Any ideas how to solve: \begin{equation} \int_0^\infty x^{n+\frac{1}{2}} (e^{a x }-1)^{-\frac{1}{2}} e^{i x t} dx \end{equation} where $a$ and $t$ are real, positive constants; $n$ is a positive integer.
I think the problem comes from having rational functions in both powers and exponential functions.
I tried to get ride of the rational power, but it didn't really help \begin{equation} \frac{\partial}{\partial q } \int_0^\infty x^{n} (e^{a x }-1)^{-\frac{1}{2}} e^{i x t+ q x^{1/2}} dx \end{equation}
Having an hint how to solve this for $t=0$ would already be useful. Thanks!
It's actually enough to resolve the $n=0$ case, since $t$-derivatives of $$F(t)=\int_0^\infty \frac{e^{i x t}}{\sqrt{e^{2x}-1}} x^{1/2}\,dx$$ will bring down more powers of $x$; I've chosen units for so that $a=1$ for convenience. Even with these simplifications, I don't know how to compute a closed-form and so will have to settle for an appropriate series expansion. We rewrite the fraction in the integrand and expand in powers of exponenetials $$\frac{e^{i x t}}{\sqrt{e^{x}-1}}=\frac{e^{i x t-x/2}}{\sqrt{1-e^{-x}}}=\sum_{k=0}^\infty \binom{2k}{k}\left(\frac{e^{-x}}{4}\right)^k e^{i x t-x/2}=\sum_{k=0}^\infty \binom{2k}{k}4^{-k} e^{i x t-(k+1/2)x}.$$
Integrating term-by-term then gives
$$F(t)=\sum_{k=0}^\infty \binom{2k}{k}4^{-k}\cdot \frac{1}{2}\pi^{1/2} [(k+\frac{1}{2})-i x]^{-3/2}=\sum_{k=0}^\infty \binom{2k}{k}\frac{ 2^{-2k-1}\pi^{1/2}}{(k+\frac{1}{2}-i t)^{3/2}}$$
which is a rather formidable result. It's possible that this can be resummed as a hypergeometric series of some sort; I'll see what I can find.