parabolic cylinder functions

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41: Bibliography F

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Generalized parabolic cylinder functions. Asymptotic Anal. 5 (6), pp. 517–531.

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External Links:

ISSN 0921-7134, MathReview (Wolfgang Bühring), ZentralBlatt

Referenced by:

§12.15.

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Tables of Weber Parabolic Cylinder Functions and Other Functions for Large Arguments. National Physical Laboratory Mathematical Tables, Vol. 4. Department of Scientific and Industrial Research, Her Majesty’s Stationery Office, London.

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External Links:

MathReview (L. J. Slater), ZentralBlatt

Referenced by:

3rd item, 2nd item.

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42: 13.8 Asymptotic Approximations for Large Parameters

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13.8.4 $$M(a,b,z)\sim b^{\frac{1}{2}a}{\mathrm{e}}^{\frac{1}{4}{\zeta }^{2}b}\left(\lambda {\left(\frac{\lambda -1}{\zeta }\right)}^{a-1}U(a-\frac{1}{2},-\zeta \sqrt{b})+\left(\lambda {\left(\frac{\lambda -1}{\zeta }\right)}^{a-1}-{\left(\frac{\zeta }{\lambda -1}\right)}^a\right)\frac{U(a-\frac{3}{2},-\zeta \sqrt{b})}{\zeta \sqrt{b}}\right)$$

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Symbols:

$M(a,b,z)$: $={}_1{}^{}F_1^{}(a;b;z)$ Kummer confluent hypergeometric function, $\sim$: asymptotic equality, $\mathrm{e}$: base of natural logarithm, $U(a,z)$: parabolic cylinder function and $z$: complex variable

Permalink:

http://dlmf.nist.gov/13.8.E4

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See also:

Annotations for §13.8(ii), §13.8 and Ch.13

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13.8.5 $$U(a,b,z)\sim b^{-\frac{1}{2}a}{\mathrm{e}}^{\frac{1}{4}{\zeta }^{2}b}\left(\lambda {\left(\frac{\lambda -1}{\zeta }\right)}^{a-1}U(a-\frac{1}{2},\zeta \sqrt{b})-\left(\lambda {\left(\frac{\lambda -1}{\zeta }\right)}^{a-1}-{\left(\frac{\zeta }{\lambda -1}\right)}^a\right)\frac{U(a-\frac{3}{2},\zeta \sqrt{b})}{\zeta \sqrt{b}}\right)$$

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Symbols:

$U(a,b,z)$: Kummer confluent hypergeometric function, $\sim$: asymptotic equality, $\mathrm{e}$: base of natural logarithm, $U(a,z)$: parabolic cylinder function and $z$: complex variable

Permalink:

http://dlmf.nist.gov/13.8.E5

Encodings:

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See also:

Annotations for §13.8(ii), §13.8 and Ch.13

… ►For the parabolic cylinder function $U$ see §12.2, and for an extension to an asymptotic expansion see Temme (1978). …

43: Bibliography J

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Parabolic cylinder functions of large order. J. Comput. Appl. Math. 190 (1-2), pp. 453–469.

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External Links:

ISSN 0377-0427, Document, MathReview Entry, ZentralBlatt

Referenced by:

§12.10(vi).

Encodings:

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44: Bibliography V

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An infinite series of Weber’s parabolic cylinder functions. Proc. Benares Math. Soc. (N.S.) 3, pp. 37.

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External Links:

MathReview (M. C. Gray), ZentralBlatt

Referenced by:

§12.13(ii).

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45: 18.30 Associated OP’s

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18.30.15 $$w(x,c)={\left|U(c-\frac{1}{2},\mathrm{i}x\sqrt{2})\right|}^{-2}.$$

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Symbols:

$\mathrm{i}$: imaginary unit, $U(a,z)$: parabolic cylinder function, $w(x)$: weight function and $x$: real variable

Permalink:

http://dlmf.nist.gov/18.30.E15

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Annotations for §18.30(iv), §18.30 and Ch.18

►For the parabolic cylinder function $U$ see §12.2(i). …

46: Software Index

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	Open Source													With Book						Commercial
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12 Parabolic Cylinder Functions
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47: 18.15 Asymptotic Approximations

… ►With $\mu =\sqrt{2n+1}$ the expansions in Chapter 12 are for the parabolic cylinder function $U(-\frac{1}{2}{\mu }^2,\mu t\sqrt{2})$, which is related to the Hermite polynomials via ►

18.15.28 $$H_n\left(x\right)=2^{\frac{1}{4}({\mu }^2-1)}{\mathrm{e}}^{\frac{1}{2}{\mu }^2{t}^2}U(-\frac{1}{2}{\mu }^2,\mu t\sqrt{2});$$

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Symbols:

$H_n\left(x\right)$: Hermite polynomial, $\mathrm{e}$: base of natural logarithm, $U(a,z)$: parabolic cylinder function, $t$: real variable, $n$: nonnegative integer and $x$: real variable

Permalink:

http://dlmf.nist.gov/18.15.E28

Encodings:

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See also:

Annotations for §18.15(v), §18.15 and Ch.18

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48: Bibliography L

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Functions on a parabolic cylinder with a negative integer index. Differ. Uravn. 21 (11), pp. 2001–2003, 2024 (Russian).

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Notes:

In Russian.

External Links:

ISSN 0374-0641, MathReview, ZentralBlatt

Referenced by:

§12.13(i).

Encodings:

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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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49: Bibliography B

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A note on integrals involving parabolic cylinder functions. SIAM J. Appl. Math. 16 (1), pp. 71–74.

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External Links:

Document, MathReview (T. Erber), ZentralBlatt

Referenced by:

§12.12.

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BibTeX

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Exponential asymptotics for an eigenvalue of a problem involving parabolic cylinder functions. Proc. Amer. Math. Soc. 114 (4), pp. 1025–1032.

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External Links:

ISSN 0002-9939, FullText, MathReview, ZentralBlatt

Referenced by:

§12.16.

Encodings:

BibTeX

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50: 2.8 Differential Equations with a Parameter

… ►For two coalescing turning points see Olver (1975a, 1976) and Dunster (1996a); in this case the uniform approximants are parabolic cylinder functions. (For envelope functions for parabolic cylinder functions see §14.15(v)). … ►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. …