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,-1,311.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.
13.Discuss the transformation w = 1/z. PART-AIMPORTANT QUESTION1.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. ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^