Exactification of the method of steepest descents: the Bessel functions of large order and argument

Abstract

The Hadamard expansion procedure applied to Laplace–type integrals taken along contours in the complex plane enables an exact description of the method of steepest descents. This mode of expansion is illustrated by the evaluation of the Bessel functions Jv(? x) and Yv(v x) of large order and argument when x is bounded away from unity. The limit x → 1, corresponding to the coalescence of the active saddles in the integral representations of the Bessel functions, translates into a progressive loss of exponential separation between the different levels of the Hadamard expansion, which renders computation in this limit more difficult. It is shown how a simple modification to this procedure can be employed to deal with the coalescence of the active saddles when x → 1.