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.

Residues/integrals. Complex Analysis.

Verify the following equations: integral from 0 to pi/2 of ( d theta/ ( a + sin^2 theta) ) = pi/2[a(a+1)]^1/2 if a > 0. Please explain every step, I want to be able to evaluate such integrals.

The Maximum Modulus Theorem.

Let f be analytic in the disk B(0;R) and for 0 =< r < R define A(r) = max { Re f(z) : |z| = r}. Show that unless f is a constant, A(r) is a strictly increasing function of r. Please justify every step and claim and show how you used all what is given. Also refer to theorems or lemmas used in the proof. The section where I ...continues

Harmonic function

(See attached file for full problem description) --- Give an example (and explain why it works) of an analytic function on a harmonic function such that the composite function is defined but NOT harmonic ---

Analytic function

Looking for solution to both parts, this problem has been bothersome for quite sometime. (See attached file for full problem description)

Problems

(See attached file for full problem description) The first part I'm having a problem with bringing (a) greater than or equal to 0 into the picture. Second part, thinking it may be infinite series, although when I tried I had a hard time with convergence