Series identity for cotangent

Asked Jun 02 '15 at 08:19

Active Jun 02 '15 at 08:19

Viewed 772 times

How to prove that $x \cot(x) = 1 - 2 \sum_{n=0}^{\infty}{\frac{x^{2}}{(n \pi)^{2}-x^{2}}}$?

First, it does not seem to be solvable, using considerations regarding Taylor series. The Fourier approach also is not effective enough (i should also take into consideration that $x \cot(x)$ has infinite discontinuites, so it narrows expansion to the interval, where the function behaves well).

Any help would be much appreciated.

asked Jun 02 '15 at 08:19

hyperkahler

2,921

1

You can start with the infinite product expression $\frac{\sin,\pi x}{\pi x}=\prod_{k=1}^\infty \left(1-\frac{x^2}{k^2}\right)$, take $\log$ on both sides and differentiate w.r.t. $x$, or see here for a nice solution. – r9m Jun 02 '15 at 08:39
possible duplicate of Find the sum of $\sum 1/(k^2 - a^2)$ when $0<a<1$ – r9m Jun 02 '15 at 15:21

Series identity for cotangent

0 Answers0

Linked