The Stokes phenomenon and certain nth-order differential equations II. The Stokes phenomenon

Abstract

1. Introduction. In paper I(2), solutions have been found for the nth-order differential equations (I, 3), (I, 9), (I, 18), namely,where ϑ = wd/dw, m is any rational fraction and a any constant. Power-series solutions have been found for these equations, together with integral representations and their asymptotic expressions valid for restricted ranges of arg z. The object of this second paper is to consider these asymptotic solutions in more detail, and to extend these expressions to all values of arg z. The Stokes phenomenon will be manifest throughout, and this will be treated in a manner suitable for further application.