(Solved):   1. Let u(x,y) = 4xy3 4x3y + x. Find a harmonic conjugate v and an analytic functi f(z) ...

 

1. Let u(x,y) = 4xy3 – 4x3y + x. Find a harmonic conjugate v and an analytic functi f(z) = u + iv satisfying the condition f(1+1) = 1 + 4i. 2. Use Cauchy's Integral formula/ Cauchy's Integral formula for derivatives to evaluate following a) Sc b) Sc cos 2z 25 2z+5 22-22 dz, C:[z] = 1. dz, C:]z + 11 = 2. 3. (a) Find the Maclaurin series for f(z) = cos 3z. (b) Use the trigonometric identity cos 3z = 4 cos’z – 3 cos z and find the Maclaurin for cos3z. 4. Locate and determine the type of singularity for the following functions. a) f(z) = 1-cosh z b) f(z) = Z 24 5. Use Cauchy's Residue theorem to evaluate ?c ztu dz, C:12 - il = 1. (22+1)