Computation of parabolic cylinder functions having complex argument

TL;DR

This paper introduces numerical methods combining asymptotic expansions, integral representations, and series to accurately compute parabolic cylinder functions with complex arguments, achieving high precision.

Contribution

It develops a comprehensive numerical approach using recent asymptotic expansions and integral representations for accurate computation of parabolic cylinder functions with complex arguments.

Findings

Achieves $5\times 10^{-13}$ relative accuracy in double precision.

Combines asymptotic, integral, and series methods effectively.

Provides numerical evidence supporting the methods' robustness.

Abstract

Numerical methods for the computation of the parabolic cylinder $U(a,z)$ for real $a$ and complex $z$ are presented. The main tools are recent asymptotic expansions involving exponential and Airy functions, with slowly varying analytic coefficient functions involving simple coefficients, and stable integral representations; these two main main methods can be complemented with Maclaurin series and a Poincar\'e asymptotic expansion. We provide numerical evidence showing that the combination of these methods is enough for computing the function with $5\times 10^{-13}$ relative accuracy in double precision floating point arithmetic.