Kummer solutions

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11: Bibliography Z

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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:

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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:

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12: 33.14 Definitions and Basic Properties

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§33.14(i) Coulomb Wave Equation

… ►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(iii) Irregular Solution $h(ϵ,\mathrm{\ell };r)$

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§33.14(iv) Solutions $s(ϵ,\mathrm{\ell };r)$ and $c(ϵ,\mathrm{\ell };r)$

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13: 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).

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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).

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Pole dynamics for elliptic solutions of the Korteweg-de Vries equation. Math. Phys. Anal. Geom. 3 (1), pp. 49–74.

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External Links:

ISSN 1385-0172, Document, MathReview (Giulio Soliani), ZentralBlatt

Referenced by:

§23.21(ii).

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Error bounds for exponentially improved asymptotic solutions of ordinary differential equations having irregular singularities of rank one. Methods Appl. Anal. 3 (1), pp. 109–134.

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External Links:

ISSN 1073-2772, Pdf, MathReview (Roderick Wong), ZentralBlatt

Referenced by:

§2.11(v).

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Convergent expansions for solutions of linear ordinary differential equations having a simple turning point, with an application to Bessel functions. Stud. Appl. Math. 107 (3), pp. 293–323.

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External Links:

ISSN 0022-2526, Document, MathReview, ZentralBlatt

Referenced by:

§10.20(ii), §2.8(i).

Encodings:

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14: Bibliography K

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Determinant structure of the rational solutions for the Painlevé II equation. J. Math. Phys. 37 (9), pp. 4693–4704.

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External Links:

ISSN 0022-2488, Document, MathReview (Andrei A. Kapaev), ZentralBlatt

Referenced by:

§32.8(ii).

Encodings:

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Determinant structure of the rational solutions for the Painlevé IV equation. J. Phys. A 31 (10), pp. 2431–2446.

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External Links:

ISSN 0305-4470, Document, MathReview (Andrei A. Kapaev), ZentralBlatt

Referenced by:

§32.8(iv).

Encodings:

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Asymptotic behavior of the solutions of the Painlevé equation of the first kind. Differ. Uravn. 24 (10), pp. 1684–1695 (Russian).

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Notes:

In Russian. English translation: Differential Equations 24(1988), no. 10, pp. 1107–1115 (1989)

External Links:

ISSN 0374-0641, MathReview, ZentralBlatt

Referenced by:

§32.11(i).

Encodings:

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The congruences of Clausen-von Staudt and Kummer for Bernoulli-Hurwitz numbers. Math. Ann. 216 (1), pp. 1–4.

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External Links:

ISSN 0025-5831, Document, MathReview (M. Basmakov), ZentralBlatt

Referenced by:

§24.16(iii).

Encodings:

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Asymptotic solution of Maxwell’s equations near caustics. Izv. Vuz. Radiofiz. 7, pp. 1049–1056.

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Referenced by:

§36.14(i).

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15: Bibliography M

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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).

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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).

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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:

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Classical solutions of the third Painlevé equation. Nagoya Math. J. 139, pp. 37–65.

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External Links:

ISSN 0027-7630, MathReview (Andrei A. Kapaev), ZentralBlatt

Referenced by:

§32.8(iii).

Encodings:

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16: 33.2 Definitions and Basic Properties

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§33.2(i) Coulomb Wave Equation

… ►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), … ►As in the case of $F_{\mathrm{\ell }}(\eta ,\rho )$, the solutions $H_{\mathrm{\ell }}^{\pm }(\eta ,\rho )$ and $G_{\mathrm{\ell }}(\eta ,\rho )$ are analytic functions of $\rho$ when . …

17: 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:

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18: 9.10 Integrals

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

9.10.8 $$\int zw(z)dz=w^{\prime }(z),$$

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Symbols:

$dx$: differential of $x$ , $\int$: integral, $z$: complex variable and $w$: ODE solution

Proof sketch:

To verify, differentiate and refer to (9.2.1).

Permalink:

http://dlmf.nist.gov/9.10.E8

Encodings:

pMML, png, TeX

See also:

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

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9.10.9 $$\int z^{2}w(z)dz=z{w}^{\prime }(z)-w(z),$$

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Symbols:

$dx$: differential of $x$ , $\int$: integral, $z$: complex variable and $w$: ODE solution

Proof sketch:

To verify, differentiate and refer to (9.2.1).

Permalink:

http://dlmf.nist.gov/9.10.E9

Encodings:

pMML, png, TeX

See also:

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

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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). …

19: 16.4 Argument Unity

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§16.4(iii) Identities

… ►The other three-term relations are extensions of Kummer’s relations for ${}_2{}^{}F_1^{}$’s given in §15.10(ii). … … ►Relations between three solutions of three-term recurrence relations are given by Masson (1991). …

20: Bibliography C

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Analytical solution of a model for complex food webs. Phys. Rev. E 65 (3), pp. (030901–1)–(030901–4).

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External Links:

Document

Referenced by:

§8.24(i).

Encodings:

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Stationary solutions of the one-dimensional nonlinear Schrödinger equation. I. Case of repulsive nonlinearity. Phys. Rev. A 62 (063610), pp. 1–10.

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External Links:

Document

Referenced by:

§22.19(iii).

Encodings:

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

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External Links:

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

Referenced by:

§16.6, §16.6.

Encodings:

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Nonclassical Symmetry Reductions and Exact Solutions for Physically Significant Nonlinear Evolution Equations. In Nonlinear and Chaotic Phenomena in Plasmas, Solids and Fluids (Edmonton, AB, 1990), W. Rozmus and J. A. Tuszynski (Eds.), pp. 72–79.

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External Links:

MathReview

Referenced by:

§29.19(ii).

Encodings:

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Special polynomials associated with rational solutions of the fifth Painlevé equation. J. Comput. Appl. Math. 178 (1-2), pp. 111–129.

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External Links:

ISSN 0377-0427, Document, MathReview (Davide Batic), ZentralBlatt

Referenced by:

§32.8(v), §32.9(ii).

Encodings:

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