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On an integral equation in diffraction theory

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On an integral equation in diffraction theory COMPOSITIO MATHEMATICA ALBERT E. HEINS On an integral equation in diffraction theory Compositio Mathematica, tome 18, no 1-2 (1967), p. 49-54. <> © Foundation Compositio Mathematica, 1967, tous droits réservés. L’accès aux archives de la revue « Compositio Mathematica » (http:// implique l’accord avec les conditions générales d’utilisation ( Toute utilisation commer- ciale ou impression systématique est constitutive d’une infraction pé- nale. Toute copie ou impression de ce fichier doit contenir la pré- sente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques 49 On an integral equation in diffraction theory by Albert E. Heins The following remarks are intended to shed further light on representation theorems for partial differential equations of a special form, in particular the two-dimensional wave equation. It has already been shown in the past decade that axially-symmetric boundary value problems of the Dirichlet or Neumann type for the three dimensional-wave equation for the disk or a disk between parallel planes may be formulated as a regular integral equation of the second kind. The advantage to such a formulation requires no further comment. These Fredholm integral equations of the second kind are derived from the Poisson representation for solutions of the wave equation and the analytically continued axis data of the Helmholtz representation. In a recent paper, J. Boersma [1] has given a similar representation for the two dimensional wave equation by accepting the three dimensional representation as a guide to construct the two dimensional one. We show here that this is again a result of the analytically continued data of the Helmholtz representation in two dimensions and the corresponding version of the Poisson represe

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