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Consider the following integrals $$ I_1(x) =\int_0^\infty\mathrm dy \frac{F(x + \mathrm iy) − F(x −\mathrm iy)}{\mathrm e^{2πy}-1}, $$

And

$$I_2(x) =\int_1^x F(t)dt$$

Where, $ F(z) = \sin^2[π\Gamma(z)/(2z)]$

I want to plot (with considerable accuracy) $$R(x)=I_1(x)/I_2(x)$$

Also , see if it's apparent from plot : $R(x)\rightarrow0$ as $x\rightarrow\infty$

I want growth rate of local minimas and Maximas of $I_1(x)$ as $x\rightarrow\infty$

See this MSE post for more details .

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    $\begingroup$ This is in essence a duplicate of a previous question asked on this forum. The problems discussed there are all the same here. $\endgroup$ – Wolfgang Bangerth May 18 at 22:31
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    $\begingroup$ @WolfgangBangerth I asked for second Integral in previous question , first Integral is new here and equally tricky . $\endgroup$ – Bambi May 19 at 8:03
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    $\begingroup$ @Bambi as far as I can tell it's "tricky"' in the same way the other integrals were. $\endgroup$ – Spencer Bryngelson May 22 at 15:02

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