- Thread starter
- #1
- Jan 31, 2012
- 253
Here is another integral representation of $\zeta(s)$ that is valid for all complex values of $s$.
It's similar to the first one, but a bit harder to derive.
$ \displaystyle \zeta(s) = 2 \int_{0}^{\infty} \frac{\sin (s \arctan t)}{(1+t^{2})^{s/2} (e^{2 \pi t} - 1)} \ dt + \frac{1}{2} + \frac{1}{s-1}$
It's similar to the first one, but a bit harder to derive.
$ \displaystyle \zeta(s) = 2 \int_{0}^{\infty} \frac{\sin (s \arctan t)}{(1+t^{2})^{s/2} (e^{2 \pi t} - 1)} \ dt + \frac{1}{2} + \frac{1}{s-1}$