parabolic cylinder functions

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31: 12.19 Tables

§12.19 Tables

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32: 14.15 Uniform Asymptotic Approximations

… ►Here we introduce the envelopes of the parabolic cylinder functions $U(-c,x)$, $\overline{U}(-c,x)$, which are defined in §12.2. For $U(-c,x)$ or $\overline{U}(-c,x)$, with $c$ and $x$ nonnegative, … ►

14.15.24 $${𝖯}_{\nu }^{-\mu }\left(x\right)=\frac{1}{{\left(\nu +\frac{1}{2}\right)}^{1/4}{2}^{(\nu +\mu )/2}\mathrm{\Gamma }\left(\frac{1}{2}\nu +\frac{1}{2}\mu +\frac{3}{4}\right)}{\left(\frac{{\zeta }^2-{\alpha }^2}{x^2-a^2}\right)}^{1/4}\left(U(\mu -\nu -\frac{1}{2},{\left(2\nu +1\right)}^{1/2}\zeta )+O\left({\nu }^{-2/3}\right)\mathrm{env}U(\mu -\nu -\frac{1}{2},{\left(2\nu +1\right)}^{1/2}\zeta )\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, $\mathrm{\Gamma }\left(z\right)$: gamma function, : Ferrers function of the first kind, $\mathrm{env}U(c,x)$: envelope of parabolic cylinder function $U(c,x)$ , $U(a,z)$: parabolic cylinder function, $x$: real variable, $\mu$: general order, $\nu$: general degree, $a$ , $\zeta$ and $\alpha$

Permalink:

http://dlmf.nist.gov/14.15.E24

Encodings:

pMML, png, TeX

See also:

Annotations for §14.15(v), §14.15 and Ch.14

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14.15.25 $${𝖰}_{\nu }^{-\mu }\left(x\right)=\frac{\pi }{{\left(\nu +\frac{1}{2}\right)}^{1/4}{2}^{(\nu +\mu +2)/2}\mathrm{\Gamma }\left(\frac{1}{2}\nu +\frac{1}{2}\mu +\frac{3}{4}\right)}{\left(\frac{{\zeta }^2-{\alpha }^2}{x^2-a^2}\right)}^{1/4}\left(\overline{U}(\mu -\nu -\frac{1}{2},{\left(2\nu +1\right)}^{1/2}\zeta )+O\left({\nu }^{-2/3}\right)\mathrm{env}\overline{U}(\mu -\nu -\frac{1}{2},{\left(2\nu +1\right)}^{1/2}\zeta )\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, $\mathrm{\Gamma }\left(z\right)$: gamma function, : Ferrers function of the second kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{env}\overline{U}(c,x)$: envelope of parabolic cylinder function $\overline{U}(c,x)$ , $\overline{U}(a,x)$: parabolic cylinder function, $x$: real variable, $\mu$: general order, $\nu$: general degree, $a$ , $\zeta$ and $\alpha$

Permalink:

http://dlmf.nist.gov/14.15.E25

Encodings:

pMML, png, TeX

See also:

Annotations for §14.15(v), §14.15 and Ch.14

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14.15.30 $${𝖯}_{\nu }^{-\mu }\left(x\right)=\frac{1}{{\left(\nu +\frac{1}{2}\right)}^{1/4}{2}^{(\nu +\mu )/2}\mathrm{\Gamma }\left(\frac{1}{2}\nu +\frac{1}{2}\mu +\frac{3}{4}\right)}{\left(\frac{{\zeta }^2+{\alpha }^2}{x^2+a^2}\right)}^{1/4}U(\mu -\nu -\frac{1}{2},{\left(2\nu +1\right)}^{1/2}\zeta )\left(1+O\left({\nu }^{-1}\ln \nu \right)\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, $\mathrm{\Gamma }\left(z\right)$: gamma function, : Ferrers function of the first kind, $\ln z$: principal branch of logarithm function, $U(a,z)$: parabolic cylinder function, $x$: real variable, $\mu$: general order, $\nu$: general degree, $a$ , $\zeta$ and $\alpha$

Referenced by:

§14.15(v)

Permalink:

http://dlmf.nist.gov/14.15.E30

Encodings:

pMML, png, TeX

See also:

Annotations for §14.15(v), §14.15 and Ch.14

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33: 32.10 Special Function Solutions

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§32.10(iv) Fourth Painlevé Equation

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

32.10.19 $$\varphi (z)=\left(C_{1}U(a,\sqrt{2}z)+C_{2}V(a,\sqrt{2}z)\right)\exp \left(\frac{1}{2}\epsilon z^2\right),$$

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Symbols:

$\exp z$: exponential function, $U(a,z)$: parabolic cylinder function, $V(a,z)$: parabolic cylinder function, $z$: real, $\epsilon =\pm 1$ , $\varphi (z)$: function, $a$ and $C$: constant

Permalink:

http://dlmf.nist.gov/32.10.E19

Encodings:

pMML, png, TeX

See also:

Annotations for §32.10(iv), §32.10 and Ch.32

… ►When $a+\frac{1}{2}$ is zero or a negative integer the $U$ parabolic cylinder functions reduce to Hermite polynomials (§18.3) times an exponential function; thus …

34: Bibliography T

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Parabolic cylinder functions $U(n,x)$ for natural $n$ and positive $x$ . Comput. Phys. Commun. 69, pp. 415–419.

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Notes:

Double-precision Fortran, minimum accuracy 14S, maximum accuracy 21D.

External Links:

Document, Code

Referenced by:

4th item.

Encodings:

BibTeX

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The numerical computation of special functions by use of quadrature rules for saddle point integrals. II. Gamma functions, modified Bessel functions and parabolic cylinder functions. Report TW 183/78 Mathematisch Centrum, Amsterdam, Afdeling Toegepaste Wiskunde.

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Notes:

Algol procedures, maximum accuracy 14S.

External Links:

FullText, ZentralBlatt

Referenced by:

§12.21(ii).

Encodings:

BibTeX

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Numerical and asymptotic aspects of parabolic cylinder functions. J. Comput. Appl. Math. 121 (1-2), pp. 221–246.

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Notes:

Numerical analysis in the 20th century, Vol. I, Approximation theory

External Links:

ISSN 0377-0427, Document, MathReview (José Luis López), ZentralBlatt

Referenced by:

§12.10(vii), §12.10(vi), §12.10(vi), §12.18.

Encodings:

BibTeX

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35: Bibliography S

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Evaluation of associated Legendre functions off the cut and parabolic cylinder functions. Electron. Trans. Numer. Anal. 9, pp. 137–146.

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Notes:

Orthogonal polynomials: numerical and symbolic algorithms (Leganés, 1998)

External Links:

ISSN 1068-9613, FullText, MathReview (Adam Marlewski), ZentralBlatt

Referenced by:

§14.30(i), 3rd item.

Encodings:

BibTeX

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On the expansion of the parabolic cylinder function in a series of the product of two parabolic cylinder functions. J. Indian Math. Soc. (N. S.) 3, pp. 226–230.

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External Links:

ZentralBlatt

Referenced by:

§12.13(i).

Encodings:

BibTeX

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On certain integrals and expansions involving Weber’s parabolic cylinder functions. J. Indian Math. Soc. (N. S.) 4, pp. 158–166.

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External Links:

MathReview (M. C. Gray), ZentralBlatt

Referenced by:

§12.13(ii).

Encodings:

BibTeX

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Parabolic Cylinder Functions and their Application in Symmetric Two-centre Shell Model. In Proceedings of the Conference on Mathematical Analysis and its Applications (Inst. Engrs., Mysore, 1977), Matscience Rep., Vol. 91, Aarhus, pp. P81–P89.

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External Links:

MathReview

Referenced by:

§12.17.

Encodings:

BibTeX

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On parabolic cylinder functions. J. Inst. Math. Appl. 4 (1), pp. 106–112.

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External Links:

Document, MathReview (F. M. Arscott), ZentralBlatt

Referenced by:

§12.16.

Encodings:

BibTeX

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36: 18.11 Relations to Other Functions

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18.11.3 $$H_n\left(x\right)=2^{n}U(-\frac{1}{2}n,\frac{1}{2},x^2)=2^{n}xU(-\frac{1}{2}n+\frac{1}{2},\frac{3}{2},x^2)=2^{\frac{1}{2}n}{\mathrm{e}}^{\frac{1}{2}{x}^2}U(-n-\frac{1}{2},2^{\frac{1}{2}}x),$$

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Symbols:

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

A&S Ref:

22.5.55, 22.5.58 ((factor $x$ missing on RHS of 22.5.55))

Referenced by:

§18.11(i), §18.15(v)

Permalink:

http://dlmf.nist.gov/18.11.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §18.11(i), §18.11(i), §18.11 and Ch.18

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18.11.4 $${\mathrm{𝐻𝑒}}_n\left(x\right)=2^{\frac{1}{2}n}U(-\frac{1}{2}n,\frac{1}{2},\frac{1}{2}{x}^2)=2^{\frac{1}{2}(n-1)}xU(-\frac{1}{2}n+\frac{1}{2},\frac{3}{2},\frac{1}{2}{x}^2)={\mathrm{e}}^{\frac{1}{4}{x}^2}U(-n-\frac{1}{2},x).$$

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Symbols:

${\mathrm{𝐻𝑒}}_n\left(x\right)$: Hermite polynomial, $U(a,b,z)$: Kummer confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, $U(a,z)$: parabolic cylinder function, $n$: nonnegative integer and $x$: real variable

A&S Ref:

22.5.59

Permalink:

http://dlmf.nist.gov/18.11.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §18.11(i), §18.11(i), §18.11 and Ch.18

►For the parabolic cylinder function $U(a,z)$, see §12.2. …

37: Bibliography G

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Integral representations for computing real parabolic cylinder functions. Numer. Math. 98 (1), pp. 105–134.

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External Links:

ISSN 0029-599X, Document, MathReview Entry, ZentralBlatt

Referenced by:

§12.18.

Encodings:

BibTeX

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Computing the real parabolic cylinder functions $U(a,x)$, $V(a,x)$ . ACM Trans. Math. Software 32 (1), pp. 70–101.

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External Links:

ISSN 0098-3500, Document, MathReview (Syamal K. Sen), ZentralBlatt

Referenced by:

§12.18.

Encodings:

BibTeX

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Algorithm 850: Real parabolic cylinder functions $U(a,x)$, $V(a,x)$ . ACM Trans. Math. Software 32 (1), pp. 102–112.

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External Links:

ISSN 0098-3500, Document, MathReview (Syamal K. Sen), ZentralBlatt

Referenced by:

§12.18, 1st item.

Encodings:

BibTeX

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Algorithm 914: parabolic cylinder function $W(a,x)$ and its derivative. ACM Trans. Math. Software 38 (1), pp. Art. 6, 5.

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External Links:

ISSN 0098-3500, Document, FullText, MathReview Entry, ZentralBlatt

Referenced by:

§12.18, §12.18, 2nd item, §12.21(ii).

Encodings:

BibTeX

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Fast and accurate computation of the Weber parabolic cylinder function $W(a,x)$ . IMA J. Numer. Anal. 31 (3), pp. 1194–1216.

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External Links:

ISSN 0272-4979, Document, FullText, MathReview (Adam Marlewski), ZentralBlatt

Referenced by:

§12.18, §12.18.

Encodings:

BibTeX

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38: 32.3 Graphics

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32.3.3 $$u\sim kU(-\nu -\frac{1}{2},x),$$ $x\to +\mathrm{\infty }$.

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Symbols:

$\sim$: asymptotic equality, $U(a,z)$: parabolic cylinder function, $x$: real, $k$: real, $\nu$: parameter and $u_k(x;\nu )$: solution

Permalink:

http://dlmf.nist.gov/32.3.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §32.3(iii), §32.3 and Ch.32

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32.3.5 $$w(x)\sim 2\sqrt{2}{k}^2{U}^2(-\nu -\frac{1}{2},\sqrt{2}x),$$ $x\to +\mathrm{\infty }$;

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Symbols:

$\sim$: asymptotic equality, $U(a,z)$: parabolic cylinder function, $x$: real, $k$: real and $\nu$: parameter

Permalink:

http://dlmf.nist.gov/32.3.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §32.3(iii), §32.3 and Ch.32

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39: Bibliography O

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Uniform asymptotic expansions for Weber parabolic cylinder functions of large orders. J. Res. Nat. Bur. Standards Sect. B 63B, pp. 131–169.

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External Links:

MathReview (N. D. Kazarinoff), ZentralBlatt

Referenced by:

§12.10(i), §12.10(ii), §12.10(iii), §12.10(iv), §12.10(v), §12.10(vii), §12.10(vii), §12.11(iii), §12.11(iii), §12.14(ix), §12.14(ix), §12.14(ix), §12.14(xi), Ch.12, §18.16(v).

Encodings:

BibTeX

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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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40: 28.8 Asymptotic Expansions for Large $q$

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28.8.4 $$U_m(\xi )\sim D_m\left(\xi \right)-\frac{1}{2^{6}h}\left(D_{m+4}\left(\xi \right)-4!\left(\genfrac{}{}{0pt}{}{m}{4}\right)D_{m-4}\left(\xi \right)\right)+\frac{1}{2^{13}{h}^2}\left(D_{m+8}\left(\xi \right)-2^5(m+2)D_{m+4}\left(\xi \right)+4!{\mathrm{\hspace{0.17em}2}}^5(m-1)\left(\genfrac{}{}{0pt}{}{m}{4}\right)D_{m-4}\left(\xi \right)+8!\left(\genfrac{}{}{0.0pt}{}{m}{8}\right)D_{m-8}\left(\xi \right)\right)+\mathrm{\cdots },$$

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Defines:

$U_m(\xi )$: function (locally)

Symbols:

$D_{\nu }\left(z\right)$: parabolic cylinder function, $\sim$: Poincaré asymptotic expansion, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $!$: factorial (as in $n!$), $m$: integer, $h$: parameter and $\xi$: variable

A&S Ref:

20.9.17 (in different form)

Permalink:

http://dlmf.nist.gov/28.8.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §28.8(ii), §28.8 and Ch.28

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28.8.5 $$V_m(\xi )\sim \frac{1}{2^{4}h}\left(-D_{m+2}\left(\xi \right)-m(m-1)D_{m-2}\left(\xi \right)\right)+\frac{1}{2^{10}{h}^2}\left(D_{m+6}\left(\xi \right)+(m^2-25m-36)D_{m+2}\left(\xi \right)-m(m-1)(m^2+27m-10)D_{m-2}\left(\xi \right)-6!\left(\genfrac{}{}{0.0pt}{}{m}{6}\right)D_{m-6}\left(\xi \right)\right)+\mathrm{\cdots },$$

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Defines:

$V_m(\xi )$: function (locally)

Symbols:

$D_{\nu }\left(z\right)$: parabolic cylinder function, $\sim$: Poincaré asymptotic expansion, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $!$: factorial (as in $n!$), $m$: integer, $h$: parameter and $\xi$: variable

A&S Ref:

20.9.18 (in slightly different form)

Referenced by:

Erratum (V1.0.16) for Equation (28.8.5)

Permalink:

http://dlmf.nist.gov/28.8.E5

Encodings:

pMML, png, TeX

Error (effective with 1.0.16):

Originally the $-$ in front of the $6!$ was given incorrectly as $+$.

Suggested 2017-02-02 by Daniel Karlsson

See also:

Annotations for §28.8(ii), §28.8 and Ch.28

… ►The approximants are elementary functions, Airy functions, Bessel functions, and parabolic cylinder functions; compare §2.8. …