In the course of solving a particular physical problem, I have derived a relationship between two unknown functions: $$ f(s) = \frac{s \sinh{\frac{\pi s}{2}}}{2 \pi i \beta} \int_{-c- i \infty}^{-c+i\infty} X(\nu) \ \Big(\frac{2}{\beta a}\Big)^\nu \ \Gamma\Big(\frac{1}{2}(\nu + i s)\Big) \Gamma\Big(\frac{1}{2}(\nu - i s)\Big) \ d\nu$$ The two functions $f(s)$ and $X(\nu)$ are unknown and $1>c>0$ specifies the vertical line along which we integrate . The two parameters $\beta$ and $a$ can be tuned such that their non-dimensional combination is either very large or very small. I am hoping to use this large/small parameter to develop some asymptotic expansion using a sum of residues. Simply getting an algebraic relationship in certain limits between the two functions could be helpful.

Although $X(\nu)$ is unknown, I do know a number of results about its structure in the complex plane:

1) It decreases exponentially along vertical lines as you move away form the real axis

2) $X(\nu)$ is meromorphic in the whole complex plane with simple poles at (and only at) $\nu = u_p +n$ where $n,p \in \mathbb{N}$ and the $u_p$ satisfy $u_p \sin{\frac{\pi u_p}{2}} + 1=0, Re(u_p)\ge0$

3) $X(\nu)$ decreases factorially in the positive real direction along horizontal lines and increases factorially in the negative real direction.

My intuition says that we can close the given integration contour and then argue either from point 1) or from the small/largeness of $\beta a$ that the closure of the contour has a vanishingly small contribution (depending on exactly how we close; I have been using box contours). The results I am getting though are not consistent with previous numerical and approximate results I have obtained, which I am very confident in.

The fact that the gamma functions and $X(\nu)$ can have poles that coincide has me stumped. This occurs because almost all of the $u_p$ are purely real, but $u_0$ is purely imaginary and so if $ s = -i u_0$ the gamma functions and $X(\nu)$ have overlapping poles.

How can I systematically approximate this integral as a sum over the residues, given the information provided above?



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