# The Asymptotic Expansion of Bessel Functions of Large Order

F. W. J. Olver

## Abstract

New expansions are obtained for the functions I$_{\nu}$($\nu$z), K$_{\nu}$($\nu$z) and their derivatives in terms of elementary functions, and for the functions J$_{\nu}$($\nu$z), Y$_{\nu}$($\nu$z), H$_{\nu}^{(1)}$($\nu$z), H$_{\nu}^{(2)}$($\nu$z) and their derivatives in terms of Airy functions, which are uniformly valid with respect to z when $|\nu|$ is large. New series for the zeros and associated values arc derived by reversion and used to determine the distribution of the zeros of functions of large order in the z-plane. Particular attention is paid to the complex zeros of Y$_{n}$(z) and the Hankel functions when the order n is an integer or half an odd integer, and for this purpose some new asymptotic expansions of the Airy functions are derived. Tables are given of complex zeros of Airy functions and other quantities which facilitate the rapid calculation of the smaller complex zeros of Y$_{n}$(z), Y$_{n}^{\prime}$(z), and the Hankel functions and their derivatives, when 2n is an integer, to an accuracy of three or four significant figures.