Operational Representations of Whittaker's Confluent Hypergeometric Function and Weber's Parabolic Cylinder Function

1932 ◽  
Vol s2-34 (1) ◽  
pp. 103-125 ◽  
Author(s):  
S. Goldstein
Keyword(s):  
Hypergeometric Function ◽  
Confluent Hypergeometric Function ◽  
Parabolic Cylinder ◽  
Parabolic Cylinder Function ◽  
Cylinder Function

Derivation of Green-type, transitional and uniform asymptotic expansions from differential equations III. The confluent hypergeometric function U ( a , c , z ) for large | c |

1965 ◽  
Vol 284 (1399) ◽  
pp. 531-539 ◽  
Keyword(s):  
Asymptotic Expansions ◽  
Confluent Hypergeometric Function ◽  
Parabolic Cylinder ◽  
Exponential Integral ◽  
Parabolic Cylinder Function ◽  
Uniform Expansions ◽  
Kummer's Equation ◽  
Uniform Expansion ◽  
Special Case ◽  
Uniform Asymptotic Expansions

The methods introduced by Jorna (1964 a , b ) are applied to Kummer’s equation, and Green-type, transitional and uniform expansions derived for solutions of the type denoted in Slater (1960) by U ( a , c , z ) which are valid for large | c |. The subsidiary function in the uniform expansion is essentially a parabolic cylinder function of order ½ a . The general exponential integral is studied as a special case. Here the uniform expansion involves as subsidiary function the extensively tabulated error integral.

The asymptotic behaviour of Pearcey’s integral for complex variables

1991 ◽  
Vol 432 (1886) ◽  
pp. 391-426 ◽  
Keyword(s):  
Asymptotic Behaviour ◽  
Oscillatory Integral ◽  
Complex Variables ◽  
Stationary Points ◽  
Parabolic Cylinder ◽  
Parabolic Cylinder Function ◽  
Cylinder Function ◽  
Arbitrary Complex ◽  
The Rich ◽  
Asymptotic Character

A deeper understanding of the rich structure of the canonical form of the oscillatory integral describing the cusp diffraction catastrophe, generally known as Pearcey’s integral P '( X , Y ), can be obtained by considering its analytic continuation to arbitrary complex variables X and Y . A new integral representation for P '( X , Y ) is given in the form of a contour integral involving a Weber parabolic cylinder function whose order is the variable of integration. It is shown how the asymptotics of P '( X , Y ) may be obtained from this representation for complex X and Y when either | X | or | Y | → ∞, without reference to the usual stationary points of the integrand. For the case | X | → ∞, Y finite the full asymptotic expansion of P '( X , Y ) is derived and its asymptotic character is found to be either exponentially large or algebraic in certain sectors of the X - plane. The case | Y | → ∞ , X finite is complicated by the presence of exponentially small subdominant terms in certain sectors of the Y - plane, and only the first terms in the expansion are given. The asymptotic behaviour of P '( X , Y ) on the caustic Y 2 + (2/3 X ) 3 = 0 is also obtained from the new representation and is shown to agree with recent results of D. Kaminski. The various properties of the Weber parabolic cylinder function required in this paper are collected together in the Appendix.

The Parabolic Cylinder Function D v (x)

2008 ◽  
pp. 471-484
Author(s):  
Keith B. Oldham ◽  
Jan C. Myland ◽  
Jerome Spanier
Keyword(s):  
Parabolic Cylinder ◽  
Parabolic Cylinder Function ◽  
Cylinder Function
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