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 got this problem from, talks about The Maximum Principle. The Maximum Modulus Theorem ( first, second, and 3rd versions).

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