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The Keldysh-Sedov problem for functions holomorphic in a bi-halfplane

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Abstract

We pose and solve the Keldysh-Sedov problem: determine a function holomorphic in a bi-halfplane where its real and imaginary parts are given on mutually nonintersecting rectangles which exhaust the plane and which are the frame of the boundary of the bi-halfplane. Necessary and sufficient conditions for this problem to be solvable are given.

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Literature cited

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    A. M. Lavrent'ev and B. V. Shabat, Methods of the Theory of Functions of a Complex Variable [in Russian], Fizmatgiz, Moscow (1973).

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    V. A. Kakichev, “Boundary properties of a Cauchy type integral in several variables,” Uch. Zapiski Shakhtinskogo Ped. In-ta,2, No. 6, 25–90 (1959).

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    Le-din' Zon, “A Hubert problem for functions analytic in the bi-halfplane,” Sb. Funktsional'nyi Analiz, Teoriya Funktsii i Ikh Prilozheniya, No. 1, Makhachkala, 117–124 (1974).

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    N. I. Muskhelishvili, Singular Integral Equations [in Russian], Fizmatgiz, Moscow (1962).

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Additional information

Translated from Matematicheskii Zametki, Vol. 19, No. 5, pp. 681–690, May, 1976.

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Kakichev, V.A., Zon, L. The Keldysh-Sedov problem for functions holomorphic in a bi-halfplane. Mathematical Notes of the Academy of Sciences of the USSR 19, 410–415 (1976). https://doi.org/10.1007/BF01142561

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Keywords

  • Imaginary Part