Welcome to mathimatikoi.org forum; Enjoy your visit here.

## Gamma, trigonometric and iterated integral

Calculus (Integrals, Series)
Tolaso J Kos
Articles: 2
Posts: 855
Joined: Sat Nov 07, 2015 6:12 pm
Location: Larisa
Contact:

### Gamma, trigonometric and iterated integral

The result $$\int_{-\infty}^\infty\frac{dx}{\left(e^x-x+1\right)^2+\pi^2}=\frac{1}{2}$$ holds and can be easily extracted via residues.

Use the above to prove that: $$\int_0^\infty x^{-x} e^{-x}\, \Gamma(x) \sin (\pi x)\, dx= \frac {\pi}{ 2}$$ whereas $\Gamma$ is the Gamma function defined as $\displaystyle \Gamma(x)=\int_0^\infty t^{x-1}e^{-t} \, dx$ and $x^{-x}$ is the iterated function.
Imagination is much more important than knowledge.
whitexlotus
Articles: 0
Posts: 10
Joined: Sun Sep 04, 2016 5:08 am

### Re: Gamma, trigonometric and iterated integral

This is very beautiful, can you give me some hint ?
Civil Engineer