residue

few questions on solutions (1st attachment) question pertaining to residue(2nd attachment)

Imaginary powers/ residue

(See attached file for full problem description)

Bounds for analytic functions

If p(z)=a0+a1z+.....+anz^n ia a polynomial and max|p(z)|=M for |z|=1, show that each coefficient ak is bounded by M. Note:(a0 means a subscript 0, a1z means a subscript 1 times z, anz^n means a subscript n times z to the n power, and ak means a subscript k)

Cauchy principal value, residue

Verify the integral formula with the aid of residues. 1.) Show that the p.v. of the integral of (x^2+1)/(x^4+1) from 0 to infinite = (pi)/(sqrt 2). Note: p.v.=principal value; pi is approximately 3.14; sqrt 2=square root of 2 Please show all work and explain the steps, especially how you found the zeros of the function.

Open mapping theorem. Complex Analysis

Let P : C -> R be defined by P(z) = Re z; show that P is an open map but it is not a closed map. ( Hint: Consider the set F = { z : Imz = ( Re z)^-1 and Re z doesn't equal to 0}.) Please explain every step and justify.