in terms of parabolic cylinder functions

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1: 12.16 Mathematical Applications

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2: 18.24 Hahn Class: Asymptotic Approximations

… ►This expansion is in terms of the parabolic cylinder function and its derivative. … ►Both expansions are in terms of parabolic cylinder functions. … ►This expansion is uniformly valid in any compact $x$-interval on the real line and is in terms of parabolic cylinder functions. …

3: Bibliography O

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Whittaker functions with both parameters large: Uniform approximations in terms of parabolic cylinder functions. Proc. Roy. Soc. Edinburgh Sect. A 86 (3-4), pp. 213–234.

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External Links:

ISSN 0308-2105, MathReview (P. F. Hsieh), ZentralBlatt

Referenced by:

§13.20(iii), §13.20(iv).

Encodings:

BibTeX

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4: Bibliography L

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Integral representation of the Hankel function in terms of parabolic cylinder functions. Quart. J. Mech. Appl. Math. 23 (3), pp. 315–327.

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External Links:

Document, MathReview (T. Erber), ZentralBlatt

Referenced by:

§12.12, §12.16.

Encodings:

BibTeX

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5: 32.10 Special Function Solutions

… ► ${\text{P}}_{\text{IV}}$ has solutions expressible in terms of parabolic cylinder functions (§12.2) iff either …

6: 12.10 Uniform Asymptotic Expansions for Large Parameter

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§12.10(ii) Negative $a$,

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§12.10(vi) Modifications of Expansions in Elementary Functions

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§12.10(vii) Negative $a$, . Expansions in Terms of Airy Functions

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Modified Expansions

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7: Bibliography C

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Expansions in terms of parabolic cylinder functions. Proc. Edinburgh Math. Soc. (2) 8, pp. 50–65.

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External Links:

MathReview (I. I. Hirschman, Jr.), ZentralBlatt

Referenced by:

§12.16.

Encodings:

BibTeX

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8: 12.14 The Function $W(a,x)$

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Positive $a$,

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Airy-type Uniform Expansions

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9: 2.8 Differential Equations with a Parameter

… ►For further examples of uniform asymptotic approximations in terms of parabolic cylinder functions see §§13.20(iii), 13.20(iv), 14.15(v), 15.12(iii), 18.24. …

10: 16.18 Special Cases

§16.18 Special Cases

►The ${}_1{}^{}F_1^{}$ and ${}_2{}^{}F_1^{}$ functions introduced in Chapters 13 and 15, as well as the more general ${}_p{}^{}F_q^{}$ functions introduced in the present chapter, are all special cases of the Meijer $G$-function. …As a corollary, special cases of the ${}_1{}^{}F_1^{}$ and ${}_2{}^{}F_1^{}$ functions, including Airy functions, Bessel functions, parabolic cylinder functions, Ferrers functions, associated Legendre functions, and many orthogonal polynomials, are all special cases of the Meijer $G$-function. Representations of special functions in terms of the Meijer $G$-function are given in Erdélyi et al. (1953a, §5.6), Luke (1969a, §§6.4–6.5), and Mathai (1993, §3.10).