I'm trying to evaluate $$ \int_0^\infty \frac{x^a}{1+x} \: dx, \: -1<a<0 $$
using contour integrals. Actually, I have found the correct answer using a "Pac-Man" contour and residues. My only question concerns the correct way to take the limit of the contour. Suppose it has inner radius $\epsilon \downarrow 0$, outer radius $R \to \infty$, and it "eats" the positive x-axis with a "bite angle" of $2 \alpha \downarrow 0$. It seems to me that the right way to do things is to let $\alpha \downarrow 0$, then $\epsilon \downarrow \infty, R \to \infty$. Is this correct?