Shabalin P.L. Hilbert’s boundary value problem for a class of generalized analytic functions with a singular line
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https://doi.org/10.15688/mpcm.jvolsu.2022.4.2
Pavel L. Shabalin
Doctor of Physical and Mathematical Sciences, Associate Professor,
Professor of the Department of Higher Mathematics,
Kazan State University of Architecture and Engineering
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https://orcid.org/0000-0002-8764-6132
Zelenaya Str.,1 – 420043 Kazan, Tatarstan Russian Federation
Abstract. In this paper, we study an inhomogeneous Hilbert boundary value problem with a finite index and a boundary condition on the real axis for one generalized equation Cauchy — Riemann with singular coefficient. A structural formula for the general solution of this equation is obtained under constraints leading to an infinite index of the accompanying Hilbert boundary value problem for analytic functions. The study of the solvability of the latter is the basis for solving the boundary value problem for generalized analytical functions. Note that here we have obtained a boundary value problem with an infinite index and two swirl points. Such a situation is not studied in the case of an unbounded domain, even for analytic functions. We have obtained a formula for the general solution of the indicated Hilbert problem in the theory of analytic functions. We carried out a complete study of the solvability of this problem in terms of the characteristics of the coefficient of the boundary condition. Note that here we have obtained a boundary value problem with an infinite index and two swirl points. Such a situation is not studied in the case of an unbounded domain, even for analytic functions. We have obtained a formula for the general solution of the indicated Hilbert problem in the theory of analytic functions. We carried out a complete study of the solvability of this problem in the class of bounded analytic functions. This study made it possible to obtain a formula for solving the Hilbert problem already for generalized analytic functions. We also formulated the conditions under which the problem has no solution, has a single solution or an infinite number of solutions.
Key words: Hilbert boundary value problem, generalized analytic functions, singular line, infinite index, entire function of given order zero.
Hilbert’s boundary value problem for a class of generalized analytic functions with a singular line by Shabalin P.L. is licensed under a Creative Commons Attribution 4.0 International License.
Citation in English: Mathematical Physics and Computer Simulation. Vol. 25 No. 4 2022, pp. 15-28
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