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dnu dn-iu dn-2u dz + P(z) + P2(Z) dn +* ? {pn(z) pnckn(Z)Iu = 0 dzn ~ dz n1 dz n2 have been studied by various mathematicians, induding Birkhoff,t Schlesinger, and Horn, with regard to the asymptotic forms of the solutions for large complex values of the parameter p. The second-order case goes back to Liouville. The essential differences between the present treatment and previous treatments of the nth-order case are the following: (1) Before this it has been necessary to assume the coefficient 4n(z) to be bounded from zero. We permit it to have a zero of any integral order v. (2) The variable z has been confined to a finrite interval of the real axis. We allow it to vary over complex values in a finite region. (3) In order that the forms be valid for all p in the neighborhood of p = 00, it has been necessary to require that the coefficient din(z) be real, except possibly for a constant imaginary factor. We dispense with this restriction. Besides proving the existence of solutions asymptotic to n simple independent expressions, we consider the following problem arising from the first of the above generalizations: What is the form of a solution determined by boundary values at the point where the coefficient 45n(z) has its zero? At a point where q5n(z) is different from zero the problem becomes elementary, since it is always possible to find a non-singular matrix of simple forms for n solutions and their first n-1 derivatives which is valid in some neighborhood including the point. But in our case the simple forms are not valid at the zero. The problem is complicated by the Stokes phenomenon, that is, the discontinuity, as a solution is taken across a "Stokes line," of the coefficients

Asymptotic solutions of certain linear differential equations in which the coefficient of the parameter may have a zero