Painlev%C3%A9%20transcendents

SearchAdvancedHelp

(0.002 seconds)

11—20 of 130 matching pages

11: 32.15 Orthogonal Polynomials

§32.15 Orthogonal Polynomials

… ►

12: 32.5 Integral Equations

§32.5 Integral Equations

…

13: 25.14 Lerch’s Transcendent

§25.14 Lerch’s Transcendent

►

§25.14(i) Definition

… ►The Hurwitz zeta function $\zeta (s,a)$ (§25.11) and the polylogarithm ${\mathrm{Li}}_s\left(z\right)$ (§25.12(ii)) are special cases: ►

25.14.2 $$\zeta (s,a)=\mathrm{\Phi }(1,s,a),$$ $\mathrm{\Re }s>1$, $a\ne 0,-1,-2,\mathrm{\dots }$,

ⓘ

Symbols:

$\zeta (s,a)$: Hurwitz zeta function, $\mathrm{\Phi }(z,s,a)$: Lerch’s transcendent, $\mathrm{\Re }$: real part, $a$: real or complex parameter and $s$: complex variable

Keywords:

specialization

Proof sketch:

Derivable from (25.11.1), (25.14.1).

Permalink:

http://dlmf.nist.gov/25.14.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §25.14(i), §25.14 and Ch.25

… ►

§25.14(ii) Properties

…

14: Alexander A. Its

… ► Matveev), published by Springer in 1994, and Painlevé Transcendents: The Riemann-Hilbert Approach (with A. …

15: Bibliography K

… ►

Replica field theories, Painlevé transcendents, and exact correlation functions. Phys. Rev. Lett. 89 (25), pp. (250201–1)–(250201–4).

ⓘ

External Links:

ISSN 0031-9007, Document, MathReview, ZentralBlatt

Referenced by:

§32.16.

Encodings:

BibTeX

►

Connection formulae for the first Painlevé transcendent in the complex domain. Lett. Math. Phys. 27 (4), pp. 243–252.

ⓘ

External Links:

ISSN 0377-9017, Document, MathReview (B. L. J. Braaksma), ZentralBlatt

Referenced by:

§32.11(i), §32.12(i).

Encodings:

BibTeX

… ►

Algorithm 737: INTLIB: A portable Fortran 77 interval standard-function library. ACM Trans. Math. Software 20 (4), pp. 447–459.

ⓘ

External Links:

ISSN 0098-3500, Document, GAMS (module 737; package TOMS), ZentralBlatt

Referenced by:

1st item.

Encodings:

BibTeX

… ►

Methods of computing the Riemann zeta-function and some generalizations of it. USSR Comput. Math. and Math. Phys. 20 (6), pp. 212–230.

ⓘ

External Links:

Document, MathReview (Matti Jutila), ZentralBlatt

Referenced by:

§25.18(i).

Encodings:

BibTeX

… ►

Elliptic asymptotics of the first and second Painlevé transcendents. Uspekhi Mat. Nauk 49 (1(295)), pp. 77–140 (Russian).

ⓘ

Notes:

In Russian. English translation: Russian Math. Surveys 49(1994), no. 1, pp. 81–150.

External Links:

ISSN 0042-1316, Document, MathReview (V. A. Golubeva), ZentralBlatt

Referenced by:

§32.11(i), §32.12(ii).

Encodings:

BibTeX

…

16: Bibliography F

… ►

ⓘ

Notes:

The “Freely Distributable LIBM” package provides implementations of standard elementary functions plus a few higher functions, e.g. gamma. Double precision, maximum accuracy 20S. Developed by Sun Microsystems.

External Links:

GAMS (package FDLIBM)

Referenced by:

§ ‣ Software Cross Index, § ‣ Software Cross Index.

Encodings:

BibTeX

… ►

Sur certaines sommes des intégral-cosinus. Bull. Soc. Math. Phys. Serbie 12, pp. 13–20 (French).

ⓘ

External Links:

MathReview (L. Carlitz), ZentralBlatt

Referenced by:

§6.15.

Encodings:

BibTeX

… ►

Continuous and Discrete Painlevé Equations. In Painlevé Transcendents: Their Asymptotics and Physical Applications, D. Levi and P. Winternitz (Eds.), NATO Adv. Sci. Inst. Ser. B Phys., Vol. 278, pp. 33–47.

ⓘ

Notes:

Proc. NATO Adv. Res. Workshop, Sainte-Adèle, Canada, 1990

External Links:

MathReview (F. Pempinelli), ZentralBlatt

Referenced by:

§32.15.

Encodings:

BibTeX

… ►

Painlevé Transcendents: The Riemann-Hilbert Approach. Mathematical Surveys and Monographs, Vol. 128, American Mathematical Society, Providence, RI.

ⓘ

External Links:

ISBN 0-8218-3651-X, MathReview Entry, ZentralBlatt

Referenced by:

§32.11(ii), §32.11(iii), §32.11(iv), §32.12(ii), §32.12(iii), §32.4(i), Profile Alexander A. Its.

Encodings:

BibTeX

… ►

On weighted polynomial approximation on the whole real axis. Acta Math. Acad. Sci. Hungar. 20, pp. 223–225.

ⓘ

External Links:

ISSN 0001-5954, Document, Link, MathReview (A. K. Varma)

Referenced by:

§18.32.

Encodings:

BibTeX

…

17: Bibliography M

… ►

Connection between quantum systems involving the fourth Painlevé transcendent and $k$-step rational extensions of the harmonic oscillator related to Hermite exceptional orthogonal polynomial. J. Math. Phys. 57 (5), pp. Paper 052101, 15 pp..

ⓘ

External Links:

ISSN 0022-2488, Document, Link, MathReview Entry

Referenced by:

§18.38(iii).

Encodings:

BibTeX

… ►

Spin Systems, Statistical Mechanics and Painlevé Functions. In Painlevé Transcendents: Their Asymptotics and Physical Applications, D. Levi and P. Winternitz (Eds.), NATO Adv. Sci. Inst. Ser. B Phys., Vol. 278, pp. 377–391.

ⓘ

Notes:

Proc. NATO Adv. Res. Workshop, Sainte-Adèle, Canada, 1990

External Links:

MathReview, ZentralBlatt

Referenced by:

§32.16.

Encodings:

BibTeX

… ►

On the second Painlevé transcendent. Proc. Roy. Soc. London Ser. A 361, pp. 277–291.

ⓘ

External Links:

ISSN 0080-4630, FullText, MathReview (Mark J. Ablowitz), ZentralBlatt

Referenced by:

§32.11(ii), §32.17, §32.3(ii).

Encodings:

BibTeX

… ►

The Second Painlevé Transcendent: A Nonlinear Airy Function. In Mechanics Today, Vol. 5, pp. 297–313.

ⓘ

External Links:

MathReview (H. E. Fettis), ZentralBlatt

Referenced by:

§32.11(ii), §32.17, §32.3(ii).

Encodings:

BibTeX

… ►

The $q$-analogue of the Laguerre polynomials. J. Math. Anal. Appl. 81 (1), pp. 20–47.

ⓘ

External Links:

ISSN 0022-247X, Document, MathReview (R. A. Askey), ZentralBlatt

Referenced by:

§18.27(v).

Encodings:

BibTeX

…

18: 8 Incomplete Gamma and Related
Functions

…

19: 28 Mathieu Functions and Hill’s Equation

…

20: Bibliography W

… ►

The Nahm equations, finite-gap potentials and Lamé functions. J. Phys. A 20 (10), pp. 2679–2683.

ⓘ

External Links:

ISSN 0305-4470, Document, MathReview (James M. D’Archangelo), ZentralBlatt

Referenced by:

§29.19(ii).

Encodings:

BibTeX

… ►

On the connection formulas of the fourth Painlevé transcendent. Anal. Appl. (Singap.) 7 (4), pp. 419–448.

ⓘ

External Links:

ISSN 0219-5305, Document, MathReview, ZentralBlatt

Referenced by:

§32.11(v), §32.11(v), §32.13(ii).

Encodings:

BibTeX

►

On the connection formulas of the third Painlevé transcendent. Discrete Contin. Dyn. Syst. 23 (1-2), pp. 541–560.

ⓘ

External Links:

ISSN 1078-0947, Document, MathReview (Galina V. Filipuk), ZentralBlatt

Referenced by:

§32.13(ii).

Encodings:

BibTeX

…