Rahul Roy, Indian Statistical Institute, New Delhi

Abstract

The classical Poisson problem is to obtain the solution of the equation $\frac{1}{2} \bigtriangleup u = -1$ for a function $u$ of a two-dimensional domain $D$ which vanishes on the boundary of $D$. We obtain an explicit solution of the above equation when $D$ is an equilateral triangle. We obtain this solution through probabilistic methods, in particular a modification of the classical ruin problem. With such methods we also give a proof of the Euler's formula $\zeta(2) = \sum_{n=1}\infty \frac{1}{n^2} = \frca{\pi^2}{6}$.

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