The asymptotic solution of linear second order differential equations in a domain containing a turning point and a regular singularity

Abstract

Asymptotic solutions of the differential equation d1 2wjdz2 = {u2z~2(z0—z) pi(z) +z ~2ql(z)} w, for large positive values of u are examined; P ₁ (z) AND Q ₁ (Z) are regular functions of the complex variable z in a domain in which P ₁ (z) does not vanish. The point z = 0 is a regular singularity of the equation and a branch-cut extending from z = 0 is taken through the point Z=Z _O which is assumed to lie on the positive real z axis. Asymptotic expansions for the solutions of the equation, valid uniformly with respect to z in domains including Z=0, Z _0+-iO are derived in terms of Bessel functions of large order. Expansions given by previous theory are not valid at all these points. The theory can be applied to the Legendre functions.