# Using orthogonality of sines to find coefficients from a given boundary condition

I'm trying to solve Laplace's equation $\nabla^2 \phi=0$ in Cartesians on $0<x<a,\; 0<y<b,\;0<z<c$ by separation of variables. A boundary condition gives the equation

$$\sum_{m=1}^{\infty} \; \sum_{n=1}^{\infty} \alpha_{m,n} \sinh\left[ \left( \frac{m^2}{a^2} + \frac{n^2}{b^2} \right)^{1/2} \pi c \right] \sin\frac{m\pi x}{a} \sin\frac{n\pi y }{b} =1.$$

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!

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$\times \int \sin \frac{p \pi x}{a} dx \int \sin \frac{q \pi y}{b} dy$
Then use orthogonality to realize that $m$ must equal $p$ and $n$ must equal $q$. You should then get a nice expression for $\alpha_{m,n}$