parabolic cylinder functions

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21: 10.16 Relations to Other Functions

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Parabolic Cylinder Functions

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22: 13.18 Relations to Other Functions

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§13.18(iv) Parabolic Cylinder Functions

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13.18.11 $$W_{-\frac{1}{2}a,\pm \frac{1}{4}}\left(\frac{1}{2}{z}^2\right)=2^{\frac{1}{2}a}\sqrt{z}U(a,z),$$

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Symbols:

$W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $U(a,z)$: parabolic cylinder function and $z$: complex variable

Permalink:

http://dlmf.nist.gov/13.18.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §13.18(iv), §13.18 and Ch.13

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13.18.12 $$M_{-\frac{1}{2}a,-\frac{1}{4}}\left(\frac{1}{2}{z}^2\right)=2^{\frac{1}{2}a-1}\mathrm{\Gamma }\left(\frac{1}{2}a+\frac{3}{4}\right)\sqrt{z/\pi }\left(U(a,z)+U(a,-z)\right),$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $U(a,z)$: parabolic cylinder function and $z$: complex variable

Permalink:

http://dlmf.nist.gov/13.18.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §13.18(iv), §13.18 and Ch.13

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13.18.13 $$M_{-\frac{1}{2}a,\frac{1}{4}}\left(\frac{1}{2}{z}^2\right)=2^{\frac{1}{2}a-2}\mathrm{\Gamma }\left(\frac{1}{2}a+\frac{1}{4}\right)\sqrt{z/\pi }\left(U(a,-z)-U(a,z)\right).$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $U(a,z)$: parabolic cylinder function and $z$: complex variable

Permalink:

http://dlmf.nist.gov/13.18.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §13.18(iv), §13.18 and Ch.13

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23: 13.6 Relations to Other Functions

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§13.6(iv) Parabolic Cylinder Functions

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13.6.12 $$U(\frac{1}{2}a+\frac{1}{4},\frac{1}{2},\frac{1}{2}{z}^2)=2^{\frac{1}{2}a+\frac{1}{4}}{\mathrm{e}}^{\frac{1}{4}{z}^2}U(a,z),$$

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Symbols:

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

Permalink:

http://dlmf.nist.gov/13.6.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §13.6(iv), §13.6 and Ch.13

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13.6.13 $$U(\frac{1}{2}a+\frac{3}{4},\frac{3}{2},\frac{1}{2}{z}^2)=2^{\frac{1}{2}a+\frac{3}{4}}\frac{{\mathrm{e}}^{\frac{1}{4}{z}^2}}{z}U(a,z).$$

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Symbols:

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

Permalink:

http://dlmf.nist.gov/13.6.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §13.6(iv), §13.6 and Ch.13

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13.6.14 $$M(\frac{1}{2}a+\frac{1}{4},\frac{1}{2},\frac{1}{2}{z}^2)=\frac{2^{\frac{1}{2}a-\frac{3}{4}}\mathrm{\Gamma }\left(\frac{1}{2}a+\frac{3}{4}\right){\mathrm{e}}^{\frac{1}{4}{z}^2}}{\sqrt{\pi }}\left(U(a,z)+U(a,-z)\right),$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $M(a,b,z)$: $={}_1{}^{}F_1^{}(a;b;z)$ Kummer confluent hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $U(a,z)$: parabolic cylinder function and $z$: complex variable

Permalink:

http://dlmf.nist.gov/13.6.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §13.6(iv), §13.6 and Ch.13

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13.6.15 $$M(\frac{1}{2}a+\frac{3}{4},\frac{3}{2},\frac{1}{2}{z}^2)=\frac{2^{\frac{1}{2}a-\frac{5}{4}}\mathrm{\Gamma }\left(\frac{1}{2}a+\frac{1}{4}\right){\mathrm{e}}^{\frac{1}{4}{z}^2}}{z\sqrt{\pi }}\left(U(a,-z)-U(a,z)\right).$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $M(a,b,z)$: $={}_1{}^{}F_1^{}(a;b;z)$ Kummer confluent hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $U(a,z)$: parabolic cylinder function and $z$: complex variable

Permalink:

http://dlmf.nist.gov/13.6.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §13.6(iv), §13.6 and Ch.13

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24: 12.10 Uniform Asymptotic Expansions for Large Parameter

§12.10 Uniform Asymptotic Expansions for Large Parameter

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§12.10(vi) Modifications of Expansions in Elementary Functions

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Modified Expansions

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25: 13.20 Uniform Asymptotic Approximations for Large $\mu$

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

13.20.16 $$W_{\kappa ,\mu }\left(x\right)={\left(\frac{1}{2}\mu \right)}^{-\frac{1}{4}}{\left(\frac{\kappa +\mu }{\mathrm{e}}\right)}^{\frac{1}{2}(\kappa +\mu )}\mathrm{\Phi }(\kappa ,\mu ,x)\left(U(\mu -\kappa ,\zeta \sqrt{2\mu })+\mathrm{env}U(\mu -\kappa ,\zeta \sqrt{2\mu })O\left({\mu }^{-\frac{2}{3}}\right)\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, $W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\mathrm{env}U(c,x)$: envelope of parabolic cylinder function $U(c,x)$ , $\mathrm{e}$: base of natural logarithm, $U(a,z)$: parabolic cylinder function, $x$: real variable and $\mathrm{\Phi }(\kappa ,\mu ,x)$: function

Referenced by:

§13.20(iv)

Permalink:

http://dlmf.nist.gov/13.20.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §13.20(iv), §13.20 and Ch.13

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13.20.18 $$W_{\kappa ,\mu }\left(x\right)={\left(\frac{1}{2}\mu \right)}^{-\frac{1}{4}}{\left(\frac{\kappa +\mu }{\mathrm{e}}\right)}^{\frac{1}{2}(\kappa +\mu )}\mathrm{\Phi }(\kappa ,\mu ,x)\left(U(\mu -\kappa ,\zeta \sqrt{2\mu })+\mathrm{env}\overline{U}(\mu -\kappa ,-\zeta \sqrt{2\mu })O\left({\mu }^{-\frac{2}{3}}\right)\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, $W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\mathrm{env}\overline{U}(c,x)$: envelope of parabolic cylinder function $\overline{U}(c,x)$ , $\mathrm{e}$: base of natural logarithm, $U(a,z)$: parabolic cylinder function, $x$: real variable and $\mathrm{\Phi }(\kappa ,\mu ,x)$: function

Referenced by:

§13.20(iv), §13.20(iv)

Permalink:

http://dlmf.nist.gov/13.20.E18

Encodings:

pMML, png, TeX

See also:

Annotations for §13.20(iv), §13.20 and Ch.13

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13.20.19 $$M_{\kappa ,\mu }\left(x\right)={\left(8\mu \right)}^{\frac{1}{4}}{\left(\frac{2\mu }{\mathrm{e}}\right)}^{2\mu }{\left(\frac{\mathrm{e}}{\kappa +\mu }\right)}^{\frac{1}{2}(\kappa +\mu )}\mathrm{\Phi }(\kappa ,\mu ,x)\left(U(\mu -\kappa ,-\zeta \sqrt{2\mu })+\mathrm{env}U(\mu -\kappa ,-\zeta \sqrt{2\mu })O\left({\mu }^{-\frac{2}{3}}\right)\right),$$

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\mathrm{env}U(c,x)$: envelope of parabolic cylinder function $U(c,x)$ , $\mathrm{e}$: base of natural logarithm, $U(a,z)$: parabolic cylinder function, $x$: real variable and $\mathrm{\Phi }(\kappa ,\mu ,x)$: function

Referenced by:

§13.20(iv)

Permalink:

http://dlmf.nist.gov/13.20.E19

Encodings:

pMML, png, TeX

See also:

Annotations for §13.20(iv), §13.20 and Ch.13

… ►For the parabolic cylinder functions $U$ and $\overline{U}$ see §12.2, and for the $\mathrm{env}$ functions associated with $U$ and $\overline{U}$ see §14.15(v). …

26: 7.18 Repeated Integrals of the Complementary Error Function

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Parabolic Cylinder Functions

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7.18.11 $${\mathrm{i}}^n\mathrm{erfc}\left(z\right)=\frac{{\mathrm{e}}^{-z^2/2}}{\sqrt{2^{n-1}\pi }}U(n+\frac{1}{2},z\sqrt{2}).$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $U(a,z)$: parabolic cylinder function, ${\mathrm{i}}^n\mathrm{erfc}\left(z\right)$: repeated integrals of the complementary error function, $z$: complex variable and $n$: nonnegative integer

A&S Ref:

7.2.13

Referenced by:

§12.7(ii)

Permalink:

http://dlmf.nist.gov/7.18.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §7.18(iv), §7.18(iv), §7.18 and Ch.7

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Probability Functions

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27: 12.11 Zeros

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§12.11(i) Distribution of Real Zeros

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§12.11(ii) Asymptotic Expansions of Large Zeros

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§12.11(iii) Asymptotic Expansions for Large Parameter

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28: Nico M. Temme

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12 Parabolic Cylinder Functions

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29: 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. …

30: 16.18 Special Cases

… ►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. …