- #1
Marin
- 193
- 0
Hi there!
I am trying to prove the following 2 identities using complex analysis methods and contour integration and I'm really stuck on defining the integration paths.
[tex]\int_{0}^{1}\frac{\log(x+1)}{x^2+1}d x=\frac{\pi\log2}{8}[/tex]
[tex]\int_{0}^{\infty}\frac{x^3}{e^x-1}d x=\frac{\pi^2}{15}[/tex]
It's interesting that the first integral is proper in both limits of integration, whereas the second - improper in both limits.
I am familiar with a proof of the second identity using series, uniform convergence and the gamma function. This proof also verifies the following generalisation of the second identity:
[tex]\int_{0}^{\infty}\frac{x^{s-1}}{e^x-1}d x=\Gamma(s)\zeta(s)[/tex]
But I'm looking for a way in the complex plain.
Any help or hints are much appreciated!
Regards, Marin
I am trying to prove the following 2 identities using complex analysis methods and contour integration and I'm really stuck on defining the integration paths.
[tex]\int_{0}^{1}\frac{\log(x+1)}{x^2+1}d x=\frac{\pi\log2}{8}[/tex]
[tex]\int_{0}^{\infty}\frac{x^3}{e^x-1}d x=\frac{\pi^2}{15}[/tex]
It's interesting that the first integral is proper in both limits of integration, whereas the second - improper in both limits.
I am familiar with a proof of the second identity using series, uniform convergence and the gamma function. This proof also verifies the following generalisation of the second identity:
[tex]\int_{0}^{\infty}\frac{x^{s-1}}{e^x-1}d x=\Gamma(s)\zeta(s)[/tex]
But I'm looking for a way in the complex plain.
Any help or hints are much appreciated!
Regards, Marin