- #1
arpon
- 235
- 16
Homework Statement
Prove that,
$$\sum _{n=1,3,5...} \frac{1}{n} e^{-nx} \sin{ny} = \frac{1}{2}\tan^{-1} (\frac{\sin{y}}{\sinh{x}})$$
Homework Equations
$$\tan^{-1}{x} = x - \frac{x^3}{3} +\frac{x^5}{5} - ... $$
3. The Attempt at a Solution
$$\sum _{n=1,3,5...} \frac{1}{n} e^{-nx} \sin{ny}$$
$$= \sum _{n=1,3,5...} \frac{1}{n} e^{-nx} Im(e^{i ny})$$
$$= \sum _{n=1,3,5...} \frac{ Im(e^{(-x+iy)n})}{n} $$
$$=Im( \sum _{n=1,3,5...} \frac{z^n}{n}) $$
where ##z=e^{-x+iy}##.
I am stuck here. Any help will be appreciated.