scholarly journals Uniform (very) sharp bounds for ratios of parabolic cylinder functions

2021 ◽  
Author(s):  
Javier Segura
Keyword(s):  
Parabolic Cylinder ◽  
Sharp Bounds ◽  
Parabolic Cylinder Functions

scholarly journals On certain infinite integrals involving Struve functions and parabolic cylinder functions

1946 ◽  
Vol 7 (4) ◽  
pp. 171-173 ◽  
Author(s):  
S. C. Mitra
Keyword(s):  
Present Note ◽  
Parabolic Cylinder ◽  
Parabolic Cylinder Functions ◽  
Struve Functions ◽  
Image Position

The object of the present note is to obtain a number of infinite integrals involving Struve functions and parabolic cylinder functions. 1. G. N. Watson(1) has proved thatFrom (1)follows provided that the integral is convergent and term-by-term integration is permissible. A great many interesting particular cases of (2) are easily deducible: the following will be used in this paper.

scholarly journals Absence of zeros and asymptotic error estimates for airy and parabolic cylinder functions

2014 ◽  
Vol 12 (1) ◽  
pp. 175-200 ◽  
Author(s):  
Felix Finster ◽  
Joel Smoller
Keyword(s):  
Error Estimates ◽  
Parabolic Cylinder ◽  
Asymptotic Error ◽  
Parabolic Cylinder Functions

Asymptotic approximations for parabolic cylinder functions

1978 ◽  
Vol 11 (18) ◽  
pp. L531-L533 ◽  
Author(s):  
C Lozano ◽  
F W J Olver
Keyword(s):  
Asymptotic Approximations ◽  
Parabolic Cylinder ◽  
Parabolic Cylinder Functions

An infinite integral involving Bessel functions and parabolic cylinder functions

1937 ◽  
Vol 33 (2) ◽  
pp. 210-211 ◽  
Author(s):  
R. S. Varma
Keyword(s):  
Bessel Functions ◽  
Parabolic Cylinder ◽  
Parabolic Cylinder Functions ◽  
Infinite Integral

The object of this paper is to evaluate an infinite integral involving Bessel functions and parabolic cylinder functions. The following two lemmas are required:Lemma 1. provided that R(m) > 0.

scholarly journals Distributions of the Ratio and Product of Two Independent Weibull and Lindley Random Variables

2020 ◽  
Vol 2020 ◽  
pp. 1-8
Author(s):  
N. J. Hassan ◽  
A. Hawad Nasar ◽  
J. Mahdi Hadad
Keyword(s):  
Hypergeometric Functions ◽  
Special Functions ◽  
Random Variables ◽  
Distribution Functions ◽  
Cumulative Distribution ◽  
Parabolic Cylinder ◽  
Generalized Hypergeometric Functions ◽  
Confluent Hypergeometric Functions ◽  
Parabolic Cylinder Functions ◽  
The Moment

In this paper, we derive the cumulative distribution functions (CDF) and probability density functions (PDF) of the ratio and product of two independent Weibull and Lindley random variables. The moment generating functions (MGF) and the k-moment are driven from the ratio and product cases. In these derivations, we use some special functions, for instance, generalized hypergeometric functions, confluent hypergeometric functions, and the parabolic cylinder functions. Finally, we draw the PDF and CDF in many values of the parameters.

scholarly journals On the squares of Weber's parabolic cylinder functions and certain integrals connected with them

1934 ◽  
Vol 4 (1) ◽  
pp. 27-32 ◽  
Author(s):  
S. C. Mitra
Keyword(s):  
Parabolic Cylinder ◽  
Parabolic Cylinder Functions ◽  
Image Position

The parabolic cylinder functions Dn(x) and D−(n+1) (± ix) are defined byfor all values of n and x.

Tables of Weber Parabolic Cylinder Functions

1957 ◽  
Vol 41 (336) ◽  
pp. 144
Author(s):  
A. Fletcher ◽  
J. C. P. Miller
Keyword(s):  
Parabolic Cylinder ◽  
Parabolic Cylinder Functions

Conical functions of purely imaginary order and argument

2013 ◽  
Vol 143 (5) ◽  
pp. 929-955
Author(s):  
T. M. Dunster
Keyword(s):  
Closed Geodesic ◽  
Asymptotic Approximations ◽  
Parabolic Cylinder ◽  
Legendre Functions ◽  
Associated Legendre Functions ◽  
Approximations And Expansions ◽  
Fourier Expansions ◽  
Parabolic Cylinder Functions ◽  
Conical Form

Associated Legendre functions are studied for the case where the degree is in conical form −½ + iτ (τ real), and the order iμ and argument ix are purely imaginary (μ and x real). Conical functions in this form have applications to Fourier expansions of the eigenfunctions on a closed geodesic. Real-valued numerically satisfactory solutions are introduced which are continuous for all real x. Uniform asymptotic approximations and expansions are then derived for the cases where one or both of μ and τ are large; these results (which involve elementary, Airy, Bessel and parabolic cylinder functions) are uniformly valid for unbounded x.

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