In elliptical coordinates, the Laplace–Dirichlet equations on an ellipse are separable. In the "angular" coordinate (parameterizing confocal ellipses), the solution satisfies the Mathieu equation, and in the "radial" coordinate (parameterizing confocal hyperbolas), the solution satisfies the modified Mathieu equation. The eigenvalues are such that the solutions to the Mathieu equation are periodic, and the solutions to the modified Mathieu equation vanish on the boundary of the ellipse.