Kummer equation

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21: Bibliography Z

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On the Computation of Zeros of Bessel and Bessel-related Functions. In Proceedings of the Sixth International Colloquium on Differential Equations (Plovdiv, Bulgaria, 1995), D. Bainov (Ed.), Utrecht, pp. 409–416.

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External Links:

MathReview, ZentralBlatt

Referenced by:

§10.74(vi).

Encodings:

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Distribution of zeros of Gauss and Kummer hypergeometric functions. A semiclassical approach. Ann. Numer. Math. 2 (1-4), pp. 457–472.

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External Links:

ISSN 1021-2655, MathReview (B. D. Agrawal), ZentralBlatt

Referenced by:

§13.9(i), §15.13.

Encodings:

BibTeX

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Doron Zeilberger’s Maple Packages and Programs Department of Mathematics, Rutgers University, New Jersey.

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Notes:

Includes hypergeometric and $q$-hypergeometric summation, solution of difference equations (for example, by Petkovšek’s algorithm), combinatorial algorithms, and (package LUC) evaluation of zonal polynomials.

External Links:

Home Page

Referenced by:

§ ‣ Software Cross Index, § ‣ Software Cross Index, L. Lapointe and L. Vinet (1996).

Encodings:

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Error bounds for asymptotic solutions of second-order linear difference equations. J. Comput. Appl. Math. 71 (2), pp. 191–212.

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External Links:

ISSN 0377-0427, Document, MathReview, ZentralBlatt

Referenced by:

§2.9(i), §2.9(ii).

Encodings:

BibTeX

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22: Bibliography D

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Computational properties of three-term recurrence relations for Kummer functions. J. Comput. Appl. Math. 233 (6), pp. 1505–1510.

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External Links:

ISSN 0377-0427, Document, FullText, MathReview (Richard B. Paris), ZentralBlatt

Referenced by:

§13.29(iv), §13.29(iv).

Encodings:

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Unification of one-dimensional Fokker-Planck equations beyond hypergeometrics: Factorizer solution method and eigenvalue schemes. Phys. Rev. E (3) 57 (1), pp. 252–275.

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External Links:

ISSN 1063-651X, Document, MathReview (Victor I. Stepanov), ZentralBlatt

Referenced by:

§31.17(ii).

Encodings:

BibTeX

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Formes canoniques des équations confluentes de l’équation de Heun. Ann. Soc. Sci. Bruxelles Sér. I 92 (1-2), pp. 53–78.

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External Links:

MathReview (A. E. Danese), ZentralBlatt

Referenced by:

§31.12.

Encodings:

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Sur les équations confluentes de l’équation de Heun. Ann. Soc. Sci. Bruxelles Sér. I 92 (3), pp. 151–189.

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External Links:

ISSN 0037-959X, MathReview, ZentralBlatt

Referenced by:

§31.12.

Encodings:

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The KP equation with quasiperiodic initial data. Phys. D 123 (1-4), pp. 123–152.

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Notes:

Nonlinear waves and solitons in physical systems (Los Alamos, NM, 1997)

External Links:

ISSN 0167-2789, Document, MathReview (Pavel Krejčí), ZentralBlatt

Referenced by:

§21.9.

Encodings:

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23: Bibliography P

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Sur les équations différentielles du second ordre à points critiques fixès. C.R. Acad. Sc. Paris 143, pp. 1111–1117.

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External Links:

ZentralBlatt

Referenced by:

§32.2(iv).

Encodings:

BibTeX

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Smoothing of the Stokes phenomenon for high-order differential equations. Proc. Roy. Soc. London Ser. A 436, pp. 165–186.

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External Links:

ISSN 0962-8444, FullText, MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§2.11(v).

Encodings:

BibTeX

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A uniform asymptotic expansion for the incomplete gamma function. J. Comput. Appl. Math. 148 (2), pp. 323–339.

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External Links:

ISSN 0377-0427, Document, MathReview (Dumitru Acu), ZentralBlatt

Referenced by:

(8.11.6), §8.11(iii), (8.12.18), §8.12, §8.12, §8.12, Erratum (V1.0.16) for Equation (8.12.18).

Encodings:

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A Kummer-type transformation for a ${}_2{}^{}F_2^{}$ hypergeometric function. J. Comput. Appl. Math. 173 (2), pp. 379–382.

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External Links:

ISSN 0377-0427, Document, MathReview (P. Anandani), ZentralBlatt

Referenced by:

§16.6.

Encodings:

BibTeX

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Remarks on computing the probability integral in one and two dimensions. In Proceedings of the Berkeley Symposium on Mathematical Statistics and Probability, 1945, 1946, pp. 63–78.

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External Links:

MathReview (L. A. Aroian), ZentralBlatt

Referenced by:

(7.8.8), §7.8, Erratum (V1.0.17) for Equation (7.8.8).

Encodings:

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24: Bibliography M

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On reducing the Heun equation to the hypergeometric equation. J. Differential Equations 213 (1), pp. 171–203.

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External Links:

ISSN 0022-0396, Document, MathReview (Wolfgang Bühring), ZentralBlatt

Referenced by:

§31.7(i).

Encodings:

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The 192 solutions of the Heun equation. Math. Comp. 76 (258), pp. 811–843.

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External Links:

ISSN 0025-5718, Document, MathReview (Wadim Zudilin), ZentralBlatt (Masaaki Yoshida)

Referenced by:

§31.3(iii).

Encodings:

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Rational solutions of the Painlevé VI equation. J. Phys. A 34 (11), pp. 2281–2294.

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External Links:

ISSN 0305-4470, Document, MathReview (Shun Shimomura), ZentralBlatt

Referenced by:

§32.8(vi).

Encodings:

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Picard and Chazy solutions to the Painlevé VI equation. Math. Ann. 321 (1), pp. 157–195.

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External Links:

ISSN 0025-5831, Document, MathReview, ZentralBlatt

Referenced by:

§32.9(iii).

Encodings:

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On a Kummer-type transformation for the generalized hypergeometric function ${}_2{}^{}F_2^{}$ . J. Comput. Appl. Math. 157 (2), pp. 507–509.

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External Links:

ISSN 0377-0427, Document, MathReview Entry, ZentralBlatt

Referenced by:

§16.6.

Encodings:

BibTeX

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25: Bibliography T

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Asymptotic expansions of Kummer hypergeometric functions with three asymptotic parameters a, b and z. Indagationes Mathematicae.

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External Links:

ISSN 0019-3577, Document, Link

Referenced by:

§13.8(iv).

Encodings:

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Asymptotic expansions of Kummer hypergeometric functions for large values of the parameters. Integral Transforms Spec. Funct. 33 (1), pp. 16–31.

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External Links:

ISSN 1065-2469, Document, Link, MathReview Entry, ZentralBlatt

Referenced by:

§13.8(iv).

Encodings:

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Rotating black holes: Separable wave equations for gravitational and electromagnetic perturbations. Phys. Rev. Lett. 29 (16), pp. 1114–1118.

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External Links:

Document

Referenced by:

§31.17(ii).

Encodings:

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Eigenfunction Expansions Associated with Second-Order Differential Equations. Clarendon Press, Oxford.

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External Links:

MathReview (H. Pollard)

Referenced by:

§1.18(x).

Encodings:

BibTeX

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Eigenfunction Expansions Associated with Second Order Differential Equations, Part 2, Partial Differential Equations. Clarendon Press, Oxford.

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External Links:

ISBN 0198533187, MathReview Entry

Referenced by:

§1.18(x).

Encodings:

BibTeX

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26: 18.5 Explicit Representations

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18.5.4_5 $${\mathrm{i}}^n{U}_n\left(\frac{1}{2\mathrm{i}}\right)=F_{n+1}.$$

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Symbols:

$U_n\left(x\right)$: Chebyshev polynomial of the second kind, $\mathrm{i}$: imaginary unit and $n$: nonnegative integer

Proof sketch:

Use (18.5.2) and (26.11.7).

Referenced by:

§18.5(i), §18.5(i), Erratum (V1.2.0) Section 18.5

Permalink:

http://dlmf.nist.gov/18.5.E4_5

Encodings:

pMML, png, TeX

Addition (effective with 1.2.0):

This equation was added.

See also:

Annotations for §18.5(i), §18.5(i), §18.5 and Ch.18

… ►In this equation $w(x)$ is as in Table 18.3.1, (reproduced in Table 18.5.1), and $F(x)$, ${\kappa }_n$ are as in Table 18.5.1. … ►For the definitions of ${}_2{}^{}F_1^{}$, ${}_1{}^{}F_1^{}$, and ${}_2{}^{}F_0^{}$ see §16.2. … ►

18.5.12 $$L_n^{(\alpha )}\left(x\right)=\sum _{\mathrm{\ell }=0}^n\frac{{\left(\alpha +\mathrm{\ell }+1\right)}_{n-\mathrm{\ell }}}{(n-\mathrm{\ell })!\mathrm{\ell }!}{(-x)}^{\mathrm{\ell }}=\frac{{\left(\alpha +1\right)}_n}{n!}{}_1{}^{}F_1^{}(\genfrac{}{}{0pt}{}{-n}{\alpha +1};x).$$

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Symbols:

${}_1{}^{}F_1^{}(a;b;z)$: $=M(a,b,z)$ notation for the Kummer confluent hypergeometric function, $L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $\mathrm{\ell }$: nonnegative integer, $n$: nonnegative integer and $x$: real variable

A&S Ref:

22.5.54

Referenced by:

§18.11(i), §18.17(v), §18.17(vi), §18.17(vii), §18.18(iii), §18.21(ii), Table 18.3.1, (18.34.1), §18.34(i), §18.5(iii), §18.6(ii)

Permalink:

http://dlmf.nist.gov/18.5.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §18.5(iii), §18.5(iii), §18.5 and Ch.18

… ►The first of each of equations (18.5.7) and (18.5.8) can be regarded as definitions of $P_n^{(\alpha ,\beta )}\left(x\right)$ when the conditions $\alpha >-1$ and $\beta >-1$ are not satisfied. …

27: Bibliography Y

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On rational solutions of the second Painlevé equation. Vesti Akad. Navuk. BSSR Ser. Fiz. Tkh. Nauk. 3, pp. 30–35 (Russian).

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Referenced by:

§32.8(ii).

Encodings:

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Computation of Kummer functions $U(a,b,x)$ for large argument $x$ by using the $\tau$-method. Trans. Inform. Process. Soc. Japan 36 (10), pp. 2335–2342 (Japanese).

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Notes:

Japanese with English summary. Double-precision Fortran.

External Links:

ISSN 0387-5806, FullText, MathReview

Referenced by:

§13.32(ii).

Encodings:

BibTeX

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28: 9.10 Integrals

… ►Let $w(z)$ be any solution of Airy’s equation (9.2.1). … ►

9.10.14 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-pt}\mathrm{Ai}\left(t\right)dt={\mathrm{e}}^{-p^3/3}\left(\frac{1}{3}-\frac{p{}_1{}^{}F_1^{}(\frac{1}{3};\frac{4}{3};\frac{1}{3}{p}^3)}{3^{4/3}\mathrm{\Gamma }\left(\frac{4}{3}\right)}+\frac{p^2{}_1{}^{}F_1^{}(\frac{2}{3};\frac{5}{3};\frac{1}{3}{p}^3)}{3^{5/3}\mathrm{\Gamma }\left(\frac{5}{3}\right)}\right),$$ $p\in ℂ$.

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Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $\mathrm{\Gamma }\left(z\right)$: gamma function, ${}_1{}^{}F_1^{}(a;b;z)$: $=M(a,b,z)$ notation for the Kummer confluent hypergeometric function, $ℂ$: complex plane, $dx$: differential of $x$ , $\in$: element of, $\mathrm{e}$: base of natural logarithm, $\int$: integral and $p$: parameter

Keywords:

Laplace transform

Source:

Gibbs (1973, Problem 72–21)

Permalink:

http://dlmf.nist.gov/9.10.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §9.10(v), §9.10 and Ch.9

… ►For the confluent hypergeometric function ${}_1{}^{}F_1^{}$ and the incomplete gamma function $\mathrm{\Gamma }$ see §§13.1, 13.2, and 8.2(i). … ►

9.10.18 $$\mathrm{Ai}\left(z\right)=\frac{3z^{5/4}{\mathrm{e}}^{-(2/3)z^{3/2}}}{4\pi }{\int }_0^{\mathrm{\infty }}\frac{t^{-3/4}{\mathrm{e}}^{-(2/3)t^{3/2}}\mathrm{Ai}\left(t\right)}{z^{3/2}+t^{3/2}}dt,$$ .

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Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\mathrm{ph}$: phase and $z$: complex variable

Proof sketch:

To prove this equation, combine (13.4.5) with (13.6.11) and use (5.5.3).

Referenced by:

(9.10.19), §9.17(iii), §9.5(ii), Erratum (V1.0.10) for Equation (9.10.18)

Permalink:

http://dlmf.nist.gov/9.10.E18

Encodings:

pMML, png, TeX

Errata (effective with 1.0.10):

The original and incorrect version,

$$\mathrm{Ai}\left(z\right)=\frac{z^{5/4}{\mathrm{e}}^{-(2/3)z^{3/2}}}{2^{7/2}\pi }{\int }_0^{\mathrm{\infty }}\frac{t^{-1/2}{\mathrm{e}}^{-(2/3)t^{3/2}}\mathrm{Ai}\left(t\right)}{z^{3/2}+t^{3/2}}dt$$

taken from Schulten et al. (1979), has been corrected.

Reported 2015-03-20 by Walter Gautschi

See also:

Annotations for §9.10(vii), §9.10 and Ch.9

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9.10.19 $$\mathrm{Bi}\left(x\right)=\frac{3x^{5/4}{\mathrm{e}}^{(2/3)x^{3/2}}}{2\pi }{⨍}_0^{\mathrm{\infty }}\frac{t^{-3/4}{\mathrm{e}}^{-(2/3)t^{3/2}}\mathrm{Ai}\left(t\right)}{x^{3/2}-t^{3/2}}dt,$$ $x>0$,

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Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $\mathrm{Bi}\left(z\right)$: Airy function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, ${⨍}_a^b$: Cauchy principal value and $x$: real variable

Proof sketch:

To prove this equation, combine (9.2.10) with (9.10.18).

Referenced by:

§9.5(i), Erratum (V1.0.10) for Equation (9.10.19)

Permalink:

http://dlmf.nist.gov/9.10.E19

Encodings:

pMML, png, TeX

Errata (effective with 1.0.10):

The original and incorrect version,

$$\mathrm{Bi}\left(x\right)=\frac{x^{5/4}{\mathrm{e}}^{(2/3)x^{3/2}}}{2^{5/2}\pi }{⨍}_0^{\mathrm{\infty }}\frac{t^{-1/2}{\mathrm{e}}^{-(2/3)t^{3/2}}\mathrm{Ai}\left(t\right)}{x^{3/2}-t^{3/2}}dt$$

taken from Schulten et al. (1979), has been corrected.

Reported 2015-03-20 by Walter Gautschi

See also:

Annotations for §9.10(vii), §9.10 and Ch.9

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29: 7.11 Relations to Other Functions

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7.11.4 $$\mathrm{erf}z=\frac{2z}{\sqrt{\pi }}M(\frac{1}{2},\frac{3}{2},-z^2)=\frac{2z}{\sqrt{\pi }}{\mathrm{e}}^{-z^2}M(1,\frac{3}{2},z^2),$$

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Symbols:

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

A&S Ref:

7.1.21

Permalink:

http://dlmf.nist.gov/7.11.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §7.11, §7.11 and Ch.7

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7.11.5 $$\mathrm{erfc}z=\frac{1}{\sqrt{\pi }}{\mathrm{e}}^{-z^2}U(\frac{1}{2},\frac{1}{2},z^2)=\frac{z}{\sqrt{\pi }}{\mathrm{e}}^{-z^2}U(1,\frac{3}{2},z^2).$$

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Symbols:

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

Referenced by:

§7.11

Permalink:

http://dlmf.nist.gov/7.11.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §7.11, §7.11 and Ch.7

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7.11.6 $$C\left(z\right)+\mathrm{i}S\left(z\right)=zM(\frac{1}{2},\frac{3}{2},\frac{1}{2}\pi \mathrm{i}{z}^2)=z{\mathrm{e}}^{\pi \mathrm{i}{z}^2/2}M(1,\frac{3}{2},-\frac{1}{2}\pi \mathrm{i}{z}^2).$$

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Symbols:

$C\left(z\right)$: Fresnel integral, $S\left(z\right)$: Fresnel integral, $M(a,b,z)$: $={}_1{}^{}F_1^{}(a;b;z)$ Kummer confluent hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit and $z$: complex variable

Permalink:

http://dlmf.nist.gov/7.11.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §7.11, §7.11 and Ch.7

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30: Software Index

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	Open Source													With Book						Commercial
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13.32(ii) $M(a,b,x)$, $U(a,b,x)$, $𝐌(a,b,x)$, $M_{\kappa ,\mu }\left(x\right)$, $W_{\kappa ,\mu }\left(x\right)$, $x,a,b\in ℝ$	✓		✓	✓	✓	✓						✓	✓	✓			✓		✓		✓	✓	a
13.32(iii) $M(a,b,z)$, $U(a,b,z)$, $𝐌(a,b,z)$, $M_{\kappa ,\mu }\left(z\right)$, $W_{\kappa ,\mu }\left(z\right)$, $z,a,b\in ℂ$	✓											✓	✓	✓					✓		✓	✓	a
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28 Mathieu Functions and Hill’s Equation
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