I suppose a stupid question but I was wondering about it for a while:
Can one apply the residue theorem to a function $f$ which is defined and holomorphic on $U-\{a_1,a_2,\dots\}$ where $U$ is simply connected open subset of the complex plane and $a_k$ for $k\geq 1$ are all simple poles of $f(z)$. In particular I am thinking about the case when $f(z)= \Gamma(z)$, the gamma function. It is known that it has simple poles at $z=-k$, $k=0,1,2,\dots$ with resdue $(-1)^k/k!$.
I hope the question is clear. Excuse me in case it is to trivial. I am not an expert in complex analysis.
The Wikipedia-article demands a finite number of points $a_k$. Will one get a problem with a suitable choosen closed contour?
To be more precise I would like to calculate the following integral:
$$\int_{-\infty}^{\infty} dx f(x)$$
I was wondering if the typical "trick" of constructing a closed half-circle in the upper-half plane would work too when the poles continue ad infinitum.
EDIT:
My initial motivation is to invert a Mellin transform using the Mellin inversion theorem, when the Mellin transform has an infinite number of isolated poles.
For example: I am able to show (by using a specific symmetry of my problem) that the Melin transform of a function $P(x)$ for $0\leq x\leq x_c$ fulfills the following equation:
$$M(s) = \frac{2x_c^s}{s-2+x_c^s}$$
where $0<x_c<1$. Now when I try to apply the Mellin inversion theorem I need to know where the poles lie. Unfortunately I think there is an infinite number poles of $M(s)$.
Best Answer
It does work for an infinite number of singularities, as long as they are all isolated.
From Ahlfors' Complex Analysis, p.150: