I am sorry in advance for the very specific question but I am not sure where else to ask.
I am currently writing an essay on the analytic continuation of the Zeta function and to help I am reading the book "Riemann's Zeta Function" by H. M. Edwards. I have to show that a function defined by a contour integral is analytic everywhere (except for at s=1). The integral in question is
$\int_{+\infty}^{+\infty} \dfrac{(-x)^{s}}{(e^{x}-1)x} dx $
Where the contour indicated begins at $+\infty$, moves to the left down the positive real axis, circles the origin once in a counterclockwise direction, and returns up the positive real axis to $+\infty$. Strictly speaking the path must be taken slightly above the real axis as it descends from $+\infty$ and slightly below as it ascends back.
I have shown that this integral converges (since $e^{x}$ grows much faster than $x^{s}$) but now need to show it is analytic. The books justification is simply "because convergence is uniform on compact domains". This seems to me to be alluding to the fact that if a sequence of analytic functions converges uniformly in a region S of the complex plane, then the limit is analytic in S, but that is about as far as I can get.
Is anyone able to shed some light on why this function is analytic and how to interpret the justification my book has given?
Thanks in advance.