Abstract
We study the analog of the Cauchy-type integral for the theory of time-harmonic electromagnetic fields in case of a piece-wise Liapunov surface of integration and we prove the Sokhotski-Plemelj theorem for it as well as the necessary and sufficient condition for the possibility to extend a given pair of vector fields from such a surface up to a solution of the time-harmonic Maxwell equations in a domain. Formula for the square of the singular Cauchy-type integral is given. The proofs of all these facts are based on intimate relations between time-harmonic solutions of the Maxwell equations and some versions of quaternionic analysis.
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Schneider, B., Shapiro, M. Some properties of the Cauchy-type integral for the time-harmonic Maxwell equations. Integr equ oper theory 44, 93–126 (2002). https://doi.org/10.1007/BF01197863
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DOI: https://doi.org/10.1007/BF01197863