On the Solution of a Generalized Wiener-Hopf Equation

This paper deals with the generalized Wiener‐Hopf equation $${\mathrm{G(\alpha )X}}_{+}{\mathrm{(\alpha )+H(\alpha )X}}_{+}{\mathrm{(-\alpha )-Y}}_{-}{\mathrm{(\alpha )+\Psi }}^{\mathrm{(i)}}{\text{(α)=0,\hspace{0.17em}τ}}_1{\mathrm{<\tau <\tau }}_2$$ , where α (= σ + iτ) is a complete variable, G(α), H(α), and Ψ⁽ⁱ⁾(α) are known functions, and X₊(α) and Y₋(α) are unknowns, analytic in upper and lower half‐planes, respectively, as indicated by their respective subscripts. This type of equation arises in a class of boundary‐value problems in electromagnetic theory, the geometries of which may be described as modified Wiener‐Hopf type. The method of approach, which is fundamentally different than those currently available in the literature, is based on a pairing of singularities in the complex α plane. This leads to a functional equation which is exactly solvable in its asymptotic form. The knowledge of this solution permits one to employ one of several rapidly converging numerical procedures available in the literature for a more accurate solution. Two examples illustrating the application of the procedure are included in the paper.