Binomial boundary value problems with shift for a piecewise analytic function and for a pair of functions analytic in the same domain

Abstract

As is known (see Section 5) a characteristic singular integral equation with Cauchy kernel (5.9) can be reduced to an equivalent binomial boundary value problem for a piecewise analytic function vanishing at infinity, namely, to the Riemann boundary value problem (5.9’). Solving this problem we determine the number l of linearly independent solutions and the number ρ of linearly independent conditions of solvability and we find these solutions and solvability conditions in explicit form. The characteristic singular integral equations with shift, considered in Section 6, can be treated as n-nomial boundary? value problems for a piecewise analytic function vanishing at infinity and, in general, n > 2. As apparent from the results of sections 6 and 5, these equations not only are not solvable in explicit form but also it is not possible, in the general case, to determine the numbers l and ρ.