Boundary Value Problems For Analytic Functions Research Articles

Riemann–Hilbert Problems on a Cut Plane

We consider the Riemann–Hilbert-type boundary-value problems in the case of several unknown functions and obtain solvability conditions and index formulas for the linear conjugation problem, the Riemann–Hilbert problem, and the Riemann–Hilbert–Poincare problem on a plane cut along several regular arcs in various weighted functional classes. We also examine the general differential boundary-value problem for analytic functions and boundary-value problems with shifts and complex conjugation on a cut plane.

Solution of a singular homogeneous Hilbert boundary-value problem for analytic function in multiply connected circular domain

We offer a new approach for solving the homogeneous Riemann-Hilbert boundary-value problem for analytic function in multiply connected circular domains. The approach is based on determination of analytic function in terms of known boundary values of its argument in a special case.

Haseman boundary value problems for metaanalytic functions with different shifts on the unit circumference

In this article, Haseman boundary value problems (BVPs) for a class of metaanalytic functions with different shifts on the unit circumference are investigated. The expression of solution and the condition of solvability for the problem are obtained by reducing the problem to an equivalent system of two Haseman BVPs for analytic functions.

Asymmetric three-dimensional Stokes flows about two fused equal spheres

Exact solutions to three-dimensional Stokes flow problems for asymmetric translation and rotation of two fused rigid spheres of equal size have been obtained in toroidal coordinates. The problems have been reduced to three-contour equations for meromorphic functions from a certain class, and then the latter have been reduced to Fredholm integral equations of the second kind by the Mehler–Fock transform of order 1. For the specified class of functions, the equivalence of the corresponding three-contour and Fredholm equations has been established in the framework of Riemann boundary-value problems for analytic functions. As an illustration for the obtained solutions, the pressure has been calculated at the surface of the body for both problems, and resisting force and torque, experienced by the body in asymmetric translation and rotation, have been computed as functions of a geometrical parameter of the body.

Hilbert Formulas for r-Analytic Functions in the Domain Exterior to Spindle

Hilbert formulas for an r-analytic function, defined by a generalized Cauchy--Riemann system in the domain exterior to the contour of a spindle in the meridional cross-section plane, have been derived. The derivation is based on the theory of Riemann boundary-value problems for analytic functions. For numerical calculations, Fourier integrals with Hilbert formulas representing the real and imaginary parts of the r-analytic function have been reduced to the form of regular integrals. The problem of the axially symmetric steady motion of a rigid spindle-shaped body in a Stokes fluid has been solved, and the pressure in the fluid has been expressed analytically based on a Hilbert formula. As an illustration, streamlines about the body, vortex and pressure functions at the contour of the body, and the drag force, exerted on the body by the fluid, have been calculated.

Analytic solution of the relativistic Coulomb problem for a spinless Salpeter equation

We construct an analytic solution to the spinless $S$-wave Salpeter equation for two quarks interacting via a Coulomb potential, $[2{(\ensuremath{-}{\ensuremath{\nabla}}^{2}+{m}^{2})}^{\frac{1}{2}}\ensuremath{-}M\ensuremath{-}\frac{\ensuremath{\alpha}}{r}] \ensuremath{\psi}(r)=0$, by transforming the momentum-space form of the equation into a mapping or boundary-value problem for analytic functions. The principal part of the three-dimensional wave function is identical to the solution of a one-dimensional Salpeter equation found by one of us and discussed here. The remainder of the wave function can be constructed by the iterative solution of an inhomogeneous singular integral equation. We show that the exact bound-state eigenvalues for the Coulomb problem are ${M}_{n}=\frac{2m}{{(1+\frac{{\ensuremath{\alpha}}^{2}}{4{n}^{2}})}^{\frac{1}{2}}}$, $n=1, 2, \dots{}$, and that the wave function for the static interaction diverges for $r\ensuremath{\rightarrow}0$ as $C{(\mathrm{mr})}^{\ensuremath{-}\ensuremath{\nu}}$, where $\ensuremath{\nu}=(\frac{\ensuremath{\alpha}}{\ensuremath{\pi}})(1+\frac{\ensuremath{\alpha}}{\ensuremath{\pi}}+\ensuremath{\cdots})$ is known exactly.