# Nature of function as $x\rightarrow\infty$

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:

https://mathoverflow.net/q/412913/145223

• $F(z)$ has to be bounded in $a\le\Re(z)\le$ if you want to apply Abel-Plana formula to the interval $[a,b]$. Jan 12, 2022 at 16:05
• @TravorLZH that's not true, Abel plana holds for much weaker conditions see FWJ Olver "Asymptotics and special functions"
– TPC
Jan 13, 2022 at 8:02