Integral Representations for Computing Real Parabolic Cylinder Functions

TL;DR

This work develops integral representations for real-valued parabolic cylinder functions $U(a,x)$, $V(a,x)$ and $W(a,x)$ and their derivatives by applying contour integrals and saddle-point (steepest-descent) methods. It treats both $a>0$ and $a<0$ across multiple regimes of the scaled parameter $t=x/(2\sqrt{|a|})$, derives Wronskian relations for consistency checks, and connects the integral forms to uniform asymptotic expansions. The representations are designed to be stable for quadrature-based numerical evaluation, including large-parameter regimes, and provide a foundation for future algorithms and implementation details. The work also clarifies how these integral forms interface with existing uniform expansions and the properties of the associated $W$-function, laying groundwork for robust, transferable numerical routines in applied contexts.

Abstract

Integral representations are derived for the parabolic cylinder functions $U(a,x)$, $V(a,x)$ and $W(a,x)$ and their derivatives. The new integrals will be used in numerical algorithms based on quadrature. They follow from contour integrals in the complex plane, by using methods from asymptotic analysis (saddle point and steepest descent methods), and are stable starting points for evaluating the functions $U(a,x)$, $V(a,x)$ and $W(a,x)$ and their derivatives by quadrature rules. In particular, the new representations can be used for large parameter cases. Relations of the integral representations with uniform asymptotic expansions are also given. The algorithms will be given in a future paper.