The Dirichlet problem for Poisson’s equation in a right-isosceles triangle

We derive an explicit series representation for the Dirichlet problem for Poisson’s equation in a right-isosceles triangle. Linearity is used to decompose the boundary data into three harmonic subproblems, each carrying the prescribed values on a single side, and a fourth Poisson subproblem with homogeneous Dirichlet conditions. The two leg-driven harmonic fields admit simple sine-hyperbolic expansions in the original coordinates, while the hypotenuse-driven field is obtained after a 45 ∘ rotation and scaling that maps the domain to a symmetric triangle. The remaining forced problem is treated by an eigenfunction expansion of the Dirichlet Laplacian, with coefficients obtained by orthogonal projection. The resulting formula provides a convenient analytic benchmark for assessing numerical solvers on polygonal domains.