Singularities and the Rayleigh Hypothesis for Solutions to the Helmholtz Equation

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Abstract
Possible and actual singularities in the analytic continuation of the solution to the Dirichlet problem for the two-dimensional Helmholtz equation are studied in order to investigate the Rayleigh hypothesis of scattering theory. The procedure uses a representation for the solution to the analytic Cauchy problem and relates one type of singularity in the Schwarz function associated with the boundary curve to singularities in the solution. The results provide confiration of criteria for the validity of the Rayleigh hypothesis that have been given by van den Berg and Fokkema.