Hurwitz%20zeta%20function

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1: 25.11 Hurwitz Zeta Function

§25.11 Hurwitz Zeta Function

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§25.11(i) Definition

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§25.11(ii) Graphics

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§25.11(vi) Derivatives

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§25.11(ix) Integrals

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2: Bibliography N

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On an integral transform involving a class of Mathieu functions. SIAM J. Math. Anal. 20 (6), pp. 1500–1513.

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External Links:

ISSN 0036-1410, Document, MathReview (K. C. Gupta), ZentralBlatt

Referenced by:

§28.26(ii).

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Reduction and evaluation of elliptic integrals. Math. Comp. 20 (94), pp. 223–231.

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External Links:

FullText, MathReview (I. A. Stegun), ZentralBlatt

Referenced by:

§19.37(iv).

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BibTeX

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Error bounds for the asymptotic expansion of the Hurwitz zeta function. Proc. A. 473 (2203), pp. 20170363, 16.

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External Links:

ISSN 1364-5021, Document, Link, MathReview Entry, ZentralBlatt

Referenced by:

§8.15.

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A table of integrals of the error functions. J. Res. Nat. Bur. Standards Sect B. 73B, pp. 1–20.

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Notes:

See for additions and corrections same journal, Sect. B 75, 149–163 (1975)

External Links:

MathReview, ZentralBlatt

Referenced by:

§7.14(iii), §7.21, §7.7(iii).

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Hypergeometric functions. Acta Math. 94, pp. 289–349.

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External Links:

Document, MathReview (A. Erdélyi), ZentralBlatt

Referenced by:

§16.10, §16.8(ii), §16.8(ii).

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3: Software Index

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	Open Source													With Book						Commercial
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20 Theta Functions
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►‘✓’ indicates that a software package implements the functions in a section; ‘a’ indicates available functionality through optional or add-on packages; an empty space indicates no known support. … ►In the list below we identify four main sources of software for computing special functions. … ►

Such software ranges from a collection of reusable software parts (e.g., a library) to fully functional interactive computing environments with an associated computing language. Such software is usually professionally developed, tested, and maintained to high standards. It is available for purchase, often with accompanying updates and consulting support.

… ►The following are web-based software repositories with significant holdings in the area of special functions. …

4: 25.12 Polylogarithms

… ►The notation ${\mathrm{Li}}_2\left(z\right)$ was introduced in Lewin (1981) for a function discussed in Euler (1768) and called the dilogarithm in Hill (1828): … ►The special case $z=1$ is the Riemann zeta function: $\zeta \left(s\right)={\mathrm{Li}}_s\left(1\right)$. … ►Further properties include …and … ►In terms of polylogarithms …

5: Bibliography K

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The Riemann Zeta-Function. de Gruyter Expositions in Mathematics, Vol. 5, Walter de Gruyter & Co., Berlin.

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Notes:

Translated from the Russian by Neal Koblitz

External Links:

ISBN 3-11-013170-6, MathReview (Matti Jutila), ZentralBlatt

Referenced by:

Ch.25.

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

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External Links:

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

Referenced by:

1st item.

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

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External Links:

Document, MathReview (Matti Jutila), ZentralBlatt

Referenced by:

§25.18(i).

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Complex zeros of an incomplete Riemann zeta function and of the incomplete gamma function. Math. Comp. 24 (111), pp. 679–696.

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External Links:

FullText, MathReview, ZentralBlatt

Referenced by:

§8.13(iii), §8.22(ii).

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Complex zeros of two incomplete Riemann zeta functions. Math. Comp. 26 (118), pp. 551–565.

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External Links:

FullText, MathReview, ZentralBlatt

Referenced by:

§8.22(ii).

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6: Bibliography C

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The Riemann zeta-function and its derivatives. Proc. Roy. Soc. London Ser. A 450, pp. 477–499.

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External Links:

ISSN 0962-8444, FullText, MathReview, ZentralBlatt

Referenced by:

§25.18(i).

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Chebyshev approximations for the Riemann zeta function. Math. Comp. 25 (115), pp. 537–547.

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External Links:

ISSN 0025-5718, FullText, MathReview, ZentralBlatt

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1st item.

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On some series representations of the Hurwitz zeta function. J. Comput. Appl. Math. 216 (1), pp. 297–305.

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External Links:

ISSN 0377-0427, Document, FullText, MathReview, ZentralBlatt

Referenced by:

§25.11(iv), §25.11(iv).

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An efficient algorithm for the Hurwitz zeta and related functions. J. Comput. Appl. Math. 225 (2), pp. 338–346.

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External Links:

ISSN 0377-0427, Document, FullText, MathReview, ZentralBlatt

Referenced by:

§25.18(i), §25.18(i).

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Validated computation of certain hypergeometric functions. ACM Trans. Math. Software 38 (2), pp. Art. 11, 20.

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External Links:

ISSN 0098-3500, Document, FullText, MathReview Entry, ZentralBlatt

Referenced by:

§13.29(v), §15.19(v).

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7: Bibliography B

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A program for computing the Riemann zeta function for complex argument. Comput. Phys. Comm. 20 (3), pp. 441–445.

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Notes:

Double-precision Fortran, maximum accuracy 10S.

External Links:

Document, Code, ZentralBlatt

Referenced by:

1st item.

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Coulomb functions (negative energies). Comput. Phys. Comm. 20 (3), pp. 447–458.

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Notes:

Double-precision Fortran code.

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Document, Code

Referenced by:

1st item, 4th item.

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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).

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A short table of the functions ${\mathrm{Ki}}_n(x)$, from $n=1$ to $n=16$ . Phil. Mag. Series 7 20, pp. 343–347.

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External Links:

Document, ZentralBlatt

Referenced by:

§10.43(iii), 2nd item.

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Bessel functions and modular relations of higher type and hyperbolic differential equations. Comm. Sém. Math. Univ. Lund [Medd. Lunds Univ. Mat. Sem.] 1952 (Tome Supplementaire), pp. 12–20.

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External Links:

MathReview (S. C. van Veen), ZentralBlatt

Referenced by:

§35.5(i).

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8: Bibliography P

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Tables of Digamma and Trigamma Functions. In Tracts for Computers, No. 1, K. Pearson (Ed.),

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External Links:

ZentralBlatt

Referenced by:

§5.1, Notations F.

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An asymptotic representation for $\zeta (\frac{1}{2}+it)$ . Methods Appl. Anal. 4 (4), pp. 449–470.

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External Links:

ISSN 1073-2772, MathReview (Matti Jutila), ZentralBlatt

Referenced by:

§25.9, §8.22(ii).

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The Stokes phenomenon associated with the Hurwitz zeta function $\zeta (s,a)$ . Proc. Roy. Soc. London Ser. A 461, pp. 297–304.

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External Links:

ISSN 1364-5021, Document, MathReview (Marc Prevost), ZentralBlatt

Referenced by:

(25.11.43), §25.11(xii).

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Automatic computation of Bessel function integrals. Comput. Phys. Comm. 25 (3), pp. 289–295.

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Notes:

Double-precision Fortran, maximum accuracy 20S.

External Links:

ISSN 0010-4655, Document, Code

Referenced by:

6th item.

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Chebyshev polynomial expansions of the Riemann zeta function. Math. Comp. 26 (120), pp. G1–G5.

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External Links:

ISSN 0025-5718, FullText, MathReview

Referenced by:

2nd item, 3rd item.

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9: Bibliography M

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Rational approximations, software and test methods for sine and cosine integrals. Numer. Algorithms 12 (3-4), pp. 259–272.

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Notes:

Includes Fortran code, maximum accuracy 20S.

External Links:

ISSN 1017-1398, Document, MathReview, ZentralBlatt

Referenced by:

§6.13, 4th item, §6.21(ii).

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Contribution no. 14. The Riemann zeta function. BIT 5, pp. 138–141.

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Referenced by:

§25.21(ii).

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Calculation of the modified Bessel functions of the second kind with complex argument. Math. Comp. 20 (95), pp. 407–412.

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External Links:

ISSN 0025-5718, FullText, MathReview (H. E. Fettis), ZentralBlatt

Referenced by:

§10.74(iii).

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Derivatives of the Hurwitz zeta function for rational arguments. J. Comput. Appl. Math. 100 (2), pp. 201–206.

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External Links:

ISSN 0377-0427, Document, MathReview, ZentralBlatt

Referenced by:

(25.11.21), (25.11.22), (25.11.23), §25.11, §25.11(vi), §25.18(i), (25.6.15).

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

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External Links:

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

Referenced by:

§18.27(v).

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BibTeX

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