  We consider the Dirichlet and Neumann eigenvalues of the Laplacian for a
simply connected, planar domain. The eigenvalues are the characteristic values
of layer potential operators involving the Hankel function. First, we obtain an
asymptotic formula for the Laplace eigenvalues with respect to the perturbation
of the domain. The results are based on the Gohberg--Sigal theory for operator
valued functions. Second, we propose a numerical computation scheme to compute
the Laplacian eigenvalues based on the matrix formulation of the layer
potential operators with a geometric basis associated with the exterior
conformal mapping of the domain. This provides a way to compute Laplace
eigenvalues, $\lambda$, by finding roots of polynomials in $\lambda$ and
$\log\lambda$. We also derive a fully computable a priori error estimate with
no assumption on the domain's convexity. This shows the relation between the
domain's regularity and the convergence rate of the proposed method to compute
Laplace eigenvalues.
