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\begin{document}

For a box, a x b x c, with \( \phi =0 \) on all sides except one where \( \phi =V(x,y) \).

Solve laplaces equation: with \( \phi =F(x)G(y)H(z) \). Then \( \frac{1}{F}F_{xx}+\frac{1}{G}G_{yy}+\frac{1}{H}H_{yy}=0 \) so \( F(x)\propto e^{\pm i\alpha x} \), \( G(y)\propto e^{\pm i\beta y} \), and \( H(z)\propto e^{\pm \gamma z} \). 

checking with B.C.'s you get: \( \alpha _{n}=n\frac{\pi }{a} \) and \( \beta _{m}=m\frac{\pi }{a} \). The full solution is then:
\( \phi =\sum _{n,m}A_{nm}sin(\alpha _{n}x)sin(\beta _{m}y)sinh(\gamma _{nm}z) \) where \( A_{nm}=\frac{4}{absinh(\gamma _{mn}c)}\int _{0}^{a}dx*sin(\alpha _{n}x)\int _{0}^{b}dy*sin(\beta _{n}y)V(x,y) \). The solution can be found for arbitrary B.C.'s by superpositions
of this solution.



\end{document}
