analytic functions mcq +notes

8/17/2019 Analytic Functions MCQ +Notes

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

SRM UNIVERSITYRAMAPURAM PART- VADAPALANI CAMPUS, CHENNAI – 600 026

Department of MathematicsSub Title: ADVANCED CALCULUS AND COMPLEX ANALYSIS

Sub Code:15 MA102

Unit -IV - ANALYTIC FUNCTIONS

Part – A 1. Cauchy-Riemann equations are

(a) y x vu and x y vu (b) y x vu and x y vu (c) x x vu and y y vu

(d) y x vu and x y vu Ans : (a)

2. If ivu z f )( in polar form is analytic then r u

is

(a)

v

(b)

vr (c)

v

r

1 (d)

v

Ans : (c)

3. If ivu z f )( in polar form is analytic then

u

is

(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 yy xx uu (b) 0 yx xy uu (c) 0 y x uu (d) 022

y x uu Ans : (a)

5. A function )( z f is analytic function if

(a) Real part of )( z f is analytic (b) Imaginary part of )( z f is analytic

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

6. If u and v are harmonic functions then ivu z f )( is(a) Analytic function (b) need not be analytic function(c) Analytic function only at 0 z (d) none of the above Ans : (a)

7. If )()( cybxiay x z f 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 ||)( z z f is a non-constant

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


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10. A function v is called a conjugate harmonic function for a harmonic function u in whenever(a) ivu f is analytic (b) u is analytic (c) v is analytic (d) ivu f is analytic

Ans : (a)

11. The function 3223)( cybxy yax xiy x f 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 x y z f 2)( (b) Re x y z f 2)( 2 (c) Re 22)( x y z f (d) Re x y z f )(

Ans : (b)

13. If ye ax cos is harmonic, then a is(a) i (b) 0 (c) -1 (d) 2 Ans : (a)

14. The harmonic conjugate of 23 32 xy x x is

(a)323 y y x x (b)

3232 y y x y (c)323 y y x y (d) 3232 y y x y Ans : (b)

15. The harmonic conjugate of )1(2),( y x y xu is

(a) C x y x 222

(b) C y y x 222

(c) C y y x 222

(d) C y y x 222

Ans : (d)

16. harmonic conjugate of xe y xu y cos),( is

(a) C ye x cos (b) C ye x sin (c) C xe y sin (d) C xe y sin Ans : (d)

17. If the real part of an analytic function )( z f is ,22 y y x then the imaginary part is

(a) xy2 (b) xy x 22

(c) y xy2 (d) x xy2 Ans : (d)

18. If the imaginary part of an analytic function )( z f is ,2 y xy then the real part is

(a) y y x 22

(b) x y x 22 (c) x y x 22 (d) y y x 22 Ans : (c) 19. z z f )( is differentiable

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

20. 2

)( z z f is differentiable

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

21. 2

)( z z f is

(a) differentiable and analytic everywhere(b) not differentiable at 0 z but analytic at 0 z (c) differentiable at 1 z and not analytic at 1 z only(d) differentiable at 0 z but not analytic at 0 z Ans : (d)

22. If,0,0

;0,)()( 22

zif

zif y x xy

z f then )( z f is

(a) continuous but not differentiable at 0 z (b) differentiable at 0 z (c) analytic everywhere except at 0 z (d) not differentiable at 0 z Ans : (d)

23. ze z f )( is analytic


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

24. )sin(cos yi ye x is

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

25. If )( z f is analytic, then )( z f is

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

26. The points at which)23(

)()( 2

2

z z

z z z f 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 which1

1)( 2 z

z f 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 y xu is

(a) 22 y x

x (b) 22

y x

y (c)

y

x1tan (d)

x

y1tan Ans : (d)

29. If ),2()( z z z f then )1( i f (a) 0 (b) i (c) -i (d) 2 Ans : (b)

30. If z z f )( then )43( i f

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

31. Critical points of the bilinear transformationdzc

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. dzcbzaw is a bilinear transformation when

(a) 0bcad (b) 0bcad (c) 0cd ab (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 transformationAns : (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 )( z f is conformal at all points z except at points where

(a) 0)(' z f (b) 0)(' z f (c) 0)(' z f (d) 0)(' z f Ans : (a)


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39. The fixed points of the transformation 2 zw 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 of11

z z

w are

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

42. The mapping z

zw 1

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

1 y in z-plane is

(a) circle 0422

vvu (b) circle 422

vu (c) circle 222

vu (d) none of theseAns : (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 z z z of z - plane into 1,0, wwiw ofw 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-iyu=x v=-yux =1 vx =-1uy =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 =0uy =2y v y = 0ux = vy and u y = - vx are not satisfied everywhere except at z=0


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So f (z) may be differentiable only at z=0. Now u x,vx,u y,vy are continuous everywhere and inparticular at (0,0).

3. Test the analyticity of the function w=sin z.Solution: w=f (z) =sin zu+iv = sin(x+iy)=sin x cosiy+ cos x siniy

= sin x coshy+i cos x sinhyu= sin x cushy v= cos x sinhy

ux = cosx cushy v x = -sinx sinhyuy = sinx sinhy v y = cosx cushyux = vy and u y = - 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 v x = 2xuy = 2x vy = -2y

ux vy and u y - 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 v x = siny

uy = siny v y = cosy

ux = vy and u y = - vxThe 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. u x = vy ------------------- (1) and u y = - vx------------------------------(2)To prove v-iu and –v+iu are also analyticFor this, we have to show that

(i) ux = vy and -u y = vx (ii) ux = vy and u y = - vx


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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) = c 1and v(x,y) =c 2 where c 1andc2 are constants are orthogonal to each other.

Solution: If u(x,y) = c 1 , then du = 0But by total differential operator we have

du =

(Say)

Similarly, for the curve v(x,y) =c 2 we have

(Say)

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

u(x,y) = c 1and v(x,y) =c 2 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=


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Now u x = vy and u y = - vx2 =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 ( )

=


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f (z) = 2log z +cTo 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=3zu+iv = 3(x+iy)u=3x v=3y

x= y=

Given


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.

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 fromf (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


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=

=

(i). =

= 2

= 2

= 2

=2 = 0(ii)

=

=

=

=

=2f’ (z)

(iii). =

=

=

= 4 =

2.

Prove that the function u = satisfies laplace’s equation and find thecorresponding analytic function f (z) = u+iv.Solution: Given u =

+


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+

=

u satisfies Laplace equation.

To find f (z): u is givenStep 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:


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Integrating f (z) = -z

To prove v is harmonic

+y

=

4. Prove that the function u = +1 satisfies laplace’s equation and find thecorresponding analytic function f (z) = u+iv. Solution: Given u = +1

= -6x-6

u satisfies Laplace equation.


Step 2:

Step3:


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Integrating f (z) =

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

Solution: Given u=


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+iVWhere F (z) = , U= V =

Given v =

Step 1:

Step 2:

Step3:


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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+iVWhere 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, -irespectively. Hence find the image of

Solution: The bilinear transformations is given by


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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.


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11.Show that the transformation w = 1/z transforms circles and straight line in the z-plane intocircles 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 passingthrough 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 inthe w-plane passing through the origin. Thus lines in the z-plane not passing through the originmap 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 theorigin in the z- plane map into the lines through the origin in the w- plane.


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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


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u is harmonic.To find the harmonic conjugateLet 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, 3irespectively.

Solution: The bilinear transformations is given by

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.Solution: The bilinear transformations is given by


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Since the above relation becomes.

2w+2=-zw+izW (z+2) = iz-2

w = is the required bilinear transformation.

analytic functions mcq +notes

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