error%20and%20related%20functions

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11: Bibliography K

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

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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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Document, MathReview (Matti Jutila), ZentralBlatt

Referenced by:

§25.18(i).

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Cyclic identities for Jacobi elliptic and related functions. J. Math. Phys. 44 (4), pp. 1822–1841.

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ISSN 0022-2488, Document, MathReview (Zhi Guo Liu), ZentralBlatt

Referenced by:

§22.9(iv), §22.9(v).

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Theta relations and projective normality of Abelian varieties. Amer. J. Math. 98 (4), pp. 865–889.

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FullText, MathReview (T. Oda), ZentralBlatt

Referenced by:

§21.6(ii).

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Algorithm 421. Complex gamma function with error control. Comm. ACM 15 (4), pp. 271–272.

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ISSN 0001-0782, Document

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§5.24(iv).

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12: Bibliography G

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Algorithm 363: Complex error function. Comm. ACM 12 (11), pp. 635.

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Notes:

Includes Algol program, maximum accuracy 10S. For certification see same journal v. 15 (1972), pp. 465–466

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ISSN 0001-0782, Document

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§7.25(iii).

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Recursive computation of the repeated integrals of the error function. Math. Comp. 15 (75), pp. 227–232.

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FullText, MathReview (J. C. P. Miller), ZentralBlatt

Referenced by:

§7.22(iii).

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Efficient computation of the complex error function. SIAM J. Numer. Anal. 7 (1), pp. 187–198.

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Document, MathReview (P. Barrucand), ZentralBlatt

Referenced by:

§7.22(i), §7.25(iii).

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A table of integrals of the error function. II. Additions and corrections. J. Res. Nat. Bur. Standards Sect. B 75B, pp. 149–163.

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Notes:

See for the first part same journal, Sect. B 73, 1-20 (1969)

External Links:

MathReview, ZentralBlatt

Referenced by:

§7.14(iii).

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Algorithm 939: computation of the Marcum Q-function. ACM Trans. Math. Softw. 40 (3), pp. 20:1–20:21.

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Notes:

Includes Fortran program claiming 12S accuracy

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

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3rd item, §10.77(ix).

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13: Bibliography

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Exact linearization of a Painlevé transcendent. Phys. Rev. Lett. 38 (20), pp. 1103–1106.

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Document, MathReview (Mark Adler)

Referenced by:

§32.11(ii), §32.13(i), §32.13(ii), §32.13(iii), §32.5.

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Some series of the zeta and related functions. Analysis (Munich) 18 (2), pp. 131–144.

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ISSN 0174-4747, MathReview (Jack Quine), ZentralBlatt

Referenced by:

(25.11.37), (25.11.38), (25.11.39), (25.11.40), §25.11, §25.11(xi), (25.8.1), (25.8.4), §25.8.

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Repeated integrals and derivatives of $K$ Bessel functions. SIAM J. Math. Anal. 20 (1), pp. 169–175.

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ISSN 0036-1410, Document, MathReview (A. Haimovici), ZentralBlatt

Referenced by:

§10.43(iii).

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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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Integral equations and relations for Lamé functions. Quart. J. Math. Oxford Ser. (2) 15, pp. 103–115.

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Document, MathReview (I. Marx), ZentralBlatt

Referenced by:

§29.8.

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14: Bibliography C

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The inverse of the error function. Pacific J. Math. 13 (2), pp. 459–470.

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MathReview (D. P. Banerjee), ZentralBlatt

Referenced by:

§7.17(ii).

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Table of integrals of squared Jacobian elliptic functions and reductions of related hypergeometric $R$-functions. Math. Comp. 75 (255), pp. 1309–1318.

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ISSN 0025-5718, Document, MathReview, ZentralBlatt

Referenced by:

§19.25(v), §19.29(i), §22.14(ii).

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Rational Chebyshev approximations for the error function. Math. Comp. 23 (107), pp. 631–637.

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FullText, MathReview, ZentralBlatt

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2nd item.

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Performance evaluation of programs related to the real gamma function. ACM Trans. Math. Software 17 (1), pp. 46–54.

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ISSN 0098-3500, Document, MathReview, ZentralBlatt

Referenced by:

§5.24(ii), §5.24(iii).

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

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

Referenced by:

§13.29(v), §15.19(v).

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15: 19.36 Methods of Computation

… ►where the elementary symmetric functions $E_s$ are defined by (19.19.4). … ►Legendre’s integrals can be computed from symmetric integrals by using the relations in §19.25(i). … ►If the iteration of (19.36.6) and (19.36.12) is stopped when ($M$ and $T$ being approximated by $a_s$ and $t_s$, and the infinite series being truncated), then the relative error in $R_F$ and $R_G$ is less than $r$ if we neglect terms of order $r^2$. … ►

§19.36(iii) Via Theta Functions

… ►For computation of Legendre’s integral of the third kind, see Abramowitz and Stegun (1964, §§17.7 and 17.8, Examples 15, 17, 19, and 20). …