## Abstract

The asymptotic behaviour of solutions of the fourth order differential equation $\frac{\text{d}^{4}u}{\text{d}z^{4}}$ + [$z^{2}$$\frac{\text{d}^{2}u}{\text{d}z^{2}}$ + $az$$\frac{\text{d}u}{\text{d}z}$ + $bu$] = 0, in which the constants $a$ and $b$ are supposed real and positive, is examined for large real values of $z$ and $b$. Exact solutions are given in terms of generalized hypergeometric functions and some special cases of $a$ and $b$ are mentioned for which solutions may be expressed in terms of simpler known functions. The differential equation possesses four transition points and for positive values of $b$ one of these transition points lies on the positive real z-axis. The asymptotic discussion is centred around an integral representation which involves a modified Bessel function of the second kind whose order is purely imaginary for large positive values of $b$. Asymptotic forms as $|z|$ $\rightarrow $ $\infty $ and about the transition point on the positive real z-axis are given. A qualitative discussion of the zeros of the Bessel function $K$$_{\mu}$($z$) for imaginary order and complex argument is presented in the appendix.