representation via Schottky group

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11: Bibliography W

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Asymptotics of orthogonal polynomials via recurrence relations. Anal. Appl. (Singap.) 10 (2), pp. 215–235.

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External Links:

ISSN 0219-5305, Document, Link, MathReview (Li Zhong Peng), ZentralBlatt

Referenced by:

§2.9(iii), §2.9(iii).

Encodings:

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Uniform asymptotic expansion of $J_{\nu }(\nu a)$ via a difference equation. Numer. Math. 91 (1), pp. 147–193.

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External Links:

ISSN 0029-599X, Document, MathReview (J. P. Singhal), ZentralBlatt

Referenced by:

§10.20(ii).

Encodings:

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Uniform asymptotics of the Stieltjes-Wigert polynomials via the Riemann-Hilbert approach. J. Math. Pures Appl. (9) 85 (5), pp. 698–718.

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External Links:

ISSN 0021-7824, Document, MathReview (Rabah Khaldi), ZentralBlatt

Referenced by:

§18.29.

Encodings:

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Asymptotic Approximations to Truncation Errors of Series Representations for Special Functions. In Algorithms for Approximation, A. Iske and J. Levesley (Eds.), pp. 331–348.

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External Links:

ISBN 3-540-33283-9, Document, MathReview Entry, ZentralBlatt

Referenced by:

§2.10(i).

Encodings:

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Group Theory and its Application to the Quantum Mechanics of Atomic Spectra. Pure and Applied Physics. Vol. 5, Academic Press, New York.

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Notes:

Expanded and improved ed. Translated from the German by J. J. Griffin.

External Links:

ISBN 0-12-750550-4, MathReview (A. S. Wightman), ZentralBlatt

Referenced by:

§34.12.

Encodings:

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12: 26.19 Mathematical Applications

§26.19 Mathematical Applications

… ►Partitions and plane partitions have applications to representation theory (Bressoud (1999), Macdonald (1995), and Sagan (2001)) and to special functions (Andrews et al. (1999) and Gasper and Rahman (2004)). …

13: 15.19 Methods of Computation

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§15.19(iii) Integral Representations

►The representation (15.6.1) can be used to compute the hypergeometric function in the sector . … ►Initial values for moderate values of $|a|$ and $|b|$ can be obtained by the methods of §15.19(i), and for large values of $|a|$, $|b|$, or $|c|$ via the asymptotic expansions of §§15.12(ii) and 15.12(iii). …

14: 28.36 Software

… ►Citations in bulleted lists refer to papers for which research software has been made available and can be downloaded via the Web. … ►See also Clemm (1969), Delft Numerical Analysis Group (1973), Rengarajan and Lewis (1980), and Schäfke and Schmidt (1966). … ►See also Clemm (1969), Delft Numerical Analysis Group (1973), Rengarajan and Lewis (1980), Van Buren and Boisvert (2007), and Ziener et al. (2012).

15: 26.22 Software

… ►Citations in the bulleted list refer to sources from which research software can be found and downloaded via the Web. … ►

RISC Combinatorics Group (website).

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16: Bibliography K

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Series expansions for the third incomplete elliptic integral via partial fraction decompositions. J. Comput. Appl. Math. 207 (2), pp. 331–337.

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External Links:

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

Referenced by:

§19.12, §19.5.

Encodings:

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Quantum Groups. Graduate Texts in Mathematics, Vol. 155, Springer-Verlag, New York.

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External Links:

ISBN 0-387-94370-6, MathReview (Yu. N. Bespalov), ZentralBlatt

Referenced by:

§17.17.

Encodings:

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Closed-form representations of the Lambert $W$ function. Fract. Calc. Appl. Anal. 7 (2), pp. 177–190.

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External Links:

ISSN 1311-0454, MathReview (Lewis H. Ryder), ZentralBlatt

Referenced by:

§4.13.

Encodings:

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Jacobi Functions and Analysis on Noncompact Semisimple Lie Groups. In Special Functions: Group Theoretical Aspects and Applications, pp. 1–85.

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Notes:

Reprinted by Kluwer Academic Publishers, Boston, 2002.

External Links:

ISBN 90-277-1822-9; 1-4020-0319-6, MathReview (Jacques Faraut), ZentralBlatt

Referenced by:

§14.31(ii), §15.17(iii), §15.9(ii).

Encodings:

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Compact quantum groups and $q$-special functions. In Representations of Lie Groups and Quantum Groups, Pitman Res. Notes Math. Ser., Vol. 311, pp. 46–128.

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Notes:

Sections 3 and 4 also available as arXiv:math/9403216

External Links:

MathReview (Eric M. Opdam)

Referenced by:

(18.27.14_3).

Encodings:

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17: 8.15 Sums

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8.15.2 $$a\sum _{k=1}^{\mathrm{\infty }}\left(\frac{{\mathrm{e}}^{2\pi \mathrm{i}k(z+h)}}{{\left(2\pi \mathrm{i}k\right)}^{a+1}}\mathrm{\Gamma }(a,2\pi \mathrm{i}kz)+\frac{{\mathrm{e}}^{-2\pi \mathrm{i}k(z+h)}}{{\left(-2\pi \mathrm{i}k\right)}^{a+1}}\mathrm{\Gamma }(a,-2\pi \mathrm{i}kz)\right)=\zeta (-a,z+h)+\frac{z^{a+1}}{a+1}+\left(h-\frac{1}{2}\right)z^a,$$ $h\in [0,1]$.

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Symbols:

$\zeta (s,a)$: Hurwitz zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $[a,b]$: closed interval, $\in$: element of, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\mathrm{\Gamma }(a,z)$: incomplete gamma function, $z$: complex variable, $a$: parameter and $k$: nonnegative integer

Proof sketch:

Start with and $z>0$. For $\mathrm{\Gamma }(a,\pm 2\pi \mathrm{i}kz)$ take integral representation (8.2.2) and use the substitution $t=\pm 2\pi \mathrm{i}kz(1+\tau )$. The sum and the integral can be interchanged, and the sum can be evaluated via (4.6.1). Use integration by parts. This will result in $\left(h-\frac{1}{2}\right)z^a$ plus two integrals with infinitely many poles. The residue theorem (§1.10(iv)) will give us an infinite series which can be identified via (25.11.1). For other values of $a$ and $z$ use analytic continuation.

Referenced by:

§25.11(x), §25.11(x), §8.15, Erratum (V1.1.3) for Additions

Permalink:

http://dlmf.nist.gov/8.15.E2

Encodings:

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Addition (effective with 1.1.3):

This equation was added.

See also:

Annotations for §8.15 and Ch.8

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18: 9.11 Products

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§9.11(iii) Integral Representations

… ►For an integral representation of the Dirac delta involving a product of two $\mathrm{Ai}$ functions see §1.17(ii). ►For further integral representations see Reid (1995, 1997a, 1997b). … ►

9.11.19 $${\int }_0^{\mathrm{\infty }}\frac{dt}{{\mathrm{Ai}}^2\left(t\right)+{\mathrm{Bi}}^2\left(t\right)}={\int }_0^{\mathrm{\infty }}\frac{tdt}{\mathrm{Ai}_{}^{\prime }{}_{}{}^2\left(t\right)+\mathrm{Bi}_{}^{\prime }{}_{}{}^2\left(t\right)}=\frac{{\pi }^2}{6}.$$

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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 $x$ , $\int$: integral, $M\left(z\right)$: Airy modulus function, $N\left(z\right)$: Airy modulus function, $\varphi \left(z\right)$: Airy phase function, $\theta \left(z\right)$: Airy phase function and $x$: real variable

Proof sketch:

Extend the definitions of §9.8(i) to positive values of $x$, obtain the indefinite integrals of $1/M^2\left(x\right)$ and $x/N^2\left(x\right)$ via the first two of (9.8.14), then combine the values of $\theta \left(0\right)$ and $\varphi \left(0\right)$ given in §9.8(i) with $\theta \left(+\mathrm{\infty }\right)=\varphi \left(+\mathrm{\infty }\right)=0$ obtained from (9.8.4), (9.8.8), and §9.7(ii). (Communicated by M.E. Muldoon.)

Permalink:

http://dlmf.nist.gov/9.11.E19

Encodings:

pMML, png, TeX

See also:

Annotations for §9.11(v), §9.11 and Ch.9

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19: 5.9 Integral Representations

§5.9 Integral Representations

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Binet’s Formula

… ►For additional representations see Whittaker and Watson (1927, §§12.31–12.32). … ► …

20: 10.64 Integral Representations

§10.64 Integral Representations

… ►See Apelblat (1991) for these results, and also for similar representations for ${\mathrm{ber}}_{\nu }\left(x\sqrt{2}\right)$, ${\mathrm{bei}}_{\nu }\left(x\sqrt{2}\right)$, and their $\nu$-derivatives. …