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 ( I'm just a student so, please excuse me if anything is trivial):

Consider the following function :


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 that We can't do anything beyond $x=3$ numerically.

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    $\begingroup$ Why is this question of interest? $\endgroup$ Jan 1, 2022 at 18:03
  • $\begingroup$ @GeraldEdgar it just pops out in my study of Abel plana summation for different test functions. $\endgroup$
    – TPC
    Jan 1, 2022 at 18:25
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    $\begingroup$ Tryng different methods for numerical integration, no problem for $0.218 \leq x \leq 3.381$ but outside this range, all of them just explode $\endgroup$ Jan 3, 2022 at 15:35
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    $\begingroup$ Please be aware that every edit of a question or of one of its answers bumps the thread to the front page. This has happened for this thread already more than 10 times, and this is a nuisance for other users. Please refrain from unnecessary edits to your posts. -- Usually, the vast majority of minor edits can be avoided by writing and proofreading a question or an answer carefully before posting it. $\endgroup$
    – Stefan Kohl
    Feb 28, 2022 at 22:15
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    $\begingroup$ A short comment. The nature of the $I(x)$ depends on the nature of the integrand's numerator $Im[F(x+iy)-F(x-iy)]$. For x passing some point beyond the (first?) zero of I(x) the integrand behaves wildly. You can plot it. Working with Pari/GP the breakpoint is $x_0=3.208917736262569989928994080...$ for which $I(x_0)=0$. I think Abel-Plana is not a good tool for summation/integration of such kind of functions. You should try an adequate variable transformation first. What if $u=\Gamma(x)$?. $\endgroup$ Mar 1, 2022 at 15:08


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