representation via Hurwitz system

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1: 25.11 Hurwitz Zeta Function

§25.11 Hurwitz Zeta Function

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§25.11(iii) Representations by the Euler–Maclaurin Formula

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§25.11(iv) Series Representations

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§25.11(vii) Integral Representations

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§25.11(x) Further Series Representations

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2: 21.10 Methods of Computation

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Belokolos et al. (1994, Chapter 5) and references therein. Here the Riemann surface is represented by the action of a Schottky group on a region of the complex plane. The same representation is used in Gianni et al. (1998).

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Tretkoff and Tretkoff (1984). Here a Hurwitz system is chosen to represent the Riemann surface.

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Deconinck and van Hoeij (2001). Here a plane algebraic curve representation of the Riemann surface is used.

3: 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

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Addition (effective with 1.1.3):

This equation was added.

See also:

Annotations for §8.15 and Ch.8

►For the Hurwitz zeta function $\zeta (s,a)$ see §25.11(i). …

4: 18.38 Mathematical Applications

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Quadrature

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Quadrature “Extended” to Pseudo-Spectral (DVR) Representations of Operators in One and Many Dimensions

►The basic ideas of Gaussian quadrature, and their extensions to non-classical weight functions, and the computation of the corresponding quadrature abscissas and weights, have led to discrete variable representations, or DVRs, of Sturm–Liouville and other differential operators. … ►

Group Representations

… ►Algebraic structures were built of which special representations involve Dunkl type operators. …

5: Wolter Groenevelt

… ►Groenevelt’s research interests is in special functions and orthogonal polynomials and their relations with representation theory and interacting particle systems. …

6: Bibliography H

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Lie groups, Lie algebras, and representations. Second edition, Graduate Texts in Mathematics, Vol. 222, Springer, Cham.

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Notes:

An elementary introduction

External Links:

ISBN 978-3-319-13466-6; 978-3-319-13467-3, Document, Link, MathReview Entry

Referenced by:

§1.2(vi).

Encodings:

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Analytic evaluation of two-center STO electron repulsion integrals via ellipsoidal expansion. Internat. J. Quantum Chem. 88 (6), pp. 701–734.

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External Links:

Document

Referenced by:

§8.24(iii).

Encodings:

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Numerical Tools for the Study of Finite Gap Solutions of Integrable Systems. Ph.D. Thesis, Technischen Universität Berlin.

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Notes:

(All relevant material, plus corrections, are included in Deconinck et al. (2004).)

External Links:

ZentralBlatt

Referenced by:

§21.5(i).

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Sums of powers of integers via generating functions. Fibonacci Quart. 34 (3), pp. 244–256.

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External Links:

ISSN 0015-0517, MathReview (Noriaki Kimura), ZentralBlatt

Referenced by:

§24.4(iii).

Encodings:

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Bernoulli numbers and polynomials via residues. J. Number Theory 76 (2), pp. 178–193.

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External Links:

ISSN 0022-314X, Document, MathReview (L. Skula), ZentralBlatt

Referenced by:

§24.14(ii).

Encodings:

BibTeX

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7: 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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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:

BibTeX

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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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Comment on “The incomplete beta function law for parallel tempering sampling of classical canonical systems” [J. Chem. Phys. 120, 4119 (2004)]. J. Chem. Phys. 121 (2), pp. 1167.

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External Links:

Document

Referenced by:

§8.24(ii).

Encodings:

BibTeX

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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:

BibTeX

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8: 9.17 Methods of Computation

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§9.17(iii) Integral Representations

►Among the integral representations of the Airy functions the Stieltjes transform (9.10.18) furnishes a way of computing $\mathrm{Ai}\left(z\right)$ in the complex plane, once values of this function can be generated on the positive real axis. … ►

§9.17(iv) Via Bessel Functions

… ►Zeros of the Airy functions, and their derivatives, can be computed to high precision via Newton’s rule (§3.8(ii)) or Halley’s rule (§3.8(v)), using values supplied by the asymptotic expansions of §9.9(iv) as initial approximations. …

9: Bibliography B

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Integral representations and summations of the modified Struve function. Acta Math. Hungar. 141 (3), pp. 254–281.

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External Links:

ISSN 0236-5294, Document, FullText, MathReview (Eugenia N. Petropoulou), ZentralBlatt

Referenced by:

§11.7(v), §11.7(v).

Encodings:

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On the Hurwitz zeta-function. Rocky Mountain J. Math. 2 (1), pp. 151–157.

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External Links:

MathReview (K. Thanigasalam), ZentralBlatt

Referenced by:

(25.11.26).

Encodings:

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A new asymptotic representation for $\zeta (\frac{1}{2}+it)$ and quantum spectral determinants. Proc. Roy. Soc. London Ser. A 437, pp. 151–173.

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External Links:

ISSN 0962-8444, MathReview (Jack Quine), ZentralBlatt

Referenced by:

§25.9.

Encodings:

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From Klein to Painlevé via Fourier, Laplace and Jimbo. Proc. London Math. Soc. (3) 90 (1), pp. 167–208.

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External Links:

ISSN 0024-6115, Document, MathReview (Andrei A. Kapaev), ZentralBlatt

Referenced by:

§32.9(iii).

Encodings:

BibTeX

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Riemann zeta function: Rapidly converging series and integral representations. Appl. Math. Lett. 5 (2), pp. 83–88.

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External Links:

ISSN 0893-9659, Document, MathReview (Jacek Fabrykowski), ZentralBlatt

Referenced by:

§25.18(i).

Encodings:

BibTeX

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10: Bibliography R

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A non-negative representation of the linearization coefficients of the product of Jacobi polynomials. Canad. J. Math. 33 (4), pp. 915–928.

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External Links:

ISSN 0008-414X, Document, Link, MathReview (R. A. Askey), ZentralBlatt

Referenced by:

§18.18(v).

Encodings:

BibTeX

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Notes:

Symbolic and extended-precision general purpose computer algebra system.

External Links:

Home Page

Referenced by:

§ ‣ Software Cross Index.

Encodings:

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Integral representations for products of Airy functions. Z. Angew. Math. Phys. 46 (2), pp. 159–170.

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External Links:

ISSN 0044-2275, Document, MathReview (P. K. Banerji), ZentralBlatt

Referenced by:

§9.11(iii), §9.11(v), §9.11(v), (9.5.6), §9.5(ii).

Encodings:

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Integral representations for products of Airy functions. II. Cubic products. Z. Angew. Math. Phys. 48 (4), pp. 646–655.

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External Links:

ISSN 0044-2275, Document, MathReview (E. R. Love), ZentralBlatt

Referenced by:

(9.11.16), (9.11.17), §9.11(iii), §9.11(v).

Encodings:

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Integral representations for products of Airy functions. III. Quartic products. Z. Angew. Math. Phys. 48 (4), pp. 656–664.

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External Links:

ISSN 0044-2275, Document, MathReview (E. R. Love), ZentralBlatt

Referenced by:

§9.11(iii), §9.11(v).

Encodings:

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