The Stokes phenomenon and certain nth-order differential equations

Abstract

In papers I(1) and II(2), the author has investigated the Stokes phenomenon associated with the arbitrary constants that occur in the asymptotic expressions of solutions of the nth-order differential equationand has provided rules for tracing a given linear combination of n such asymptotic expressions across certain defined lines, designated as Stokes lines, anti-Stokes lines and branch-cuts. The object of these investigations is to apply the results to the approximate solution of more complicated linear differential equations of the nth order, and to this end it is an advantage to develop the theory using matrix techniques. In the present paper, the equations are recast into a special matrix form, and it is shown how the results obtained in II may be used in a systematic manner.