Computation of parabolic cylinder functions having complex argument

10/30/2022

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by   T. M. Dunster, et al.

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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 Poincaré asymptotic expansion. We provide numerical evidence showing that the combination of these methods is enough for computing the function with 5× 10^-13 relative accuracy in double precision floating point arithmetic.