# Numerical estimates on asymptotic of given functional as $x\rightarrow \infty$

Consider the following functional :

$$I(x,s) =\int_0^\infty\mathrm dy \frac{F(x + \mathrm iy, s) − F(x −\mathrm iy, s)}{\mathrm e^{2πy}-1},$$ where $$F(z, s) = \dfrac{\sin^2[π\Gamma(z)/(2z)]}{z^s}$$.

Let us restrict $$s\in[0,1]$$

In Mathematica

f[z_, s_] = Sin[πGamma[z]/2z]^2/z^s;
II[x_, s_] := Integrate[(f[x + I y, s] - f[x - I y, s])/(Exp[2 Pi y] - 1),
{y, 0,Infinity}]


Can we get sharp numerical asymptotic of $$I(x)$$ as as $$x\rightarrow \infty$$?

Also, can we get quantitative upper and lower bound estimations on the functional ?

Update : Also try for $$F(z)= \log(z){\sin^2[π\Gamma(z)/(2z)]}$$

• Have you tried Series[]? – Michael E2 Feb 25 '20 at 19:20
• @Michael E2 Thank you for comment, sir . I've not used it. Actually this is my first day and second question on this site . I'm not used to Mathematica . – bambi Feb 25 '20 at 19:25
• I’m away from my computer, I can’t try it. It may or may not work on Integrate[] – Michael E2 Feb 25 '20 at 20:30
• @MichaelE2 doesn't seem to work trivially at least. The function f is rather pathological near the origin. !Mathematica graphics – chris Feb 25 '20 at 20:33
• The most important cases for my purpose is $s=1$ or very near to 1 and $s=0$ or very near to zero – bambi Feb 26 '20 at 8:05