relation%20to%20error%20functions

SearchAdvancedHelp

(0.010 seconds)

1—10 of 16 matching pages

1: Bibliography N

… ►

Tables Relating to Mathieu Functions: Characteristic Values, Coefficients, and Joining Factors. 2nd edition, National Bureau of Standards Applied Mathematics Series, U.S. Government Printing Office, Washington, D.C..

ⓘ

External Links:

MathReview, ZentralBlatt

Referenced by:

§28.1, §28.26(i), 6th item, Notations S, Notations S.

Encodings:

BibTeX

… ►

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

ⓘ

External Links:

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

Referenced by:

§8.15.

Encodings:

BibTeX

… ►

Error bounds and exponential improvement for Hermite’s asymptotic expansion for the gamma function. Appl. Anal. Discrete Math. 7 (1), pp. 161–179.

ⓘ

External Links:

ISSN 1452-8630, Document, FullText, MathReview (Junesang Choi), ZentralBlatt

Referenced by:

§5.11(ii), §5.11(ii).

Encodings:

BibTeX

… ►

Error bounds and exponential improvement for the asymptotic expansion of the Barnes $G$-function. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 470 (2172), pp. 20140534, 14.

ⓘ

External Links:

ISSN 1364-5021, MathReview (Mehdi Hassani), ZentralBlatt

Referenced by:

§5.17, §5.17.

Encodings:

BibTeX

… ►

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

ⓘ

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).

Encodings:

BibTeX

…

2: Bibliography G

… ►

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

ⓘ

Notes:

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

External Links:

ISSN 0001-0782, Document

Referenced by:

§7.25(iii).

Encodings:

BibTeX

… ►

Some elementary inequalities relating to the gamma and incomplete gamma function. J. Math. Phys. 38 (1), pp. 77–81.

ⓘ

External Links:

MathReview (H. O. Pollak), ZentralBlatt

Referenced by:

§5.6(i), §6.8, §7.8, §8.10.

Encodings:

BibTeX

… ►

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

ⓘ

External Links:

Document, MathReview (P. Barrucand), ZentralBlatt

Referenced by:

§7.22(i), §7.25(iii).

Encodings:

BibTeX

… ►

Questions of Numerical Condition Related to Polynomials. In Studies in Numerical Analysis, G. H. Golub (Ed.), pp. 140–177.

ⓘ

External Links:

MathReview Entry, ZentralBlatt

Referenced by:

§3.8(vi).

Encodings:

BibTeX

… ►

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

ⓘ

Notes:

Includes Fortran program claiming 12S accuracy

External Links:

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

Referenced by:

3rd item, §10.77(ix).

Encodings:

BibTeX

…

3: Bibliography V

… ►

An integral transform involving Heun functions and a related eigenvalue problem. SIAM J. Math. Anal. 17 (3), pp. 688–703.

ⓘ

External Links:

ISSN 0036-1410, Document, MathReview (William L. Perry), ZentralBlatt

Referenced by:

§31.10(i).

Encodings:

BibTeX

►

Airy Functions and Applications to Physics. Second edition, Imperial College Press, London.

ⓘ

External Links:

ISBN 978-1-84816-548-9; 1-84816-548-X, MathReview Entry, ZentralBlatt

Referenced by:

§1.17(ii), §9.10(ii), §9.10(ix), §9.11(v), §9.16, §9.16, §9.5(i).

Encodings:

BibTeX

… ►

Calculation of Special Functions: The Gamma Function, the Exponential Integrals and Error-Like Functions. CWI Tract, Vol. 10, Stichting Mathematisch Centrum, Centrum voor Wiskunde en Informatica, Amsterdam.

ⓘ

Notes:

A reprint was published in 1986.

External Links:

ISBN 90-6196-277-3, MathReview (John P. Coleman), ZentralBlatt

Referenced by:

§5.21, §6.18(iv), §7.22(v), §7.6(i), §7.7(i), §7.7(ii).

Encodings:

BibTeX

… ►

Integral relations for Lamé functions. SIAM J. Math. Anal. 13 (6), pp. 978–987.

ⓘ

External Links:

ISSN 0036-1410, Document, MathReview (F. M. Arscott), ZentralBlatt

Referenced by:

§29.8, §29.8.

Encodings:

BibTeX

… ►

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

ⓘ

External Links:

ISSN 0176-4276, Document, MathReview, ZentralBlatt

Referenced by:

§30.16(i), §30.16(ii), §30.16(ii).

Encodings:

BibTeX

…

4: Bibliography F

… ►

ⓘ

Notes:

The “Freely Distributable LIBM” package provides implementations of standard elementary functions plus a few higher functions, e.g. gamma. Double precision, maximum accuracy 20S. Developed by Sun Microsystems.

External Links:

GAMS (package FDLIBM)

Referenced by:

§ ‣ Software Cross Index, § ‣ Software Cross Index.

Encodings:

BibTeX

… ►

Complex zeros of the error function and of the complementary error function. Math. Comp. 27 (122), pp. 401–407.

ⓘ

External Links:

FullText, MathReview, ZentralBlatt

Referenced by:

§7.13(i), §7.13(ii), 1st item.

Encodings:

BibTeX

… ►

Numerical calculation of singular integrals related to Hankel transform. Comput. Math. Appl. 21 (2-3), pp. 87–94.

ⓘ

Notes:

Fortran, minimum precision 6D.

External Links:

ISSN 0898-1221, Document, MathReview, ZentralBlatt

Referenced by:

§10.77(ix).

Encodings:

BibTeX

… ►

Error bounds for asymptotic expansions of the ratio of two gamma functions. SIAM J. Math. Anal. 18 (3), pp. 890–896.

ⓘ

External Links:

ISSN 0036-1410, Document, FullText, MathReview (P. Anandani), ZentralBlatt

Referenced by:

§5.11(iii), §5.11(iii).

Encodings:

BibTeX

… ►

Error bounds for the asymptotic expansion of the ratio of two gamma functions with complex argument. SIAM J. Math. Anal. 23 (2), pp. 505–511.

ⓘ

External Links:

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

Referenced by:

§5.11(iii), §5.11(iii).

Encodings:

BibTeX

…

5: 28.8 Asymptotic Expansions for Large $q$

… ►For recurrence relations for the coefficients in these expansions see Frenkel and Portugal (2001, §3). For error estimates see Kurz (1979), and for graphical interpretation see Figure 28.2.1. … ►For recurrence relations for the coefficients in these expansions see Frenkel and Portugal (2001, §4 and §5). … ►They are derived by rigorous analysis and accompanied by strict and realistic error bounds. … ►For related results see Langer (1934) and Sharples (1967, 1971). …

6: Bibliography S

… ►

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

ⓘ

External Links:

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

Referenced by:

§22.19(i), §22.20(vi).

Encodings:

BibTeX

… ►

Chebyshev expansions for the error and related functions. Math. Comp. 32 (144), pp. 1232–1240.

ⓘ

External Links:

FullText, Document, MathReview (L. Fox), ZentralBlatt

Referenced by:

3rd item.

Encodings:

BibTeX

… ►

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

ⓘ

External Links:

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

Referenced by:

§28.8(iv).

Encodings:

BibTeX

… ►

Automatic computing methods for special functions. IV. Complex error function, Fresnel integrals, and other related functions. J. Res. Nat. Bur. Standards 86 (6), pp. 661–686.

ⓘ

Notes:

Includes Fortran code, maximum accuracy 18S.

External Links:

ISSN 0022-4340, MathReview, ZentralBlatt

Referenced by:

§7.25(iii).

Encodings:

BibTeX

… ►

On the calculation of the inverse of the error function. Math. Comp. 22 (101), pp. 144–158.

ⓘ

External Links:

FullText, MathReview, ZentralBlatt

Referenced by:

§7.17(ii).

Encodings:

BibTeX

…

7: Bibliography

… ►

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

ⓘ

External Links:

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.

Encodings:

BibTeX

… ►

Application of the combined nonlinear-condensation transformation to problems in statistical analysis and theoretical physics. Comput. Phys. Comm. 150 (1), pp. 1–20.

ⓘ

Notes:

Includes algorithms for Lerch’s transcendent. C and Mathematica codes by Aksenov and Jentschura are available.

External Links:

Code, Document

Referenced by:

1st item.

Encodings:

BibTeX

… ►

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

ⓘ

External Links:

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

Referenced by:

§10.43(iii).

Encodings:

BibTeX

… ►

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

ⓘ

External Links:

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.

Encodings:

BibTeX

… ►

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

ⓘ

External Links:

Document, MathReview (I. Marx), ZentralBlatt

Referenced by:

§29.8.

Encodings:

BibTeX

…

8: Bibliography B

… ►

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

ⓘ

External Links:

Document, ZentralBlatt

Referenced by:

§10.43(iii), 2nd item.

Encodings:

BibTeX

… ►

Tables Relating to the Radial Mathieu Functions. Vol. 1: Functions of the First Kind. U.S. Government Printing Office, Washington, D.C..

ⓘ

External Links:

MathReview (C. J. Bouwkamp)

Referenced by:

1st item.

Encodings:

BibTeX

►

Tables Relating to the Radial Mathieu Functions. Vol. 2: Functions of the Second Kind. U.S. Government Printing Office, Washington, D.C..

ⓘ

External Links:

MathReview (C. J. Bouwkamp)

Referenced by:

2nd item, 1st item.

Encodings:

BibTeX

… ►

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.

ⓘ

External Links:

MathReview (S. C. van Veen), ZentralBlatt

Referenced by:

§35.5(i).

Encodings:

BibTeX

… ►

Concerning the zeros of some functions related to Bessel functions. J. Mathematical Phys. 10 (9), pp. 1729–1744.

ⓘ

External Links:

Document, MathReview (C. V. Coffman), ZentralBlatt

Referenced by:

§10.21(xi).

Encodings:

BibTeX

…

9: Bibliography D

… ►

Computational properties of three-term recurrence relations for Kummer functions. J. Comput. Appl. Math. 233 (6), pp. 1505–1510.

ⓘ

External Links:

ISSN 0377-0427, Document, FullText, MathReview (Richard B. Paris), ZentralBlatt

Referenced by:

§13.29(iv), §13.29(iv).

Encodings:

BibTeX

… ►

Complex zeros of cylinder functions. Math. Comp. 20 (94), pp. 215–222.

ⓘ

Notes:

For a numerical error in one of the zeros see Amos (1985).

External Links:

ISSN 0025-5718, FullText, MathReview, ZentralBlatt

Referenced by:

2nd item.

Encodings:

BibTeX

… ►

Uniform asymptotic expansions for Whittaker’s confluent hypergeometric functions. SIAM J. Math. Anal. 20 (3), pp. 744–760.

ⓘ

Notes:

(This reference has several typographical errors.)

External Links:

ISSN 0036-1410, Document, MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§13.21(ii), §13.21(iii), §13.21(iv).

Encodings:

BibTeX

… ►

Asymptotic approximations for the Jacobi and ultraspherical polynomials, and related functions. Methods Appl. Anal. 6 (3), pp. 21–56.

ⓘ

External Links:

ISSN 1073-2772, Pdf, MathReview (Walter Van Assche), ZentralBlatt

Referenced by:

§15.12(iii), §18.15(i), §18.15(vi).

Encodings:

BibTeX

… ►

Olver’s error bound methods applied to linear ordinary differential equations having a simple turning point. Anal. Appl. (Singap.) 12 (4), pp. 385–402.

ⓘ

External Links:

ISSN 0219-5305, Document, Link, MathReview (Thomas Mbah Acho), ZentralBlatt

Referenced by:

5th item, §14.32.

Encodings:

BibTeX

…

10: Software Index

… ►‘✓’ 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. … ►

These are collections of software (e.g. libraries) or interactive systems of a somewhat broad scope. Contents may be adapted from research software or may be contributed by project participants who donate their services to the project. The software is made freely available to the public, typically in source code form. While formal support of the collection may not be provided by its developers, within active projects there is often a core group who donate time to consider bug reports and make updates to the collection.

… ►

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.

… ►

A cross index of mathematical software in use at NIST.