Given $m \ge 0$ and $n$ odd, $ s >0 $ is there a way, for computing asymptotic expansion of following integral: $$ \int_{\mathcal{C}} \int_{\mathcal{C}} \frac{ \Gamma(m + s_1) \Gamma(m + s_2) \Gamma(1 - \frac{n}{2} + s_1) \Gamma(1 - \frac{n}{2} + s_2) \Gamma(-s_1) \Gamma(-s_2) \Gamma(m + \frac{n}{2} + s_1 + s_2) }{ \Gamma(m + \frac{n}{2} + s_1) \Gamma(m + \frac{n}{2} + s_2) \Gamma(m + \frac{n}{2} + s_1 + s_2 + s + 1)}\;ds_1\,ds_2\,, $$ where $C$ is contour avoiding all singularities? I tried to sum up all possible combinations of residues arising from $s_1$ and $s_2$, unfortunately this does not lead to anything useful. Is there at least way to show, that residues emerging from $(s_1,s_2) \in \mathbb{N_0}\times\mathbb{N_0}$ are dominant in total sum? Thanks.
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$\begingroup$ This contour $C$ starts where and ends where? Asymptotic expansion in what? $\endgroup$Michael Engelhardt– Michael Engelhardt2025-05-13 13:14:10 +00:00Commented May 13 at 13:14
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