Digital Object Identifier (DOI) 10.1007/s002110200399 Numer. Math. (2003) 94: 107–146 Numerische Mathematik Analytic continuation of Dirichlet-Neumann operators David P. Nicholls, Fernando Reitich School of Mathematics, University of Minnesota, Minneapolis, MN 5545, USA; e-mail: {nicholls,reitich}@math.umn.edu Received October 10, 2000 / Revised version received January 21, 2002 / Published online June 17, 2002 – c Springer-Verlag 2002 Summary. The analytic dependence of Dirichlet-Neumann operators (DNO) with respect to variations of their domain of definition has been successfully used to devise diverse computational strategies for their es- timation. These strategies have historically proven very competitive when dealing with small deviations from exactly solvable geometries, as in this case the perturbation series of the DNO can be easily and recursively eval- uated. In this paper we introduce a scheme for the enhancement of the domain of applicability of these approaches that is based on techniques of analytic continuation. We show that, in fact, DNO depend analytically on variations of arbitrary smooth domains. In particular, this implies that they generally remain analytic beyond the disk of convergence of their power se- ries representations about a canonical separable geometry. And this, in turn, guarantees that alternative summation mechanisms, such as Pad´ e approxi- mation, can be effectively used to numerically access this extended domain of analyticity. Our method of proof is motivated by our recent development of stable recursions for the coefficients of the perturbation series. Here, we again utilize this recursion as we compare and contrast the performance of our new algorithms with that of previously advanced perturbative methods. The numerical results clearly demonstrate the beneficial effect of incorporat- ing analytic continuation procedures into boundary perturbation methods. Moreover, the results also establish the superior accuracy and applicability of our new approach which, as we show, allows for precise calculations corresponding to very large perturbations of a basic geometry.