Eigenvalues of the Laplacian on an elliptic domain

The importance of eigenvalue problems concerning the Laplacian is well documented in classical and modern literature. Finding the eigenvalues for various geometries of the domains has posed many challenges which include infinite systems of algebraic equations, asymptotic methods, integral equations etc. In this paper, we present a comprehensive account of the general solutions to Helmholtz’s equations (defined on simply connected regions) using complex variable techniques. We consider boundaries of the form zz̄=f(z±z̄) or its inverse z±z̄=g(zz̄). To illustrate the theory, we reduce the problem on elliptic domains to equivalent linear infinite algebraic systems, where the coefficients of the infinite matrix are known polynomials of the eigenvalues. We compute truncations of the infinite system for numerical values. These values are compared to approximate values and some inequalities available in literature.