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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