THE CENTRAL 2-POINT CONNECTION PROBLEM OF HEUNS CLASS OF DIFFERENTIAL-EQUATIONS

Boundary-value problems of ordinary, linear, homogeneous second-order differential equations belong to the most important and thus well-inve stigated problems in mathematical physics. This statement is true only as long as irregular singularities of the differential equation at ha nd are not involved. If singular points of irregular type enter the pr oblem one will hardly find a systematic investigation of such a topic from a practical point of view. This paper is devoted to an outline of an approach to boundary-value problems of the class of Heun's differe ntial equation when irregular singularities may be located at the endp oints of the relevant interval. We present an approach to the central two-point connection problem for all of these equations in a quite uni form manner. The essential point is an investigation of the Birkhoff s ets of irregular difference equations, which, on the one hand, gives a detailed insight into the structure of the singularities of the under lying differential equation and, on the other hand, yields the basis o f quite convenient algorithms for numerical investigations of the boun dary values.