About new approach to solution of Riemann's boundary value problem with condition on the half-line in case of infinite index

R. B. Salimov

Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 16 (2016) no. 1, pp. 29-33.

Résumé

To solve a homogeneous Riemann boundary value problem with infinite index and condition on the half-line we propose a new approach based on the reduction of the considered problem to the corresponding task with the condition on the real axis and finite index. It is required to define a function $\Phi(z)$, analytic and bounded in the complex plane $z$, cut down on positive real semi-axis $L^+$, if the edge condition $\Phi^{+}(t)=G(t) \Phi^{-}(t)$, $t\in L^{+}$ is fulfilled, where $\Phi^{+}(t)$, $\Phi^{-}(t)$ are limit values of the function $\Phi(z)$, as $z\to t$ correspondingly on the left and on the right, $G(t)$ is a given function, for which argument $\arg G(t)=\nu^{-}t^{\rho}+\nu(t)$, $t\in L^{+}$ holds, here $\nu^{-}$, $\rho$ are given numbers, $\nu^{-}>0$, $\frac{1}{2}\rho1$, and $\ln|G(t)|$, $\nu(t)$ are functions which satisfy the Holder condition. It is admitted that $G(t)=1$ at $t\in(-\infty,0)$. The functions $E^{+}(z)=e^{(\alpha+i\beta)z^{\rho}}$, $0\le \arg z \le \pi$, $E^{-}(z)=e^{(\alpha-i\beta)z^{\rho}}$, $-\pi\le \arg z \le 0$ are used to avoid infinite gap of the $\arg G(t)$, by the selection of real numbers $\alpha$, $\beta$.

Bibliographie

[1] Salimov R. B., Karabasheva E. N., “The new approach to solving the Riemann boundary value problem with infinite index”, Izv. Saratov Univ. (N.S.), Ser. Math. Mech. Inform., 14:2 (2014), 155–164 (in Russian)

[2] Gakhov F. D., Boundary value problems, Nauka, M., 1977, 640 pp. (in Russian)

[3] Markushevich A. I., The theory of analytic functions, in 2 vol., v. 2, Nauka, M., 1968, 624 pp. (in Russian)

[4] Govorov N. V., Riemann's boundary problem with infinite index, Nauka, M., 1986, 239 pp. (in Russian)