I'm studying the limits and applicability of Abel Plana summation for different test functions ( class of functions ) . In doing so this just pops out and couldn't handle the said integral so asked here:
Consider the following function :
$$F(x)=\frac{\sin^2(\Gamma(x))\Gamma'(x)}{e^{\sin^2(\Gamma(x))}}$$
Now consider the following function :
$$I(x) =-i\int_0^\infty\mathrm dy \frac{F(x + \mathrm iy) − F(x −\mathrm iy)}{\mathrm e^{2πy}-1}$$
What is the nature of the $I(x)$ as $x\rightarrow\infty$?
( Is $I(x)\rightarrow 0$ as $x\rightarrow\infty$ true?)
( Is there an analytic way to show this to be true or false?)
Some values I computed :
$x=0.3, I= -0.4596$
$x=0.5, I= 0.3347$
$x=0.7, I= 0.1407$
$x=0.9, I= 0.0706$
$x=1 , I= 0.05211$
$x=1.5, I=0.02101$
$x=2 , I= 0.02518$
$x=3, I=0.06752$
It seems after $x=3$ we are unable to compute numerically
I also asked this question on MathOverflow with no response as follows: