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Suppose a nonnegative function $f$ on the positive real line and $f(x)\neq\Gamma(x)$. Do we know any closed formula of the following integral $$\int_{\mathbb{R}^+} \frac{\alpha^x}{\Gamma(x)}f(x)dx,$$ where $\alpha>0$?

Thanks in advance.

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  • $\begingroup$ There is a table entry for the Laplace transform ($\alpha -> s $) of the integral. Which is slightly simpler but ... involves the Laplace xform of f(x)->F(s); as F(ln(s)). So would involve a somewhat complex inversion. Would that entry be of any help? $\endgroup$ – rrogers May 3 at 15:22
  • $\begingroup$ Are you saying that consider this integral as a Laplace transform with parameter -Ln(\alpha)? Yeah, this seems to be a good way. Then, which entry shall I start from? $\endgroup$ – gouwangzhangdong May 4 at 1:39
  • $\begingroup$ Yes that's a possibility and I might have a paper somewhere that helps but I was refering to $$\mathcal{L}_{t\rightarrow s}\left({\displaystyle \intop_{u=0}^{\infty}\frac{t^{u-1}}{\Gamma\left(u\right)}f(u)du}\right)=\left[\mathcal{L}_{u\rightarrow t}\left(f(u)\right)\right]_{t=ln\left(s\right)}$$ From Caltech, a compilation of Bateman's work: authors.library.caltech.edu/43489 In particular Vol 1 Tables of Integral Transforms -> Laplace Transform -> General Formulas -.(29) . $\endgroup$ – rrogers May 4 at 11:54
  • $\begingroup$ But the notion of F(ln(s)) seems to be tied to a Laplace->Mellin mapping. If you're interested, I have a reference but would have to study it to use it. "Generalized Integral Transforms" by A.H. Zeeman, Theorem 4.2-1. I will insert it here if you like but it might end up being irrelevant. $\endgroup$ – rrogers May 4 at 11:58
  • $\begingroup$ Now, it is crystal. Many thanks for this idea.I have not heard this book before. So, yes, could you please insert it here. Really nice to meet you. $\endgroup$ – gouwangzhangdong May 4 at 14:23

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