Wow, nice work. I like it. And, er, I have difficulty in solving another question/exercise and I wanna go to seek help from you, if you'd like to.
This semester I serve as TA in Complex Analysis. Students ask me one difficult problem as follows, and I write it in the language of Tex

f(z) \epsilon H(D) \prod C(\bar{ D }), D={0< Im Z<1}. And
lim_{Im z=0, Re z \rightarrow \infty} f(z)=A exists. Prove that \forall 0<r<1, it holds that
lim_{Im z=r, Re z \rightarrow \infty} f(z)=A as well.

I feel strange that, it seems r=1 OK, but I cannot prove it. Too difficult.