I'm trying to solve Laplace's equation $\nabla^2 \phi=0$ in Cartesians on $0
$\sum_{m=1}^{\infty} \; \sum_{n=1}^{\infty} \alpha_{m,n} \sinh\left[ \left( \frac{m^2}{a^2} + \frac{n^2}{b^2} \right)^{1/2} \pi c \right] \sin\frac{m\pi x}{a} \sin\frac{n\pi y }{b} =1.$
I should like to find the coefficients $\alpha_{m,n}$ by using orthogonality of the sine functions. I feel the result ought to be given by some double integral $\int_0^a ... dx \int_0^b ...dy$ but I'm not at all sure how to work out the details.
[I didn't normalise my eigenfunctions in $x$ or $y$ -- does this make it significantly messier than it needs to be?]
Thanks!