## Abstract

The differential equation $\frac{\text{d}^{2}w}{\text{d}z^{2}}$ = $\left\{uz^{n}+\frac{\gamma}{z^{2}}+g(z)\right\}$w, where n is an integer ($\geq $ -1), u a parameter and $\gamma $ a constant, has the formal solution w = P(z)$\left\{1+\sum_{s=1}^{\infty}\frac{A_{s}(z)}{u^{s}}\right\}$ + $\frac{P^{\prime}(z)}{u}\sum_{s=0}^{\infty}\frac{B_{s}(z)}{u^{s}}$, where P(z) is a solution of the equation $\frac{\text{d}^{2}P}{\text{d}z^{2}}$ = $\left(uz^{n}+\frac{\gamma}{z^{2}}\right)$P. The coefficients A$_{s}$(z) and B$_{s}$(z) are given by recurrence relations. It is shown that they are analytic at z = 0 if, and only if, the differential equation for w can be transformed into a similar equation with n = 0, $\gamma $ = 0, or n = 1, $\gamma $ = 0, or n = -1. The first two cases (for which P is an exponential and Airy function respectively) have been treated in a previous paper. The third case, for which P is a Bessel function of order $\pm $ (1 + 4$\gamma $)$^{\frac{1}{2}}$, is examined in detail in the present paper. It is proved that for large positive u, solutions exist whose asymptotic expansions in Poincare's sense are given by the formal series, and that these expansions are uniformly valid with respect to the complex variable z.