New method for investigating the Hilbert boundary value problem with an infinite logarithmic order index

R. B. Salimov ; E. N. Khasanova

Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 20 (2020) no. 3, pp. 297-309.

Résumé

We consider the problem of identification of the analytical in the complex upper half plane by boundary condition on the entire real axis, according to which, the real part of the product, by the given on the real axis complex function and the boundary values of the desired analytical function equal zero everywhere on the real axis. It is assumed that the argument of the coefficient of the boundary condition turns to infinity as one or another degree of the logarithm of the module of the coordinate of the axis point with unlimited distance of this point from the origin in one or another direction. Derived the formula that defines an analytical function in the upper half-plane, the imaginary part of which, when the coordinate of the axis point of the positive half-axis tends to infinity, is infinitely large of the same order as the argument of the coefficient of the boundary condition. Then derived a similar analytical function, the imaginary part of which turns to infinity of the same order as the argument of the coefficient of the boundary condition, when the points of the negative real axis are removed to infinity. We eliminate the infinite gap of the argument of the coefficient of the boundary condition by using these two functions. So the problem reduced to a finite index problem by techniques similar to F. D. Gakhov method. The method of F. D. Gakhov is used to solve the last problem. The solution depends on an arbitrary integer function of zero order, whose module satisfy to an additional condition.

Bibliographie

[1] I. E. Sandrygailo, “On Hilbert's boundary-value problem with infinite index for the half-plane”, Izv. Akad. nauk Belorus. SSR. Ser. fiz. mat. nauk, 1974, no. 6, 16–23 (in Russian) | MR

[2] P. Alekna, “The Hilbert boundary-value problem with infinite index of logarithmic orden in the half-plane”, Lith. Math. J., 17 (1978), 1–6 | DOI | MR | Zbl

[3] N. I. Muskheshvili, Singular Integral Equations, Nauka, M., 1968, 511 pp. (in Russian) | MR

[4] N. V. Govorov, Riemann Boundary Problem with Infinite Index, Nauka, M., 1986, 289 pp. (in Russian)

[5] R. B. Salimov, P. L. Shabalin, “To the solution of the Hilbert problem with infinite index”, Math. Notes, 73:5 (2003), 680–689 | DOI | DOI | MR | Zbl

[6] F. D. Gakhov, Boundary-Value Problems, Nauka, M., 1977, 640 pp. (in Russian)

[7] E. N. Karabasheva, “On solvability of homogeneous Hilbert problem with countable set of points discontinuities and of a different order two-side curling at infinity”, News of the KSUAE, 2014, no. 1 (27), 242–252 (in Russian)

[8] P. G. Yurov, “The homogeneous Riemann boundary value problem with an infinite index of logarithmic type”, Izv. Vyssh. Uchebn. Zaved. Mat., 1966, no. 2, 158–163 (in Russian)

[9] R. B. Salimov, E. N. Khasanova, “The Solution of the Homogeneous Boundary Value Problem of Riemann with Infinite Index of Logarithmic Order on the Beam by a New Method”, Izv. Saratov Univ. (N. S.), Ser. Math. Mech. Inform., 17:2 (2017), 160–171 (in Russian) | DOI | MR | Zbl

[10] A. I. Markushevich, The theory of analytic functions, in 2 vols., v. 1, Nauka, M., 1968, 486 pp. (in Russian) | MR

[11] A. I. Markushevich, The theory of analytic functions, in 2 vols., v. 2, Nauka, M., 1968, 624 pp. (in Russian) | MR

[12] B. Ya. Levin, Distribution of Zeros of Entire Functions, Gostechizdat, M., 1956, 632 pp. (in Russian)