CONVERGENCE THEOREM AND OBTAINED EXPANSION FOR THE VECTOR FUNCTION OF SECOND ORDER DIFFERENTIAL EQUATIONS

Abstract

The first quarter of Nineteenth Century is considered as a basis to the theory of boundary value problems, which involves the determination of solutions of differential equations that satisfy prescribed boundary conditions. By the application of the method of separation of variable to partial differential equations of mathematical physics, one was led to the expansion of an arbitrary function in terms of a system of functions known as 'proper functions' or 'eigen functions' of a differential equation for corresponding 'proper values' or 'eigne values' of an involved parameter. In this paper contains a convergence theorem and obtained expansion for the vector function of the type which is continuous in some suitable interval and bounded variation in that interval, when p(x) and q(x) tends to +  or -, which will be suitable in the work.