incomplete%20Riemann%20zeta%20function

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1: 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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Double-precision Fortran, maximum accuracy 10S.

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Document, Code, ZentralBlatt

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1st item.

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

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Double-precision Fortran code.

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Document, Code

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1st item, 4th item.

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The Riemann-Siegel expansion for the zeta function: High orders and remainders. Proc. Roy. Soc. London Ser. A 450, pp. 439–462.

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ISSN 0962-8444, FullText, MathReview (Christian Grosche), ZentralBlatt

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§25.10(ii).

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Computational strategies for the Riemann zeta function. J. Comput. Appl. Math. 121 (1-2), pp. 247–296.

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ISSN 0377-0427, Document, MathReview (Cem Y. Yildirim), ZentralBlatt

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§27.18.

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

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§25.18(i).

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2: Bibliography P

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An asymptotic formula for the incomplete gamma function. Ž. Vyčisl. Mat. i Mat. Fiz. 5, pp. 118–121 (Russian).

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English translation in U.S.S.R. Computational Math. and Math. Phys. 5(1), pp. 162–166

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Document, MathReview (E. Lukacs), ZentralBlatt

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§8.11(iv).

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Error bounds for the uniform asymptotic expansion of the incomplete gamma function. J. Comput. Appl. Math. 147 (1), pp. 215–231.

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ISSN 0377-0427, Document, MathReview, ZentralBlatt

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§8.12.

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Tables of the Incomplete Beta-function. 2nd edition, Published for the Biometrika Trustees at the Cambridge University Press, Cambridge.

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MathReview, ZentralBlatt

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1st item.

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

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Double-precision Fortran, maximum accuracy 20S.

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ISSN 0010-4655, Document, Code

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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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ISSN 0025-5718, FullText, MathReview

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2nd item, 3rd item.

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3: 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

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

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Ch.25.

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Asymptotic expansions of certain $q$-series and a formula of Ramanujan for specific values of the Riemann zeta function. Acta Arith. 107 (3), pp. 269–298.

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ISSN 0065-1036, Document, Link, MathReview (Bruce C. Berndt), ZentralBlatt

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§17.2(i).

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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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Document, MathReview (Matti Jutila), ZentralBlatt

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§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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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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FullText, MathReview, ZentralBlatt

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§8.22(ii).

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4: 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. …

5: Bibliography

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Theory of Incomplete Cylindrical Functions and Their Applications. Springer-Verlag, Berlin.

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Notes:

Translated from the Russian by H. E. Fettis, J. W. Goresh and D. A. Lee, Die Grundlehren der mathematischen Wissenschaften, Band 160

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MathReview, ZentralBlatt

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1st item.

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Numerical Calculation of the Riemann Zeta Function and Generalizations by Means of the Trapezoidal Rule. In Numerical and Applied Mathematics, Part II (Paris, 1988), C. Brezinski (Ed.), IMACS Ann. Comput. Appl. Math., Vol. 1, pp. 467–472.

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MathReview

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§25.18(i).

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On some inequalities for the incomplete gamma function. Math. Comp. 66 (218), pp. 771–778.

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ISSN 0025-5718, Document, MathReview, ZentralBlatt

Referenced by:

§8.10.

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Dirichlet series related to the Riemann zeta function. J. Number Theory 19 (1), pp. 85–102.

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ISSN 0022-314X, Document, MathReview (Kai Man Tsang), ZentralBlatt

Referenced by:

(25.16.10), (25.16.11), (25.16.12), (25.16.14), (25.16.15), (25.16.5), (25.16.6), (25.16.7), (25.16.8), (25.16.9), §25.16(ii), §25.16(ii), §25.16.

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Formulas for higher derivatives of the Riemann zeta function. Math. Comp. 44 (169), pp. 223–232.

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ISSN 0025-5718, FullText, MathReview (Aleksandar Ivić), ZentralBlatt

Referenced by:

(25.11.19), (25.11.20), (25.11.6), §25.11(vi), §25.18(i), (25.4.5), (25.4.6), §25.4, (25.6.11), (25.6.12), (25.6.13), (25.6.14), §25.6(ii), §25.6.

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6: Bibliography C

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Generalized incomplete gamma functions with applications. J. Comput. Appl. Math. 55 (1), pp. 99–124.

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ISSN 0377-0427, Document, MathReview, ZentralBlatt

Referenced by:

§8.16.

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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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ISSN 0962-8444, FullText, MathReview, ZentralBlatt

Referenced by:

§25.18(i).

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Incomplete Hankel and modified Bessel functions: A class of special functions for electromagnetics. IEEE Trans. Antennas and Propagation 52 (12), pp. 3373–3389.

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ISSN 0018-926X, Document, MathReview (T. Erber), ZentralBlatt

Referenced by:

§10.46.

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

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ISSN 0025-5718, FullText, MathReview, ZentralBlatt

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1st item.

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Zeros of the Hankel function of real order out of the principal Riemann sheet. J. Comput. Appl. Math. 37 (1-3), pp. 89–99.

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ISSN 0377-0427, Document, MathReview (Boro Döring), ZentralBlatt

Referenced by:

§10.21(ix).

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7: 5.11 Asymptotic Expansions

… ►Wrench (1968) gives exact values of $g_k$ up to $g_{20}$. … ►where $\zeta \left(K\right)$ is as in Chapter 25. In the case $K=1$ the factor $1+\zeta \left(K\right)$ is replaced with 4. … ►For re-expansions of the remainder terms in (5.11.1) and (5.11.3) in series of incomplete gamma functions with exponential improvement (§2.11(iii)) in the asymptotic expansions, see Berry (1991), Boyd (1994), and Paris and Kaminski (2001, §6.4). … ►

8: Bibliography M

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Algorithm AS 63. The incomplete beta integral. Appl. Statist. 22 (3), pp. 409–411.

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Notes:

Double-precision Fortran. For remark see same journal 26 (1977), pp. 111-114

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4th item.

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

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

Referenced by:

§18.27(v).

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9: Bibliography V

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On the zeros of the Riemann zeta function in the critical strip. IV. Math. Comp. 46 (174), pp. 667–681.

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ISSN 0025-5718, FullText, Document, MathReview (Jürgen G. Hinz), ZentralBlatt

Referenced by:

§25.10(ii), §25.18(ii).

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On the series expansion method for computing incomplete elliptic integrals of the first and second kinds. Math. Comp. 23 (105), pp. 61–69.

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FullText, MathReview, ZentralBlatt

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§19.12, §19.5.

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A new estimate of the function $\zeta (1+it)$ . Izv. Akad. Nauk SSSR. Ser. Mat. 22, pp. 161–164 (Russian).

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MathReview (H. Davenport), ZentralBlatt

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§27.12.

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Error estimates for Rayleigh-Ritz approximations of eigenvalues and eigenfunctions of the Mathieu and spheroidal wave equation. Constr. Approx. 20 (1), pp. 39–54.

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ISSN 0176-4276, Document, MathReview, ZentralBlatt

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§30.16(i), §30.16(ii), §30.16(ii).

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Über die Riemann’sche Primzahlfunction. Math. Ann. 55, pp. 441–464 (German).

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ISSN 0025-5831; 1432-1807/e, Document, MathReview Entry, ZentralBlatt

Referenced by:

§27.12.

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10: Bibliography S

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Transformations of the Jacobian amplitude function and its calculation via the arithmetic-geometric mean. SIAM J. Math. Anal. 20 (6), pp. 1514–1528.

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ISSN 0036-1410, Document, MathReview (J. M. H. Peters), ZentralBlatt

Referenced by:

§22.19(i), §22.20(vi).

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Uniform asymptotic forms of modified Mathieu functions. Quart. J. Mech. Appl. Math. 20 (3), pp. 365–380.

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Document, MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§28.8(iv).

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Some identities involving the Riemann zeta function. II. Indian J. Pure Appl. Math. 17 (10), pp. 1175–1186.

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ISSN 0019-5588, MathReview (Jean-Marie De Koninck), ZentralBlatt

Referenced by:

§24.14(ii).

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Sums of certain series of the Riemann zeta function. J. Math. Anal. Appl. 134 (1), pp. 129–140.

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ISSN 0022-247X, Document, MathReview (Aleksandar Ivić), ZentralBlatt

Referenced by:

(25.8.3), §25.8.

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A Maple package for symmetric functions. J. Symbolic Comput. 20 (5-6), pp. 755–768.

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Notes:

For software see John Stembridge’s home page. The SF package contains Maple software for zonal, Jack, and Macdonald polynomials.

External Links:

Home Page, ISSN 0747-7171, Document, MathReview, ZentralBlatt

Referenced by:

3rd item.

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