Can you elaborate your argument, especially why you can apply Cauchy's integral formula? How do you deal with the poles at ##z=\pm i\pi## which are inside ##U_5(0).##
Wouldn't just be quicker to say "...by Residue Theorem"? You always get me with a shorter proof, back at you!
So I've been not getting how #7 parts a) and b) are related, the only way I can think to solve b) is to use CR-eqns but I figure you're going to say "Why did you do all that calculus when you could've just applied Theorem X [which relates part a) to b)]?". So should I thumb through theorems and exercises until I find Theorem X? Or do I have your blessing to go ahead and write out the calculus?