## Abstract

An investigation is made of the differential equations $\frac{\text{d}^{2}w}{\text{d}z^{2}}$ = {u$^{2}$ + f(u, z)}w, $\quad \frac{\text{d}^{2}w}{\text{d}z^{2}}$ = {u$^{2}$z + f(u, z)}w, $\quad \frac{\text{d}^{2}w}{\text{d}z^{2}}$ = $\frac{1}{z}\frac{\text{d}w}{\text{d}z}$ + $\left\{u^{2}+\frac{\mu ^{2}-1}{z^{2}}+f(u,\mu,z)\right\}$w, in which u is a large complex parameter, $\mu $ is a real or complex parameter independent of u, and z is a complex variable whose domain of variation may depend on arg u and $\mu $, and need not be bounded. General conditions are obtained under which solutions exist having the formal series w = P(z) $\sum_{s=0}^{\infty}\frac{A_{s}}{u^{2s}}$ + $\frac{P^{\prime}(z)}{u^{2}}\sum_{s=0}^{\infty}\frac{B_{s}}{u^{2s}}$ as their asymptotic expansions for large $|$u$|$, uniformly valid with respect to z, arg u and $\mu $. Here P(z) is respectively an exponential function, Airy function or Bessel function of order $\mu $, and the coefficients A$_{s}$ and B$_{s}$ are given by recurrence relations.