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(First of all, this is my first Mathematica question. I'm not used to Mathematica that much. So, apologies in advance.)

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{\sin^2[π\Gamma(z)/(2z)]}{z^s} $.

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

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

The reason for the question is that the functional gives very massive values ( upto 10^100) after the value x=6 which I think are not correct. I don't know how to resolve this issue.So I'm posting this question for bigger accurate values

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    $\begingroup$ you defined $F$ as taking one argument $z$ but you are calling in using 2 arguments? What is s? $\endgroup$
    – Nasser
    Commented Feb 24, 2020 at 20:15
  • $\begingroup$ @Nasser thanks for correcting see denominator $(z^s)$ $\endgroup$
    – bambi
    Commented Feb 24, 2020 at 20:17
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    $\begingroup$ If you looking for analytical integration, Mathematica can not do the integral at all f[z_, s_] := Sin[Gamma[z]/z]^2/z^s; Integrate[(f[x + I y, s] - f[x - I y, s])/(Exp[2 Pi y] - 1), y] For numerical integration, need numerical values for the parameters involved. $\endgroup$
    – Nasser
    Commented Feb 24, 2020 at 20:23
  • $\begingroup$ @Nasser I'm looking for sharp approximation of the integral $\endgroup$
    – bambi
    Commented Feb 25, 2020 at 5:51
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    $\begingroup$ sharp approximation of the integral I assume you mean by a numerical approximation. For this, numerical values are needed for all parameters other than the integration variable itself. If you provide example of such values, may be someone could help. $\endgroup$
    – Nasser
    Commented Feb 25, 2020 at 10:07

1 Answer 1

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You could do it numerically?

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

Then

ParallelTable[{x, II[x, s] // Im}, {s, 1, 2, 1/2}, {x, 1, 5, 0.05}] //
  ListLinePlot[#, PlotRange -> All] &

Mathematica graphics

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  • $\begingroup$ thank you for the answer but as you can see the functional gives very massive values after the value x=6 which I think are not correct. So I'm posting this question for bigger accurate values $\endgroup$
    – bambi
    Commented Feb 26, 2021 at 15:28

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