## Abstract

The asymptotic expansion of solutions of the fourth order differential equation u<latex>$^{\text{iv}}$</latex>+<latex>$\lambda ^{2}$</latex>[(z<latex>$^{2}$</latex>+c)u<latex>$^{\prime \prime}$</latex>+azu<latex>$^{\prime}$</latex>+bu] = 0 are investigated for <latex>$|z|\rightarrow \infty $</latex> where the parameters a, b, c and <latex>$\lambda $</latex> are supposed to be arbitrary complex constants with <latex>$\lambda $</latex>, b <latex>$\neq $</latex> 0. Exact solutions in the form of Laplace and Mellin--Barnes integrals, involving a Whittaker function and a Gauss hypergeometric function respectively, are used to define a fundamental system of solutions. The asymptotic expansion of these solutions is obtained in a full neighbourhood of the point at infinity and their asymptotic character is found to be either exponentially large or algebraic in certain sectors of the z-plane. The expansions corresponding to certain special values of the parameters a and b which yield logarithmic expansions are also treated. Linear combinations of these fundamental solutions which possess an exponentially small expansion for <latex>$|z|\rightarrow \infty $</latex> in a certain sector are discussed.

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