Computation of asymptotic expansions of turning point problems via Cauchy's integral formula: Bessel functions
This provides a practical computational tool for mathematicians and physicists who need high-order asymptotic expansions for differential equations with turning points.
The paper introduces a method using Cauchy's integral formula to compute coefficient functions in Airy-type asymptotic expansions for turning point problems, achieving high accuracy for Bessel functions of complex argument.
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 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.