I need to plot the following functional with accuracy:

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

And let us restrict $s\in[0,1]$

What is the most efficient way of computing this integral?

Nature of functional as $x\rightarrow\infty$ from the computation ?

I computed some relatively small values using Mathematica which suggests the function is oscillatory with damping. But I need big values greater than x=100 and at s=1

  • 1
    $\begingroup$ For the first question, I guess before looking for the most efficient way to compute the integral maybe one should ask what is a possible way to do it. What have you tried? For the second question, I guess we'd need first to find what is the asymptotic limit for the function F(z,s) when Re(z)$\to \infty$. $\endgroup$ Feb 9, 2021 at 3:24
  • $\begingroup$ @MaximUmansky I used mathematica's NIntegrate to integrate the functional $\endgroup$
    – bambi
    Feb 26, 2021 at 11:22


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