How to evaluate this complex integral !?

Question

We have the following complex integral : $$\frac{1}{2\pi i}\int_{-i\infty}^{i\infty}e^{-\frac{\pi}{2}\cot\left(\frac{\pi}{s}\right)}\frac{x^{s}}{s}ds$$ Where $x\in\mathbb{R}:x>1$. i tried closing the contour to the left, and computing the residues at the singularities of the integrand $\left( s=-\frac{1}{n}\right)$ , but this proves to be very hard to do !!

Answer 1

Your integral, if it existed, would be equal to $$\frac{1}{2\pi i} \int_{-\infty}^{\infty} e^{-i (\pi/2) \!\coth(\pi/s)} \frac{x^{is}}{s}\,ds.$$

But

$$ \lim_{s \to 0^+} e^{-i (\pi/2) \!\coth(\pi/s)} = -i $$

and

$$ \lim_{s \to 0^-} e^{-i (\pi/2) \!\coth(\pi/s)} = i, $$

so that near $s=0$ the integrand looks like

$$ \frac{-i}{|s|}. $$

This is not integrable in the standard sense and, further, the integral does not admit a principal value.