## 2000), Numerical and asymptotic aspects of parabolic cylinder functions

Venue: | J. Comp. Appl. Math |

Citations: | 7 - 6 self |

### BibTeX

@ARTICLE{Temme_2000),numerical,

author = {Nico M. Temme},

title = {2000), Numerical and asymptotic aspects of parabolic cylinder functions},

journal = {J. Comp. Appl. Math},

year = {},

pages = {221--246}

}

### OpenURL

### Abstract

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. 1991 Mathematics Subject Classification: 33C15, 41A60, 65D20.

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Citation Context ...r in harmonic analysis; see [20]. The solutions are called parabolic cylinder functions and are entire functions of z. Many properties in connection with physical applications are given in [4]. As in =-=[1]-=-, Chapter 19, we denote two standard solutions of (1.1) by U(a; z); V (a; z). These solutions are given by the representations U(a; z) = p 2 \Gamma 1 2 a 2 4 2 \Gamma 1 4 y 1 (a; z) \Gamma( 3 4 + 1 2 ... |

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Citation Context ...d from dOE(w) dw = w 2 + w \Gammasw = 0; (4.4) giving w 0 = 1 2 h p 1 + 4 \Gamma 1 i : (4.5) We consider z as the large parameter. Whensis bounded away from 0 we can use Laplace's method (see [11] or =-=[22]-=-). When a and z are such thats! 0 Laplace's method cannot be applied. However, we can use a method given in [15] that allows small values of . To obtain a standard form for this Laplace-type integral,... |

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Citation Context ...thats! 0 Laplace's method cannot be applied. However, we can use a method given in [15] that allows small values of . To obtain a standard form for this Laplace-type integral, we transform w ! t (see =-=[16]-=-) by writing OE(w) = t \Gammasln t + A; (4.6) where A does not depend on t or w, and we prescribe that w = 0 should correspond with t = 0 and w = w 0 with t = , the saddle point in the t\Gammaplane. T... |

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Citation Context ...s paper. 1.1 Recent literature on numerical algorithms Recent papers on numerical algorithms for the parabolic cylinder functions are given in [14] (Fortran; U(n; x) for natural n and positive x) and =-=[13]-=- (Fortran; U(a; x); V (a; x), a integer and half-integer and xs0). The methods are based on backward and forward recursion. [2] gives programs in C for U(a; x); V (a; x), and uses representations in t... |

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Citation Context ...asymptotic expansions. They refer also to [3] for the evaluation of U(\Gammaia; ze 1 4 i ) for real a and z (this function is a solution of the differential equation y 00 + ( 1 4 z 2 \Gamma a)y = 0). =-=[19]-=- uses series expansions and numerical quadrature; Fortran and C programs are given, and Mathematica cells to make graphical and numerical objects. Maple [5] has algorithms for hypergeometric functions... |

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Citation Context ...l and Applied Mathematics 1. Introduction The solutions of the differential equation d 2 y dz 2 \Gamma i 1 4 z 2 + a j y = 0 (1.1) are associated with the parabolic cylinder in harmonic analysis; see =-=[20]-=-. The solutions are called parabolic cylinder functions and are entire functions of z. Many properties in connection with physical applications are given in [4]. As in [1], Chapter 19, we denote two s... |

2 |
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Citation Context ...ns y 1 (a; z) and y 2 (a; z) are the simplest even and odd solutions of (1.1) and the Wronskian of this pair equals 1. From a numerical point of view, the pair fy 1 ; y 2 g is not a satisfactory pair =-=[8]-=-, because they have almost the same asymptotic behavior at infinity. The behavior of U(a; z) and V (a; z) is, for large positive z and z AE jaj: U(a; z) = e \Gamma 1 4 z 2 z \Gammaa\Gamma 1 2 \Theta 1... |

2 |
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Citation Context ...he expansions. Only real parameters are considered in this paper. 1.1 Recent literature on numerical algorithms Recent papers on numerical algorithms for the parabolic cylinder functions are given in =-=[14]-=- (Fortran; U(n,x) for natural n and positive x) and [13] (Fortran; U(a,x),V (a,x), a integer and half-integer and x ≥ 0). The methods are based on backward and forward recursion. [2] gives programs in... |

1 |
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Citation Context ...; z); V (a; z) with real orders and real argument, and for half-integer order and complex argument, The methods are based on recursions, Maclaurin series and asymptotic expansions. They refer also to =-=[3]-=- for the evaluation of U(\Gammaia; ze 1 4 i ) for real a and z (this function is a solution of the differential equation y 00 + ( 1 4 z 2 \Gamma a)y = 0). [19] uses series expansions and numerical qua... |

1 |
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(Show Context)
Citation Context ...he expansions. Only real parameters are considered in this paper. 1.1 Recent literature on numerical algorithms Recent papers on numerical algorithms for the parabolic cylinder functions are given in =-=[14]-=- (Fortran; U(n; x) for natural n and positive x) and [13] (Fortran; U(a; x); V (a; x), a integer and half-integer and xs0). The methods are based on backward and forward recursion. [2] gives programs ... |