# How to obtain the analytic representation of the given infinite series?

This series is given in Griffiths Introduction to Electrodynamics chapter 3 in an example explaining seperation of variables.

$$V(x,y) = \frac{4V_0}{\pi} \sum_{n=1,3,5\ldots} \frac{1}{n}e^{-n\pi x/a}\sin(n\pi y/a)$$ $$V(x,y) = \frac{2V_0}{\pi} \tan^{-1}\left(\frac{\sin(\pi y/a)}{\sinh(\pi x/a)} \right)$$

I don't have any idea how to get the analytic representation of the series. I tried replacing $$n$$ by $$-n$$ to get hyperbolic sine but it also introduces summation over negative odd integers and I don't think they can be added over. Also, it will form hyperbolic sine in the numerator rather than the denominator. How to proceed?

• Replacing $n$ by $-n$ doesn’t make the sine turn into a hyperbolic sine. Jul 22, 2019 at 16:07
• A sum over negative integers is perfectly valid, but I don't think it helps here. Jul 22, 2019 at 16:09
• In the old days one spend a lot more time poring over math tables and just knew which section to go look in...
– Jon Custer
Jul 23, 2019 at 20:31
• Put $n=2k-1$, use $\sin\,x =\frac {e^{ix}-e^{-ix}} {2i}$ and you will end up with a series of the type $\sum \frac {r^{k}} {2k-1}$ with $|r| <1$. Can you handle this series? Jul 26, 2019 at 23:13
• Yeah, it's solved. Thanks. Jul 27, 2019 at 6:29