About the solution in closed form of generalized markushevich boundary value problem in the class of analytical functions

Abstract

The paper is devoted to the investigation of the problem of obtaining piecewise analytical functions F(z) = {F⁺ (z), F⁻ (z)} with the jump line L, vanishing on the infinity and satisfying on L the boundary condition

where α(t) is the preserving orientation homeomorphism of L onto itself and G(t), b(t), g(t) are given on L functions of Holder class and G(t) ? 0 on L.

The algorithm for the solution of this problem was obtained and particular cases, when it is solvable in closed form are determined.

Darbe pateikiamas algoritmas Markuševičiaus uždavinio, kai ieškomos dalimis analizines funkcijos F(z) = {F⁺ (z), F⁻ (z)} nykstančioje begalybeje, savo šuoliu linijoje L tenkinančios salyga

kur G(t), b(t), g(t) ‐ apibrežtos kontūre L funkcijos Golderio klases, o α(t) ‐ homemor‐fizmas kontūro i save. Atvejui α (t) = t uždavini suformulavo A.I. Markuševičius 1946 m. Irodyta, kad uždavinio sprendimas suvedamas i integralines antrosios rūšies Fredholmo tipo lygties sprendima. Pateikiamas pavyzdys, iliustruojantis gautus teorinius rezultatus.

Key words:

• bianalytical function
• boundary value problem
• plane with slots
• index