Abstract
Diffraction of a plane electromagnetic wave (E-polarization) by two orthogonal electrically resistive half-planes is analyzed. The physical problem reduces to a Riemann-Hilbert problem in the real axis for four pairs of analytic functions \(\Phi^+_j(\eta)(\eta\ \in\ \rm C^+)$$ and $$\Phi^-_j(\eta) = \Phi^+_j (-\eta)(\eta\ \in\ {\rm C}^-),j = 1,2,3,4,\) where ℂ+ and ℂ are the upper and lower half-planes. It is shown that the problem is equivalent to two scalar Riemann-Hilbert problems on a plane and a Riemann-Hilbert problem on a genus-3 hyperelliptic surface subject to a certain symmetry condition. A closed-form solution is derived in terms of singular integrals and the genus-3 Riemann Theta function.
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Work partly funded by National Science Foundation through grant DMS0707724
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Antipov, Y.A. A Genus-3 Riemann-Hilbert Problem and Diffraction of a Wave by Two Orthogonal Resistive Half-Planes. Comput. Methods Funct. Theory 11, 439–462 (2012). https://doi.org/10.1007/BF03321871
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DOI: https://doi.org/10.1007/BF03321871