Solution of a second order difference equation using the bilinear relations of Riemann. (English) Zbl 1059.39002

Summary: A recently proposed technique to solve a class of second order functional difference equations arising in electromagnetic diffraction theory is further investigated by applying it to a case of intermediate complexity. The proposed approach is conceptually simple and relies on first obtaining well-defined branched solutions to a pair of associated first order difference equations. The construction of these branched expressions leads to an equation system whose solution requires relationships akin to Riemann’s bilinear relations for differentials of the first and third kinds; their derivation necessitates the application of Cauchy’s theorem on Riemann surfaces of, in this particular instance, genera one and three. Branch-free solutions of the second order difference equation are then obtained by taking appropriate linear combinations of the branched solutions of the first order equations. Analysis and computation demonstrate that the resulting expressions have the desired analytical properties and recover known solutions in the appropriate limit.


39A10 Additive difference equations
30E20 Integration, integrals of Cauchy type, integral representations of analytic functions in the complex plane
78A45 Diffraction, scattering
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