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Integral representations for computing real parabolic cylinder functions. (English) Zbl 1054.65023
Summary: Integral representations are derived for the parabolic cylinder functions $U\left(a,x\right)$, $V\left(a,x\right)$ and $W\left(a,x\right)$ 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 $U\left(a,x\right)$, $V\left(a,x\right)$ and $W\left(a,x\right)$ 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.
##### MSC:
 65D20 Computation of special functions, construction of tables 33C10 Bessel and Airy functions, cylinder functions, ${}_{0}{F}_{1}$ 33F05 Numerical approximation and evaluation of special functions
##### References:
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