1. UNIT- III ANALYTIC FUNCTIONS
PART - B
IMPORTANT QUESTIONS.
1. If f (z) is a regular function of z, prove that =4 .
2. If f(z) = u+iv is a regular function of z, then show that =
.
3. If f (z) = u+iv is analytic, prove that log|
4. The real and imaginary parts of an analytic function W=u(x,y)+iv(x,y), satisfies the Laplace
equations ie.,
5. If W=u(x,y)+iv(x,y) analytic function, the curves of family u(x,y)=c1 and v(x,y)=c2 cuts
orthogonally where c1 and c2 are constants.
6. Find the analytic function f(z)=u+iv whose real part is
(i) u = (xcosy-ysiny)
(ii) u = [(
(iii) u = and u+v = .
7. Find the analytic function f(z)=u+iv whose imaginary part is
(i) v = .
(ii) v =
(iii) v = .
8. Determine the analytic function of u+v = 2u+v = .
9. Prove that the function u = is harmonic. Find the conjugate
harmonic function and corresponding f(z).
10. Find the bilinear transformation that maps,
(i) z = 1,i,-1 & w = i,0,-i
(ii) z = -2,0,2 & w = 0,i,-i
(iii) z= ,I,0 & w = 0,I, .
(iv) z = 0,1, & w = -5,-1,3
11.Find the image of |z-2i|=2 under the transformation w = 1/z.
12.Find the image of the strips (i)1/4<y<1/2 (ii) 0<y<1/2 under the transf., w = 1/z.

2. 13.Discuss the transformation w = 1/z.
PART-A
IMPORTANT QUESTION
1.Define analytic function?
2.State sufficient conditions of C-R equation to be analytic?
3.Write down the polar form of C-R equation?
4.Test whether the function w =2xy +i( is analytic.
5. Find a,b,c if f(z)=(x-2ay)+i(bx-cy) is analytic.
6.Check f(z)= is analytic & find dw/dz.
7.Verify whether f(z)= sinhz is analytic using C-R equ.,
8. Check f(z)= is analytic & find dw/dz.
9.Test the analytic function of f(z)= and find its derivatives.
10.Check it f(z)= | is analytic or not.
11.S.T the function with constant real part is constant.
12.S.T an analytic function with constant modulus is constant.
13.Define harmonic function?
14.P.T u = is a harmonic function.
15.find the conjugate harmonic of u =
16.Define bilinear transformation?
17.Find the invariant point of z-1/z+1.
18.Define conformal mapping?
19.Find the critical point of (i) w = z +1/z, (ii) w = sinz.
20.Find the image of the circle |z|= under the transf., w = 5z.
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