Seem to be encountering many interesting results from integrals in the form: $\int_0^\infty \frac{f(a,x)}{e^{2\pi x}-1}\text{d}x$

Asked Jan 11 '16 at 03:02

Active Jan 26 '16 at 15:19

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I have found interesting results from integrals in the form: $$I=\int_0^\infty \frac{f(a,x)}{e^{2\pi x}-1}\text{d}x$$

A few examples of interesting functions here are:

$$f(a,x)=\sin(ax)\implies I=\frac{1}{4}\coth \frac{a}{2}-\frac{1}{2a}$$

and... $$f(a,x)=2\arctan (x/a)\implies I=\log\Gamma(a)-a\log a+a-\frac{1}{2}\log \frac{2\pi}{a}$$

to even...

$$f(a,x)=\frac{2\sin(a \arctan x)}{(x^2+1)^{a/2}}\implies I= \zeta(a)-\frac{1}{2}-\frac{1}{a-1}$$

Does there exist a non-trivial function $f$ such that $I=\operatorname{Bi} (a)+\text{some extra stuff}$?

Non-trivial means that you can't have a factor of the numerator being $(e^{2\pi x} -1)$

edited Jan 11 '16 at 15:18

asked Jan 11 '16 at 03:02

It should be $\text{Bi}(a)$, I believe. – Jack D'Aurizio Jan 11 '16 at 11:50
@JackD'Aurizio Fixed – Jan 11 '16 at 15:19
Here you can find some integral representations of the Airy function. Although none is in the exact form you desire, perhaps they are nevertheless helpful. – user103093 Jan 14 '16 at 19:02
It looks like https://en.wikipedia.org/wiki/Abel%E2%80%93Plana_formula may be relevant – Dark Malthorp Feb 20 '24 at 19:06

0 Answers0