## Abstract

We study the uniform asymptotic expansion for a large parameter u of solutions of second-order linear differential equations in domains of the complex plane which include a turning point. The standard theory of Olver demonstrates the existence of three such expansions, corresponding to solutions of the form $Ai(u^\frac{2}{3}\zeta e^{\frac{2}{3}\alpha\pi i }) \sum^n_{s=0} \frac{A_s(\zeta)}{u^{2s}} + u^{-2} \frac{d}{d\zeta} Ai(u^\frac{2}{3}\zeta e^{\frac{2}{3}\alpha\pi i}) \sum^{n-1}_{s=0} \frac{B_s(\zeta)}{u^{2s}} + \epsilon^{(\alpha)}_n (u,\zeta)$ for $\alpha$ = 0, 1, 2, with bounds on $\epsilon^{(\alpha)}_n$. We proceed differently, by showing that the set of all solutions of the differential equation is of the form $\mathscr{A}i(u^{\frac{2}{3}}\zeta) \mathscr{A} (u, \zeta) + u^{-2}(d/d\zeta) \mathscr{A}i (u^\frac{2}{3}\zeta) \mathscr{B} (u, \zeta),$ where $\mathscr{A}$i denotes any solution of Airy's equation. The coefficient functions $\mathscr{A}$ (u, $\zeta$) and $\mathscr{B}$ (u, $\zeta$) are the focus of our attention: we show that for sufficiently large u they are holomorphic functions of $\zeta$ in a domain including the turning point, and as functions of u are described asymptotically by descending power series in u$^2$, with explicit error bounds. We apply our theory to Bessel functions.