Residues and Examples

Define residues and give formulae to calculate residues. Two examples are given Please see the attachment

Complex analysis/ singularities/ the arguement principle.

Let f be meromorphic on the region G and not constant; show that neither the poles nor the zeros of f have a limit point in G. In your solution, please refer to theorems or certain lemmas. Justify your claims and steps. I want to learn not just have the right answer. Thanks.

Complex Analysis/the argument principle.

Suppose f is analytic on B(bar) (0;1) and satisfies |f(z)| < 1 for |z| = 1. Find the number of solutions ( counting multiplicities) of the equation f(z) = z^n, where n is an integer larger than or equal to 1. Please justify every step and claim and refer to any theorems you use.

The argument principle. Complex Analysis.

Is a nonconstant meromorphic function on a region G an open mapping of G into C? Is it an open mapping of G into C_oo ( C is complex plane, oo infinity, C_oo means the extended complex plane ( C U {oo} ) ).

The argument principle. Complex analysis.

Let f be analytic in a neighborhood of D = B(bar)(0;1). If |f(z)|=<1 for |z|=1. What can you say? Justify every step and claim and refer to any theorems you use please.