  We consider the Laplace operator with Dirichlet boundary conditions on a
planar domain and study the effect that performing a scaling in one direction
has on the spectrum. We derive the asymptotic expansion for the eigenvalues and
corresponding eigenfunctions as a function of the scaling parameter around
zero. This method allows us, for instance, to obtain an approximation for the
first Dirichlet eigenvalue for a large class of planar domains, under very mild
assumptions.
