Computation of Asymptotic Expansions of Turning Point Problems via Cauchy’s Integral Formula: Bessel Functions

T M Dunster,J Segura,A Gil

doi:10.1007/s00365-017-9372-8

Abstract

Linear second-order differential equations having a large real parameter and turning point in the complex plane are considered. Classical asymptotic expansions for solutions involve the Airy function and its derivative, along with two infinite series, the coefficients of which are usually difficult to compute. By considering the series as asymptotic expansions for two explicitly defined analytic functions, Cauchy’s integral formula is employed to compute the coefficient functions to a high order of accuracy. The method employs a certain exponential form of Liouville–Green expansions for solutions of the differential equation, as well as for the Airy function. We illustrate the use of the method with the high accuracy computation of Airy-type expansions of Bessel functions of complex argument.

Highlights

In this paper we study linear second order differential equations having a simple turning point
The asymptotic expansions (2.20) and (2.21) are in theory valid close to the turning point ξ = ζ = 0 (z = z0) as u → ∞
As expected, the LiouvilleGreen expansion (3.15) loses accuracy close to the turning point and for real values of z with |z| > 1. It is worth noting than in the neighborhood of the turning point we can consider our expansions with coefficients computed via Cauchy integrals that we are discussing while for z > 1 we can compute Jν(νz) using its relation with Hankel functions and the LiouvilleGreen expansions for these functions or, alternatively, we can use (4.4) with coefficients computed from its asymptotic approximation

Summary

Introduction

In this paper we study linear second order differential equations having a simple turning point. Where u is positive and large, f (z) has a simple zero (turning point) at z = z0 (say), and f (z) and g(z) are analytic in an unbounded domain containing the turning point. This is a classical problem, with applications to numerous special functions. These coefficients are difficult to compute, primarily due to the requirement of repeated integrations They show cancellations near the turning point. We shall employ Cauchy’s integral formula to do so, and our results will be valid for real and complex ζ lying in a bounded (but not necessarily small) domain containing the turning point ζ = 0. We illustrate the use of the method with the high accuracy computation of Airy-type expansions of Bessel functions of complex argument

General method

Bessel’s equation: preliminary transformations

Bessel’s equation: turning point coefficient functions

Numerical results

Testing of the new Liouville-Green approximations

Airy-type expansions via Cauchy’s integral formula

A Appendix