

24. )sin(cos yiyex is

(a) analytic (b) not analytic (c) analytic when 0z (d) analytic when iz Ans : (b)

25. If )(zf is analytic, then )(zf is

(a) analytic (b) not analytic (c) analytic when 0z (d) analytic when 1z Ans : (a)

26. The points at which )23(

)()(

2

2

zz

zzzf is not analytic are

(a) 0 and 1 (b) 1 and -1 (c) i and 2 (d) 1 and 2 Ans : (d)

27. The points at which 1

1)(

2

zzf is not analytic are

(a) 1 and -1 (b) i and -i (c) 1 and i (d) -1 and -i Ans : (b)

28. The harmonic conjugate of 22log yxu is

(a) 22 yx

x

(b)

22 yx

y

(c)

y

x1tan (d)

x

y1tan Ans : (d)

29. If ),2()( zzzf then )1( if

(a) 0 (b) i (c) -i (d) 2 Ans : (b)

30. If zzf )( then )43( if

(a) 0 (b) 5 (c) -5 (d) 12 Ans : (b)

31. Critical points of the bilinear transformation dzc

bzaw

are

(a) a,c (b) ,d

c (c) ,

d

c (d) None of these Ans : (c)

32. The points coincide with their transformations are known as

(a) fixed points (b) critical points (c) singular points (d) None of these Ans : (a)

33. dzc

bzaw

is a bilinear transformation when

(a) 0bcad (b) 0bcad (c) 0cdab (d) None of these Ans : (b)

34. z

w1

is known as

(a) inversion (b) translation (c) rotation (d) None of these Ans : (a)

35. zw is known as

(a) inversion (b) translation (c) rotation (d) None of these Ans : (b)

36. A translation of the type zw where and are complex constants, is known as a

(a) translation (b) magnification (c) linear transformation (d) bilinear transformation

Ans : (c)

37. A mapping that preserves angles between oriented curves both in magnitude and in sense is called a/an .....

mapping.

(a) informal (b) isogonal (c) conformal (d) formal Ans : (c)

38. The mapping defined by an analytic function )(zf is conformal at all points z except at points where

(a) 0)(' zf (b) 0)(' zf (c) 0)(' zf (d) 0)(' zf Ans : (a)


39. The fixed points of the transformation 2zw are

(a) 0,1 (b) 0,-1 (c) -1,1 (d) –i,i Ans : (a)

40. The invariant points of the mapping z

zw

2 are

(a) 1,-1 (b) 0,-1 (c) 0,1 (d) -1,-1 Ans : (c)

41. The fixed points of 1

1

z

zw are

(a) 1 (b) i (c) 0,-1 (d) 0,1 Ans : (b)

42. The mapping z

zw1

transforms circles of constant radius into

(a) confocal ellipses (b) hyperbolas (c) circles (d) parabolas Ans : (a)

43. Under the transformations ,1

zw the image of the line

4

1y in z-plane is

(a) circle 0422 vvu (b) circle 422 vu (c) circle 222 vu (d) none of these

Ans : (a)

44. The bilinear transformation that maps the points ,,0 i respectively into ,1,0 is w

(a) z

1 (b) –z (c) –iz (d) iz Ans : (c)

45. The bilinear transformation which maps the points 1,0,1 zzz of z - plane into 1,0, wwiw of

w plane respectively is

(a) izw (b) zw (c) )1( ziw (d) none of these Ans : (a)

Part – B

1. Show that the function f (z) = is no where differentiable.

Solution: Given u+iv = x-iy

u=x v=-y

ux =1 vx =-1

uy =0 vy =-1

ux vy

C-R equations are not satisfied.

f (z) = is no where differentiable.

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

Solution: Let

= z =

v=0

ux =2x vx =0

uy =2y vy = 0

ux = vy and uy = - vx are not satisfied everywhere except at z=0


So f (z) may be differentiable only at z=0. Now ux,vx,uy,vy are continuous everywhere and in

particular at (0,0).

3. Test the analyticity of the function w=sin z.

Solution: w=f (z) =sin z

u+iv = sin(x+iy)

=sin x cosiy+ cos x siniy

= sin x coshy+i cos x sinhy

u= sin x cushy v= cos x sinhy

ux = cosx cushy vx = -sinx sinhy

uy = sinx sinhy vy = cosx cushy

ux = vy and uy = - vx

C-R equations are satisfied.

The function is analytic.

4. Verify the function 2xy+i( ) is analytic or not .

Solution: u=2xy v=

ux = 2y vx = 2x

uy = 2x vy = -2y

ux vy and uy - vx

C-R equations are not satisfied.

The function is not analytic.

5. Test the analyticity of the function f (z) = .

Solution: f (z) =

u+iv = = = (cosy+isiny)

u= cosy v= siny

ux = cosy vx = siny

uy = siny vy = cosy

ux = vy and uy = - vx

The function is analytic.

6. If u+iv = is analytic, show that v-iu and –v+iu are also analytic.

Solution: Given u+iv is analytic.

C-R equations are satisfied.

i.e. ux = vy ------------------- (1) and uy = - vx------------------------------(2)

To prove v-iu and –v+iu are also analytic

For this, we have to show that

(i) ux = vy and -uy = vx (ii) ux = vy and uy = - vx


These results follow directly from (1) & (2) by replacing u by v and –v and v –u and u respectively.

v-iu and –v+iu are analytic.

7.Give an example such that u and v are harmonic but u+iv is not analytic.

Solution: Consider the function w= = x-iy

u=x v=-y

ux vy , The function f(z) is not analytic. But and gives u and v are

harmonic.

8.If f (z) = u(x,y) +v(x,y) is an analytic function. Then the curves u(x,y) = c1and v(x,y) =c2 where c1and

c2 are constants are orthogonal to each other.

Solution: If u(x,y) = c1 , then du = 0

But by total differential operator we have

du =

(Say)

Similarly, for the curve v(x,y) =c2 we have

(Say)

For any curve gives the slope, Now the product of the slopes is

u(x,y) = c1and v(x,y) =c2 intersect at right angles (i.e) they are orthogonal to

each other.

9.Find the analytic region of f (z) =

Solution: Given f (z) =

u= v=


Now ux = vy and uy = - vx

2 =2 - 2 = -2

x-y=1 x-y=1

Analytic region of f (z) is x-y=1

10.Find a function w such that w=u+iv is analytic, if u= .

Solution: Given u=

= 0-i

f (z) = -i

11. Prove that u= satisfies Laplace’s equation.

Solution: Given u=

u satisfies Laplace’s equation.

12. If u=log ( ) find v and f (z) such that f (z) = u+iv is analytic.

Solution: Given u=log ( )

=


f (z) = 2log z +c

To find the conjugate harmonic v

We know that dv =

= - [by C – R equations]

dv = dx

Integrating

V = 2 +c

13. Find the critical points for the transformation

Solution: Given

2w

w

Critical points occur at

Also

The critical points occur at

=0

z = and z =

The critical points occur at z = , and .

14. Find the image of the circle under the transformation w=3z.

Solution: w=3z

u+iv = 3(x+iy)

u=3x v=3y

x= y=

Given


.

maps to a circle in w- plane with centre at the origin and radius 6.

15. Find the fixed points for the following transformation w

Solution: Fixed points are obtained from

f (z) = z

z =

Z = are the fixed points.

Part – C

1. If f(z) is an analytic function of z, prove that

(i) =0

(ii)

(iii)

Proof: If z = x+iy then


=

=

(i). =

= 2

= 2

= 2

=2 = 0

(ii)

=

=

=

=

=2f’ (z)

(iii). =

=

=

= 4 =

2. Prove that the function u = satisfies laplace’s equation and find the

corresponding analytic function f (z) = u+iv.

Solution: Given u =

+


+

=

u satisfies Laplace equation.

To find f (z): u is given

Step 1:

+

Step 2:

Step3:

Integrating f (z) =

=

3. Prove that the function v = is harmonic and determine the corresponding

analytic function of f(z)

Solution: Given v =

Step 1:

+y

Step 2:

Step3:


Integrating f (z) = -z

To prove v is harmonic

+y

=

4. Prove that the function u = +1 satisfies laplace’s equation and find the

corresponding analytic function f (z) = u+iv.

Solution: Given u = +1

= -6x-6

u satisfies Laplace equation.


Step 1:

Step 2:

Step3:


Integrating f (z) =

5. If u= find the corresponding analytic function f(z) u+iv.

Solution: Given u=


Step 1:

Step 2:

=

Step3:

Integrating f (z) = tan z

6. Determine the analytic function f(z)=u+iv such that v =

Solution: f (z) =u+iv ----------------------------- (1)

i f(z) = iu-v ------------------------------(2)

Adding (1) and (2)

F (z) = U+iV

Where F (z) = , U= V =

Given v =

Step 1:

Step 2:

Step3:


Integrating F (z) =

(1+i) f (z) =

7. Find the analytic function f(z) = u+iv given that

Solution: 3f (z) = 3u+3iv ---------------------- (1)

2if (2) = 2iu-2v ----------------------- (2)

Adding (1) and (2)

(3+2i) f (z) = (3u-2v) +i (2u+3v)

F (z) = U+iV

Where F (z) = (3+2i) f (z) , U= V =

Given

i.e., V =

Step 1:

Step 2:

Step3:

Integrating F (z) = i cot z

(3+2i) f (z) = i cot z

f (z)

f (z)

8. Find the bilinear transformation that maps the points z = 1, i, -1 into the points w=i, 0, -i

respectively. Hence find the image of

Solution: The bilinear transformations is given by


w=

w= is the required bilinear transformation.

To find the image of

Now w=

Since

Put w=u+iv we get

1-2u+ + 1+2u+ +

The interior of the unit circle (ie) maps into the half plane a>0 of the w- plane.

9. Find the mobius transformation that maps the points z = 0, 1, into the points w=-5, -1, 3

respectively. What are the invariant points of the transformation?


Since the above relation becomes.


w+5=3z-5

w= is the required bilinear transformation.

To get the invariant points, put w=z

z=

Solving for z,

Z =

= 1

The invariant points are z = 1

10. Find the image of under the transformation.

Solution: Given w = 1/z

z = x+iy and w = u+iv

And

=2

--------------------------- (1)

Substituting x and y values in equation (1), we get

This is the straight line equation in the w-plane.


11.Show that the transformation w = 1/z transforms circles and straight line in the z-plane into

circles or straight lines in the w-plane.

Solution: w = 1/z

z = x+iy and w = u+iv

Consider the equation, ----------------------- (1)

If a equation (1) represents a circle and if a=0, it represents a straight line, substituting the

valus of x and y in (1)

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

If d 0, equation (2) represents a circle and if d=0, it represents a straight line. The various cases

are discussed in detail.

Case (i): When a d 0

Equation (1) and (2) represents circles in the z-plane and w-plane not passing through the origin.

The transformation w =1/z transforms circles not passing through the origin into circles not

passing through the origin.

Case (ii): When a d=0

The equation (1) is circle through the origin in z-plane and (2) is a straight line; not passing

through the origin in the w-plane.

Circles passing through the origin in the z-planes maps into the straight lines, not passing

through the origin in the w-plane.

Case (iii): When a = d 0

Equation (1) represents a straight line not passing through the origin and (2) represents a circle in

the w-plane passing through the origin. Thus lines in the z-plane not passing through the origin

map into circles through the origin in the w-plane.

Case (iv): When a = d= 0

Equation (1) and (2) represents straight lines passing through the origin. Thus the lines through the

origin in the z- plane map into the lines through the origin in the w- plane.


12. If u= , v= prove that u and v are harmonic functions but u+iv is not an

analytic function.

Solution: Given u= and v=

To prove u and v are harmonic

Now

u is harmonic.

Now v=

is harmonic.

Now we show that u+iv is not analytic.

Now and

It is true from the above relation.

u+iv is not an analytic function.

13. Prove that u = is harmonic and find its conjugate harmonic.

Solution: Given u =

To prove

Consider u =

Differentiating this w.r.to x and y partially, we get


u is harmonic.

To find the harmonic conjugate

Let v (x,y) be the conjugate harmonic. Then w = u+iv is analytic.

By C-R equations, and =

We have

dv =

dv =

dv =

Integrating, we get

V =

14. . Find the bilinear transformation that maps the points z = -1, 0, 1 into w=0, i, 3i

respectively.


2w (z-1) = (w-3i) (z+1)

w [2z-2-z-1] = (z+1)(-3i)

w = is the required bilinear transformation.

15. Find the bilinear transformation that maps the points z = 0, 1, into the points

w=-1,-2-i, i respectively.



Since the above relation becomes.

2w+2=-zw+iz

W (z+2) = iz-2

w = is the required bilinear transformation.

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SRM UNIVERSITY Department of Mathematics