We construct a system of solutions for the coefficients of expansion in the system of trigonometric functions for the solution of the Helmholtz equation in a cylindrical coordinate system in the form of homogeneous polynomials in two biorthogonal systems of functions. Some properties of the biorthogonal systems of functions are proved.
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References
E. L. Ince, Ordinary Differential Equations, Longmans & Green Co., London (1927).
H. Bateman and A. Erdélyi, Higher Transcendental Functions, Vol. 2: Bessel Functions, Parabolic Cylinder Functions, and Orthogonal Polynomials, McGraw-Hill, New York (1953).
V. S. Vladimirov, Equations of Mathematical Physics [in Russian], Nauka, Moscow (1981).
N. N. Vorob’ev, Theory of Series [in Russian], Nauka, Moscow (1979).
V. F. Zheverzheev, L. A. Kal’nitskii, and N. A. Sapogov, Special Course of Higher Mathematics for Technical Colleges [in Russian], Vysshaya Shkola, Moscow (1970).
E. Kamke, Differentialgleichungen: Lösungsmethoden und Lösungen, Vol. I: Gewöhnliche Differentialgleichungen, Teubner, Leipzig (1977); https://doi.org/10.1007/978-3-663-05925-7.
M. A. Lavrent’ev and B. V. Shabat, Methods of the Theory of Functions of Complex Variable [in Russian], Nauka, Moscow (1987).
A. I. Markushevich, Selected Chapters of the Theory of Analytic Functions [in Russian], Nauka, Moscow (1976).
A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev, Integrals and Series, Vol. 1: Elementary Functions [in Russian], Nauka, Moscow (1981); English translation: P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev, Integrals and Series, Vol. 1: Elementary Functions, Gordon & Breach Sci. Publ., New York (1986).
M. A. Sukhorolsky, “Expansion of functions in a system of polynomials biorthogonal on a closed contour with a system of functions regular at infinitely remote point,” Ukr. Mat. Zh., 62, No. 2, 238–254 (2010); English translation: Ukr. Math. J., 62, No. 2, 268–288 (2010); https://doi.org/10.1007/s11253-010-0350-6.
M. A. Sukhorolsky, “Systems of solutions of the Helmholtz equation,” Visn. Nats. Univ. “L’viv. Politekhnika”. Ser. Fiz.-Mat. Nauky, No. 718, 19–34 (2011).
M. A. Sukhorolsky and V. V. Dostoyna, “One class of biorthogonal systems of functions that arise in the solution of the Helmholtz equation in the cylindrical coordinate system,” Mat. Met. Fiz.-Mekh. Polya, 55, No. 2, 52–62 (2012); English translation: J. Math. Sci., 192, No. 5, 541–554 (2013); https://doi.org/10.1007/s10958-013-1415-5.
M. A. Sukhorolsky, V. V. Dostoina, and O. V. Veselovska, “Biorthogonal systems of solutions of the Helmholtz system in the cylindrical coordinate system,” Visn. Nats. Univ. “L’viv. Politekhnika,” Ser. Fiz.-Mat. Nauky, No. 898, 56–68 (2018).
M. A. Sukhorolsky, I. S. Kostenko, and V. V. Dostoina, “Construction of the solutions of partial differential equations in the form of contour integrals,” Vestn. Kherson. Nats. Univ., No. 2(47), 323–326 (2013).
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Translated from Matematychni Metody ta Fizyko-Mekhanichni Polya, Vol. 64, No. 4, pp. 47–54, October–December, 2021.
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Veselovska, O.V., Dostoina, V.V. & Drohomyretska, K.T. Construction of Solutions of the Helmholtz Equation in a Cylindrical Coordinate System in the Form of Homogeneous Polynomials in Two Biorthogonal Systems of Functions. J Math Sci 279, 170–180 (2024). https://doi.org/10.1007/s10958-024-07003-5
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DOI: https://doi.org/10.1007/s10958-024-07003-5