Is the following integral transformation of $f$ known (for suitable $f$ and $s\in\mathbb{C}$)? $$ \int_1^\infty f(t) \frac{e^{-ts}}{1-e^{-ts}}dt $$ It resembles somewhat the Laplace transformation.
What about properties, references …?
$\begingroup$
$\endgroup$
3
-
4$\begingroup$ Expand the ratio of exponentials as a geometric series in exponentials. This is called a Lambert series. Interchange sum and integral. $\endgroup$– StoppleCommented Mar 19, 2020 at 17:07
-
$\begingroup$ Thanks a lot! Nice and simple idea. $\endgroup$– borntomathCommented Mar 20, 2020 at 8:51
-
1$\begingroup$ Also looks like it will have something to do with Bernoulli polynomials. $\endgroup$– Benedict W. J. IrwinCommented Apr 9, 2020 at 16:16
Add a comment
|
1 Answer
$\begingroup$
$\endgroup$
1
For what it worth, in the terms of divergent integrals, your transform can be rewritten as
$$\operatorname{reg} \int_1^\infty f(t)e^{-t s \omega _-}dt$$ Looks like some kind of analog of Fourier transform, if you ask me...
-
$\begingroup$ fixed the typo, the previous form was of course, meaningless $\endgroup$– AnixxCommented Feb 17, 2021 at 20:41