How to calculate the integral $$ I= \int_0^{\infty} \frac{t^{a-1}}{t-x} dt$$ using contour integration. ($x>0$ is fixed, $a \in \mathbb{R}$)
Approach: I calculated the values of $a$ for which the integral is convergent. The result was that $0<a<1$. We have two poles, at $t=0$ and $t=x$. I tried a circular key hole contour that excludes $t=0$ and $t=x$. The value of the integral integrated over the circular curves are zero I thougt. But the sum of the integrals integrated over the horizontal segments are $(1-e^{i(a-1)2\pi})I$, so the integral is zero. But I made a mistake I guess. Can someone help me? Thanks in advance.
EDIT: I tried to use the same keyhole in this other post: Integration of $\ln $ around a keyhole contour
Since the integrand is singular at $t=x$, this must be understood in the Cauchy principal value sense. Your contour should include little detours around $x=a$ as well as $0$.
The integrals over the arcs around $x$ are not $0$. For the top part, I get $-i \pi x^{a-1}$; for the bottom part, $-i \pi x^{a-1} e^{2 \pi i (a-1)}$.