On the Solution of Differential Equations by Definite Integrals

Abstract

It is well known that in many cases the solutions of a linear differential equation can be expressed as definite integrals, different solutions of the same equation being represented by integrals which have the same integrand, but different paths of integration. Thus, the various solutions of the hypergeometric differential equationcan be represented by integrals of the typethe path of integration being (for one particular solution) a closed circuit encircling the point t = 0 in the positive direction, then the point t = 1 in the positive direction, then the point t = 0 in the negative direction, and lastly the point t = 1 in the negative direction; or (for another particular solution) an arc in the t-plane joining the points t = 1 and t = ∞.