Integral Representations for Computing Real Parabolic Cylinder Functions

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.Comment: 31 pages, 3 figures. To appear in Numer. Mat

Integral representations for computing real parabolic cylinder functions

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

Computing special functions by using quadrature rules

The usual tools for computing special functions are power series, asymptotic expansions, continued fractions, differential equations, recursions, and so on. Rather seldom are methods based on quadrature of integrals. Selecting suitable integral representations of special functions, using principles from asymptotic analysis, we develop reliable algorithms which are valid for large domains of real or complex parameters. Our present investigations include Airy functions, Bessel functions and parabolic cylinder functions. In the case of Airy functions we have improvements in both accuracy and speed for some parts of Amos's code for Bessel functions

Computation of parabolic cylinder functions having complex argument

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

Numerical and Asymptotic Aspects of Parabolic Cylinder Functions

Several uniform asymptotics expansions of the Weber parabolic cylinder functions are considered, one group in terms of elementary functions, another group in terms of Airy functions. Starting point for the discussion are asymptotic expansions given earlier by F.W.J. Olver. Some of his results are modified to improve the asymptotic properties and to enlarge the intervals for using the expansions in numerical algorithms. Olver's results are obtained from the differential equation of the parabolic cylinder functions; we mention how modified expansions can be obtained from integral representations. Numerical tests are given for three expansions in terms of elementary functions. In this paper only real values of the parameters will be considered.Comment: 16 pages, 1 figur

Absence of Zeros and Asymptotic Error Estimates for Airy and Parabolic Cylinder Functions

We derive WKB approximations for a class of Airy and parabolic cylinder functions in the complex plane, including quantitative error bounds. We prove that all zeros of the Airy function lie on a ray in the complex plane, and that the parabolic cylinder functions have no zeros. We also analyze the Airy and Airy-WKB limit of the parabolic cylinder functions.Comment: 25 pages, LaTeX, 7 figures (published version

Brownian motion with dry friction: Fokker-Planck approach

We solve a Langevin equation, first studied by de Gennes, in which there is a solid-solid or dry friction force acting on a Brownian particle in addition to the viscous friction usually considered in the study of Brownian motion. We obtain both the time-dependent propagator of this equation and the velocity correlation function by solving the associated time-dependent Fokker-Planck equation. Exact results are found for the case where only dry friction acts on the particle. For the case where both dry and viscous friction forces are present, series representations of the propagator and correlation function are obtained in terms of parabolic cylinder functions. Similar series representations are also obtained for the case where an external constant force is added to the Langevin equation.Comment: 18 pages, 13 figures (in color

Eigenfunction expansions for a fundamental solution of Laplace's equation on R3\R^3 in parabolic and elliptic cylinder coordinates

A fundamental solution of Laplace's equation in three dimensions is expanded in harmonic functions that are separated in parabolic or elliptic cylinder coordinates. There are two expansions in each case which reduce to expansions of the Bessel functions $J_0(kr)$ or $K_0(kr)$, $r^2=(x-x_0)^2+(y-y_0)^2$, in parabolic and elliptic cylinder harmonics. Advantage is taken of the fact that $K_0(kr)$ is a fundamental solution and $J_0(kr)$ is the Riemann function of partial differential equations on the Euclidean plane