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A Canonical Diffraction Problem With Two Media
Author(s) -
Rottbrand K.
Publication year - 1996
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/(sici)1099-1476(199610)19:15<1217::aid-mma826>3.0.co;2-s
Subject(s) - mathematics , riemann–hilbert problem , mathematical analysis , boundary value problem , scalar (mathematics) , factorization , complex plane , pure mathematics , geometry , algorithm
A Wiener–Hopf equation in L 2 being equivalent [5] to a boundary value problem (of the first kind) for a wave‐scattering Sommerfeld half‐plane Σ=ℝ + ×{0} which faces two different media Ω ‐ : x 2 <0, Ω + : x 2 >0, as a special configuration in [3], is solved by canonical Weiner – Hopf factorization of its L 2 ‐ regular scalar symbol γ o =γ o ‐ γ o + . The factors are calculated by solving a Riemann–Hilbert boundary value problem on the semi‐infinite branch cuts of t j (ξ):=(ξ 2 − k 2 j ) 1/2 , k j ∈ℂ ++ for j =1,2: taken parallel to the imaginary axis. The procedure following this idea is known as the Wiener–Hopf–Hilbert(–Hurd) method [2] and requires the evaluation of elliptic‐type integrals. Formula (3.7) seems not to be contained in tables of integrals.