Abstract
In this paper, we obtain a generalization of the method of regularizing multipliers for the solution of the Hilbert boundary-value problem with finite index in the theory of analytic functions to the case of an infinite power-behaved index. This method is used to obtain a general solution of the homogeneous Hilbert problem for the half-plane, a solution that depends on the existence and the number of entire functions possessing mirror symmetry with respect to the real axis and satisfying some additional constraints related to the singularity characteristic of the index. To solve of the inhomogeneous problem, we essentially use a specially constructed solution of the homogeneous problem whereby we reduce the boundary condition of the Hilbert problem to a Dirichlet problem.
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REFERENCES
I. E. Sandrygailo, “On the Hilbert boundary-value problem with infinite index for the half-plane,” Izv. Akad. Nauk Belorus. SSR Ser. Fiz.-Mat. (1974), no. 6, 16–23.
P. Yu. Alekna, “The Hilbert boundary-value problem with infinite index of logarithmic order for the half-plane” Litovsk. Mat. Sb. [Lithuanian Math. J.], (1977), no. 1, 5–12.
N. I. Muskhelishvili, Singular Integral Equations [in Russian], Nauka, Moscow, 1968.
F. D. Gakhov, Boundary-Value Problems [in Russian], Nauka, Moscow, 1977.
L. I. Chibrikova, Main Boundary-Value Problems for Analytic Functions [in Russian], Izd. Kazan. Univ., Kazan, 1977.
L. I. Chibrikova and L. G. Salekhov, “To the solution of the Hilbert boundary-value problem,” in: Proc. of the Seminar on Boundary-Value Problems [in Russian], no. 8, Izd. Kazan. Univ., Kazan, 1971, pp. 155–175.
R. B. Salimov and V. V. Seleznev, “To the solution of the Hilbert boundary-value problem with discontinuous coefficients,” in: Proc. of the Seminar on Boundary-Value Problems [in Russian], no. 16, Izd. Kazan. Univ., Kazan, 1979, pp. 149–162.
N. V. Govorov, The Riemann Boundary-Value Problem with Infinite Index [in Russian], Nauka, Moscow, 1986.
B. Ya. Levin, Distribution of Roots of Entire Functions [in Russian], Gostekhizdat, Moscow, 1956.
A. I. Markushevich, The Theory of Analytic Functions [in Russian], vol. 2, Nauka, Moscow, 1968.
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Salimov, R.B., Shabalin, P.L. To the Solution of the Hilbert Problem with Infinite Index. Mathematical Notes 73, 680–689 (2003). https://doi.org/10.1023/A:1024064822157
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DOI: https://doi.org/10.1023/A:1024064822157