This is stuff I should know but am too intimidated to ask the professor.

In Bak/Newman's Complex Analysis, pg 148, it says \lim_{\epsilon\rightarrow 0, M\rightarrow\infty}\int_{C} R(z)\log z dz=\int_0^\infty R(x)\log x dx, where C is the horizontal line segment from i\epsilon to \sqrt{M^2-\epsilon^2}+i\epsilon. R(x) is a rational function with the degree of the numerator greater or equal to the degree of the denominator + 2, but I don't think this matters here.

No further details are given. I tried to work it all out completely, but had some issues. 1. I assume this limit means take one limit, then the next limit. Doesn't this sometimes depend on which limit you take first? Why does it not matter in this case? 2. It seems like we'll need two applications of either uniform convergence or the dominated convergence theorem to take care of the two limits. But it wasn't immediately obvious to me what I should do. I'm also unsure of whether one of these theorems (uniform convergence thing and dominated convergence thm) is stronger than the other or if they're completely different things.

I'll try to work it out more once I get time. Thanks for any help.