Integral Representations for Computing Real Parabolic Cylinder Functions

Details Integral Representations for Computing Real Parabolic Cylinder Functions Integral Representations for Computing Real Parabolic Cylinder Functions

2004-01-13

arxiv.org/abs/math/0401131v1

Mathematics

Numerical Analysis

31 pages, 3 figures. To appear in Numer. Math

Scientific paper

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.

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