Sri venkateswara college of engineering

SRI VENKATESWARA COLLEGE OF ENGINEERING

DEPARTMENT OF APPLIED MATHEMATICS

WORK SHEET-MA6251-MATHEMATICS – II

UNIT-IV -ANALYTIC FUNCTIONS

PART-A
1) (i)Define Analytic function (ii) State sufficient condition for f(z) to be analytic.

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

3) Show that the function is analytic every where in the complex plane.

4) Write the polar form of C-R equations and verify f(z) = zⁿ is analytic or not. [Ans: analytic]

5) Check whether is analytic everywhere. [Ans: nowhere analytic]

6) Find a, b, c if the function is analytic. [Ans: c=1,a=-b]

7) If f(z) = u+iv is an analytic function, prove that u satisfies the laplace equation.

8) Show that the function is harmonic.

9) Verify f(z) = cosz & f(z) = 1/z are analytic or not. [Ans: both are analytic]

10) If u + iv is analytic, then prove that v – iu is also analytic.

11) Show that an analytic function with (i) constant real part is constant, (ii) constant modulus

is constant.

12) Define conformal mapping.

13) Find the image of under the transformation . [Ans: 2u+v-5=0]

14) Find the image of the region y >1 under the transformation . [Ans: > 1]

15) Find the image of the circle |z| = under the transformation w = 5z. [Ans: u²+v² = 25]

16) Find the image of |z - 2i| = 2 under the mapping . [Ans:1 + 4v = 0]

17) Find the critical points of the transformation . [Ans: ]

18) Find the fixed points of the transformation . [Ans: 2,2]

19) Find the invariant points of the transformation . [Ans: i, -i]

20) Define bilinear transformation, under what condition this is conformal. [Ans: ]

PART- B

1. 
   Show that f(z) = log z is analytic every where except at the origin and find its derivative.
   

Ans: .

2. 
   If f(z) = u(x,y) + iv(x,y) is an analytic function,then prove that the curves of the family
   

u(x,y) = a and the curves of the family v(x,y) = b, cut orthogonally, where a and b are

varying constants.

3. 
   Prove that every analytic function can be expressed as a function of z alone, not as a function of
   
4. 
   Verify that the family of curves u= c₁ and v= c₂ cuts orthogonally when u+iv=z³.
   
5. 
   Show that the function is harmonic. Find the conjugate
   

harmonic and also find the analytic function.

[Ans:]

6. 
   Prove that the function is harmonic and determine the corresponding
   

analytic function. [Ans: f(z)= ze^z + ic]

7)Find the analytic function w = u+iv if and hence find v.

[Ans: ]

8)Prove that is harmonic and find its conjugate harmonic and the analytic function. [Ans: f(z)= log z +c,v = tan^-1(y/x)+c]

9) Find the analytic function f(z) = u+iv where u - v = e^x(Cosy-Siny) [Ans:f(z)= e^z + c]

10) If , find the analytic function f(z) [Ans: f(z) = ]

11)Construct the analytic function f(z)=u + iv given that

[Ans:]

12) If f(z) is a regular function of z prove that .

13)If f(z) is an analytic function ,prove that (i)(ii).

14) If f(z) is a regular function ,prove that .

15) If f(z) is a regular function ,prove that

16) If and , prove that both u and v satisfy the laplace equation, but

that (u + iv) is not a regular function of z.

17) Discuss the transformations (i) w = z + a (ii) w = az (iii) .

18) Find the image of the infinite strips (i) ¼ < y < ½ (ii) 0 < y < ½ under the transform w = 1/z.

[Ans: (i)Between the circles &

(ii)outside the circle in the lower half of the w-plane]

19) Find the image of the hyperbola under the transformation

[Ans: ]

20) Prove that the transformation maps the upper half of z plane onto the upper half of

w- plane. What is the image of under this transformation? [Ans: 1+2u=0]

21) Find the bilinear transformation that maps the points on to respectively.

[Ans: ]

22) Find the bilinear transformation that maps the points z = in to w = -5, -1, 3

respectively. What are the invariant points of this transformation?

[Ans:]

23) Find the bilinear transformation that maps the points z = 1, i, -1 onto the points w = i, 0, -i.

Hence find the image of |z| < 1. [Ans:> 0 of the w-plane]

24) Find the bilinear transformation that maps the points -2,0,2 on to ,.

[Ans:]
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