Formal power series and their applications in the mathematical theory of diffraction. (English. Russian original) Zbl 1303.78003

J. Math. Sci., New York 194, No. 1, 1-7 (2013); translation from Zap. Nauchn. Sem. POMI 409, 5-16 (2012).
Summary: Formal power series (FPS) the coefficients of which are smooth functions are considered. The FPS form an algebra on the field of complex numbers \((\mathbb C)\). One can differentiate the FPS. The FPS are series having an asymptotic nature (in accordance with the definition by V. S. Buslaev and M. M. Skriganov [Teor. Mat. Fiz. 19, 217–232 (1974; Zbl 0291.35020); translation in Theor. Math. Phys. 19(1974), 465–476 (1975; Zbl 0296.35023)]). As an example of applications of the FPS, the geometro-optical expansion for the scalar analog of Rayleigh waves is considered.


78A45 Diffraction, scattering
35C20 Asymptotic expansions of solutions to PDEs
78M35 Asymptotic analysis in optics and electromagnetic theory
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[1] V. M. Babich, ”Quasiphotons and the space-time ray method,” Zap. Nauchn. Semin. POMI, 342, 5–13 (2008). · Zbl 1176.81170
[2] V. M. Babich and A. I. Popov, ”Quasiphotons of waves on the surface of a heavy liquid,” Zap. Nauchn. Semin. POMI, 379, 5–23 (2010).
[3] V. M. Babich and V. S. Buldyrev, Asymptotic Methods in Problems of Plane Wave Diffraction [in Russian], Nauka, Moscow (1972). · Zbl 0255.35002
[4] V. P. Maslov, Complex WKB Method in Nonlinear Equations [in Russian], Nauka, Moscow (1977). · Zbl 0449.58001
[5] S. Bochner and W. T. Martin, Several Complex Variables [Russian Translation], Moscow (1951).
[6] V. S. Buslaev and M. M. Skriganov, ”Coordinate asymptotics of the solution of the scattering problem for the Schr√∂dinger equation,” Teor. Mat. Fiz., 19, No. 2, 217–232 (1974). · Zbl 0291.35020
[7] V. M. Babich, V. S. Buldyrev, and I. A. Molotkov, The Space-Time Ray Method [in Russian], Izd. Leningr. Univ., Leningrad (1985). · Zbl 0678.35002
[8] V. M. Babich and A. I. Popov, ”An asymptotic solution of the Hamilton–Jacobi equation concentrated near the surface,” Zap. Nauchn. Semin. POMI, 393, 23–29 (2011).
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