Kummer equation

SearchAdvancedHelp

(0.002 seconds)

31—39 of 39 matching pages

31: 33.14 Definitions and Basic Properties

… ►

§33.14(i) Coulomb Wave Equation

… ►

33.14.5 $$f(ϵ,\mathrm{\ell };r)={(2r)}^{\mathrm{\ell }+1}{\mathrm{e}}^{-r/\kappa }M(\mathrm{\ell }+1-\kappa ,2\mathrm{\ell }+2,2r/\kappa )/(2\mathrm{\ell }+1)!,$$

ⓘ

Symbols:

$f(ϵ,\mathrm{\ell };r)$: regular Coulomb function, $M(a,b,z)$: $={}_1{}^{}F_1^{}(a;b;z)$ Kummer confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\mathrm{\ell }$: nonnegative integer, $r$: real variable, $ϵ$: real parameter and $\kappa$: quantity

Referenced by:

§13.6(vii)

Permalink:

http://dlmf.nist.gov/33.14.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §33.14(ii), §33.14 and Ch.33

►where $M_{\kappa ,\mu }\left(z\right)$ and $M(a,b,z)$ are defined in §§13.14(i) and 13.2(i), and …This is a consequence of Kummer’s transformation (§13.2(vii)). … ►

33.14.15 $${\int }_0^{\mathrm{\infty }}{\varphi }_{m,\mathrm{\ell }}(r){\varphi }_{n,\mathrm{\ell }}(r)dr={\delta }_{m,n}.$$

ⓘ

Symbols:

${\delta }_{j,k}$: Kronecker delta, $dx$: differential of $x$ , $\int$: integral, $\mathrm{\ell }$: nonnegative integer, $r$: real variable and ${\varphi }_{n,\mathrm{\ell }}(r)$: function

Proof sketch:

Derivable from (33.14.14) by combining it with Dunkl (2003, Theorem 2.2).

Referenced by:

§13.10(vi), §13.23(v), §33.14(iv), Erratum (V1.0.24) for Equation (33.14.15), Erratum (V1.0.24) for Subsection 33.14(iv)

Permalink:

http://dlmf.nist.gov/33.14.E15

Encodings:

pMML, png, TeX

Clarification (effective with 1.0.24):

This equation, originally written as a definite integral ${\int }_0^{\mathrm{\infty }}{\varphi }_{n,\mathrm{\ell }}^2(r)dr=1$, was rewritten more generally as an orthonormality relation.

See also:

Annotations for §33.14(iv), §33.14 and Ch.33

…

32: 33.2 Definitions and Basic Properties

… ►

§33.2(i) Coulomb Wave Equation

… ►This differential equation has a regular singularity at $\rho =0$ with indices $\mathrm{\ell }+1$ and $-\mathrm{\ell }$, and an irregular singularity of rank 1 at $\rho =\mathrm{\infty }$ (§§2.7(i), 2.7(ii)). … ►where $M_{\kappa ,\mu }\left(z\right)$ and $M(a,b,z)$ are defined in §§13.14(i) and 13.2(i), and …This is a consequence of Kummer’s transformation (§13.2(vii)). … ►where $W_{\kappa ,\mu }\left(z\right)$, $U(a,b,z)$ are defined in §§13.14(i) and 13.2(i), …

33: 33.6 Power-Series Expansions in $\rho$

… ►

33.6.5 $$H_{\mathrm{\ell }}^{\pm }(\eta ,\rho )=\frac{{\mathrm{e}}^{\pm \mathrm{i}{\theta }_{\mathrm{\ell }}(\eta ,\rho )}}{(2\mathrm{\ell }+1)!\mathrm{\Gamma }\left(-\mathrm{\ell }\pm \mathrm{i}\eta \right)}\left(\sum _{k=0}^{\mathrm{\infty }}\frac{{\left(a\right)}_k}{{\left(2\mathrm{\ell }+2\right)}_{k}k!}{(\mp 2\mathrm{i}\rho )}^{a+k}\left(\ln \left(\mp 2\mathrm{i}\rho \right)+\psi \left(a+k\right)-\psi \left(1+k\right)-\psi \left(2\mathrm{\ell }+2+k\right)\right)-\sum _{k=1}^{2\mathrm{\ell }+1}\frac{(2\mathrm{\ell }+1)!(k-1)!}{(2\mathrm{\ell }+1-k)!{\left(1-a\right)}_k}{(\mp 2\mathrm{i}\rho )}^{a-k}\right),$$

ⓘ

Symbols:

${\theta }_{\mathrm{\ell }}(\eta ,\rho )$: phase of Coulomb functions, $\mathrm{\Gamma }\left(z\right)$: gamma function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\psi \left(z\right)$: psi (or digamma) function, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\mathrm{i}$: imaginary unit, $H_{\mathrm{\ell }}^{\pm }(\eta ,\rho )$: irregular Coulomb radial functions, $\ln z$: principal branch of logarithm function, $k$: nonnegative integer, $\mathrm{\ell }$: nonnegative integer, $\rho$: nonnegative real variable, $\eta$: real parameter and $a$: variable

A&S Ref:

14.1.14 (equivalent)

Referenced by:

§33.13, §33.6, §33.6, Erratum (V1.0.19) for Equation (33.6.5)

Permalink:

http://dlmf.nist.gov/33.6.E5

Encodings:

pMML, png, TeX

Errata (effective with 1.0.19):

On the right-hand side the factor in the denominator was incorrectly written as $\mathrm{\Gamma }\left(-\mathrm{\ell }+\mathrm{i}\eta \right)$. This has been corrected to $\mathrm{\Gamma }\left(-\mathrm{\ell }\pm \mathrm{i}\eta \right)$.

Reported 2018-05-15 by Ian Thompson

See also:

Annotations for §33.6 and Ch.33

…

34: 17.6 ${}_2{}^{}\varphi _1^{}$ Function

… ►

17.6.1 $${}_2{}^{}\varphi _1^{}(\genfrac{}{}{0pt}{}{a,b}{c};q,c/(ab))=\frac{{(c/a,c/b;q)}_{\mathrm{\infty }}}{{(c,c/(ab);q)}_{\mathrm{\infty }}},$$ .

ⓘ

Symbols:

${}_{r+1}{}^{}\varphi _s^{}(a_0,\mathrm{\dots },a_r;b_1,\mathrm{\dots },b_s;q,z)$ or ${}_{r+1}{}^{}\varphi _s^{}(\genfrac{}{}{0pt}{}{a_0,\mathrm{\dots },a_r}{b_1,\mathrm{\dots },b_s};q,z)$: basic hypergeometric (or $q$-hypergeometric) function, ${(a_1,a_2,\mathrm{\dots },a_r;q)}_n$: multiple $q$-Pochhammer symbol and $q$: complex base

Referenced by:

Erratum (V1.2.0) for Equation (17.6.1)

Permalink:

http://dlmf.nist.gov/17.6.E1

Encodings:

pMML, png, TeX

Addition (effective with 1.2.0):

This constraint was added.

See also:

Annotations for §17.6(i), §17.6(i), §17.6 and Ch.17

… ►

Bailey–Daum $q$-Kummer Sum

… ►

§17.6(iv) Differential Equations

… ►

$q$-Differential Equation

… ►(17.6.27) reduces to the hypergeometric equation (15.10.1) with the substitutions $a\to q^a$, $b\to q^b$, $c\to q^c$, followed by ${lim}_{q\to 1-}$. …

35: Bibliography

… ►

Asymptotics of solutions of the generalized sine-Gordon equation, the third Painlevé equation and the d’Alembert equation. Dokl. Akad. Nauk SSSR 280 (2), pp. 265–268 (Russian).

ⓘ

Notes:

English translation: Soviet Math. Dokl. 31(1985), no. 1, pp. 45–47

External Links:

ISSN 0002-3264, MathReview (M. S. Agranovich), ZentralBlatt

Referenced by:

§32.11(iv).

Encodings:

BibTeX

… ►

Nonlinear chains and Painlevé equations. Phys. D 73 (4), pp. 335–351.

ⓘ

External Links:

ISSN 0167-2789, Document, MathReview (F. Pempinelli), ZentralBlatt

Referenced by:

§32.2(v).

Encodings:

BibTeX

… ►

Numerical evaluation of the Kummer function with complex argument by the trapezoidal rule. Rend. Sem. Mat. Univ. Politec. Torino 49 (3), pp. 315–327.

ⓘ

External Links:

ISSN 0373-1243, MathReview, ZentralBlatt

Referenced by:

§13.29(iii).

Encodings:

BibTeX

… ►

The Whittaker-Hill equation and the wave equation in paraboloidal co-ordinates. Proc. Roy. Soc. Edinburgh Sect. A 67, pp. 265–276.

ⓘ

External Links:

MathReview (H. Hochstadt), ZentralBlatt

Referenced by:

§28.31(i), §28.31(ii), §28.31(ii), §28.32(ii).

Encodings:

BibTeX

… ►

Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA.

ⓘ

External Links:

ISBN 0-89871-412-5, MathReview (Michael Hanke), ZentralBlatt

Referenced by:

§3.7(i).

Encodings:

BibTeX

…

36: 8.19 Generalized Exponential Integral

… ►with $|\mathrm{ph}z|\le \pi$ in both equations. … ►again with $|\mathrm{ph}z|\le \pi$ in both equations. … ►

8.19.16 $$E_p\left(z\right)=z^{p-1}{\mathrm{e}}^{-z}U(p,p,z).$$

ⓘ

Symbols:

$U(a,b,z)$: Kummer confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, $E_p\left(z\right)$: generalized exponential integral, $z$: complex variable and $p$: parameter

Permalink:

http://dlmf.nist.gov/8.19.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §8.19(vi), §8.19 and Ch.8

►For $U(a,b,z)$ see §13.2(i). …

37: 7.18 Repeated Integrals of the Complementary Error Function

… ►

7.18.4 $$\frac{d^n}{{dz}^n}\left({\mathrm{e}}^{z^2}\mathrm{erfc}z\right)={(-1)}^n{2}^{n}n!{\mathrm{e}}^{z^2}{\mathrm{i}}^n\mathrm{erfc}\left(z\right),$$ $n=0,1,2,\mathrm{\dots }$.

ⓘ

Symbols:

$\mathrm{erfc}z$: complementary error function, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), ${\mathrm{i}}^n\mathrm{erfc}\left(z\right)$: repeated integrals of the complementary error function, $z$: complex variable and $n$: nonnegative integer

A&S Ref:

7.2.9

Permalink:

http://dlmf.nist.gov/7.18.E4

Encodings:

pMML, png, TeX

See also:

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

… ►

7.18.9 $${\mathrm{i}}^n\mathrm{erfc}\left(z\right)={\mathrm{e}}^{-z^2}\left(\frac{1}{2^n\mathrm{\Gamma }\left(\frac{1}{2}n+1\right)}M(\frac{1}{2}n+\frac{1}{2},\frac{1}{2},z^2)-\frac{z}{2^{n-1}\mathrm{\Gamma }\left(\frac{1}{2}n+\frac{1}{2}\right)}M(\frac{1}{2}n+1,\frac{3}{2},z^2)\right),$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $M(a,b,z)$: $={}_1{}^{}F_1^{}(a;b;z)$ Kummer confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, ${\mathrm{i}}^n\mathrm{erfc}\left(z\right)$: repeated integrals of the complementary error function, $z$: complex variable and $n$: nonnegative integer

A&S Ref:

7.2.12

Permalink:

http://dlmf.nist.gov/7.18.E9

Encodings:

pMML, png, TeX

See also:

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

►

7.18.10 $${\mathrm{i}}^n\mathrm{erfc}\left(z\right)=\frac{{\mathrm{e}}^{-z^2}}{2^n\sqrt{\pi }}U(\frac{1}{2}n+\frac{1}{2},\frac{1}{2},z^2).$$

ⓘ

Symbols:

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

Referenced by:

§7.18(iv), §7.18(iv), Erratum (V1.0.21) for Paragraph Confluent Hypergeometric Functions (in §7.18(iv))

Permalink:

http://dlmf.nist.gov/7.18.E10

Encodings:

pMML, png, TeX

See also:

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

…

38: Bibliography C

… ►

Théorie Générale de L’Équation de Mathieu et de quelques autres Équations différentielles de la mécanique. Masson et Cie, Paris (French).

ⓘ

External Links:

MathReview (A. Erdélyi), ZentralBlatt

Referenced by:

§28.1, Notations C, Notations I, Notations I, Notations J, Notations J, Notations S.

Encodings:

BibTeX

… ►

Elementary Differential Equations. Clarendon Press, Oxford.

ⓘ

External Links:

ISBN 0-19-853139-7, MathReview, ZentralBlatt

Referenced by:

§16.12.

Encodings:

BibTeX

… ►

An extension of a Kummer’s quadratic transformation formula with an application. Proc. Jangjeon Math. Soc. 16 (2), pp. 229–235.

ⓘ

External Links:

ISSN 1598-7264, MathReview (Petr Blaschke), ZentralBlatt

Referenced by:

§16.6, §16.6.

Encodings:

BibTeX

… ►

The fourth Painlevé equation and associated special polynomials. J. Math. Phys. 44 (11), pp. 5350–5374.

ⓘ

External Links:

ISSN 0022-2488, Document, MathReview, ZentralBlatt

Referenced by:

§32.8(iv).

Encodings:

BibTeX

… ►

Theory of ordinary differential equations. McGraw-Hill Book Company, Inc., New York-Toronto-London.

ⓘ

External Links:

MathReview (M. Zlámal)

Referenced by:

§1.18(ix), §1.18(x).

Encodings:

BibTeX

…

39: 12.14 The Function $W(a,x)$

… ►

§12.14(i) Introduction

►In this section solutions of equation (12.2.3) are considered. This equation is important when $a$ and $z$ $(=x)$ are real, and we shall assume this to be the case. … ►The differential equation … ►For properties of the modulus and phase functions, including differential equations and asymptotic expansions for large $x$, see Miller (1955, pp. 87–88). …