Show that the sum of a series converges to the function...

[Like Tony Stark]

Show that the sum of [MATH] \sum_{n=1, 3, 5,...}^\infty \frac{4V_0}{\pi} \frac{1}{n} e^{-\frac{n\pi x}{a}} sin(\frac{n\pi y}{a}) [/MATH] is equal to [MATH]\frac{2V_0}{\pi} arctg(\frac{sin(\frac{\pi y}{a})}{sinh(\frac{\pi x}{a})})[/MATH]
I tried to think it as a Fourier series, but since it has two variables ([MATH]y[/MATH] and [MATH]x[/MATH]) I thought that this wasn't the way to solve it. I don't see any clear relation between the two expressions, so I don't where I should start. Can you give me a hint or an idea to start, please?

 

[yoscar04]

Maybe to write the sin as an exponential function? the 1/n can be obtained from the integral of e^ncx, where c is a constant. Is it then related to a geometric series?

 

[yoscar04]

Ok, your problem is the solution to the Laplace equation in 2d in Cartesian coordinates. Look at Jackson, Electrodynamics, section 2.10, equation 2.65. It indeed helps to write the sin as a exp.