relation to error functions

SearchAdvancedHelp

(0.015 seconds)

11—20 of 66 matching pages

11: Notices

… ►Authors of the works appearing in the Digital Library of Mathematical Functions (DLMF) have assigned copyright to the works to NIST, United States Department of Commerce, as represented by the Secretary of Commerce. … ►As a condition of using the DLMF, you explicitly release NIST from any and all liabilities for any damage of any type that may result from errors or omissions in the DLMF. … ►If you feel you have found an error in DLMF, please see Possible Errors in DLMF. … ►The DLMF wishes to provide users of special functions with essential reference information related to the use and application of special functions in research, development, and education. … ►

Within each of the DLMF chapters themselves we will provide a list of research software for the functions discussed in that chapter. The purpose of these listings is to provide references to the research literature on the engineering of software for special functions. To qualify for listing, the development of the software must have been the subject of a research paper published in the peer-reviewed literature. If such software is available online for free download we will provide a link to the software.

In general, we will not index other software within DLMF chapters unless the software is unique in some way, such as being the only known software for computing a particular function.

…

12: 14 Legendre and Related Functions

Chapter 14 Legendre and Related Functions

…

13: 27.11 Asymptotic Formulas: Partial Sums

… ►The behavior of a number-theoretic function $f(n)$ for large $n$ is often difficult to determine because the function values can fluctuate considerably as $n$ increases. …where $F(x)$ is a known function of $x$, and $O\left(g(x)\right)$ represents the error, a function of smaller order than $F(x)$ for all $x$ in some prescribed range. …Dirichlet’s divisor problem (unsolved as of 2022) is to determine the least number ${\theta }_0$ such that the error term in (27.11.2) is $O\left(x^{\theta }\right)$ for all $\theta >{\theta }_0$. … ►Equations (27.11.3)–(27.11.11) list further asymptotic formulas related to some of the functions listed in §27.2. …The error terms given here are not necessarily the best known. …

14: 13.3 Recurrence Relations and Derivatives

§13.3 Recurrence Relations and Derivatives

►

§13.3(i) Recurrence Relations

… ►Kummer’s differential equation (13.2.1) is equivalent to … ►

§13.3(ii) Differentiation Formulas

… ►

13.3.22 $$\frac{d}{dz}U(a,b,z)=-aU(a+1,b+1,z),$$

ⓘ

Symbols:

$U(a,b,z)$: Kummer confluent hypergeometric function, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ and $z$: complex variable

A&S Ref:

13.4.21

Permalink:

http://dlmf.nist.gov/13.3.E22

Encodings:

pMML, png, TeX

See also:

Annotations for §13.3(ii), §13.3 and Ch.13

…

15: 11.10 Anger–Weber Functions

§11.10 Anger–Weber Functions

… ►

§11.10(vi) Relations to Other Functions

… ► … ►

§11.10(viii) Expansions in Series of Products of Bessel Functions

… ►

§11.10(ix) Recurrence Relations and Derivatives

…

16: 14.3 Definitions and Hypergeometric Representations

§14.3 Definitions and Hypergeometric Representations

►

§14.3(i) Interval

… ►

§14.3(ii) Interval

… ►

§14.3(iii) Alternative Hypergeometric Representations

… ►

§14.3(iv) Relations to Other Functions

…

17: 11 Struve and Related Functions

Chapter 11 Struve and Related Functions

…

18: 7.17 Inverse Error Functions

§7.17 Inverse Error Functions

►

§7.17(i) Notation

►The inverses of the functions $x=\mathrm{erf}y$, $x=\mathrm{erfc}y$, $y\in ℝ$, are denoted by … ►

§7.17(ii) Power Series

… ►

§7.17(iii) Asymptotic Expansion of $\mathrm{inverfc}x$ for Small $x$

…

19: 13.8 Asymptotic Approximations for Large Parameters

… ►For the parabolic cylinder function $U$ see §12.2, and for an extension to an asymptotic expansion see Temme (1978). … ►For an extension to an asymptotic expansion complete with error bounds see Temme (1990b), and for related results see §13.21(i). … ►uniformly with respect to bounded positive values of $x$ in each case. … ►These results follow from Temme (2022), which can also be used to obtain more terms in the expansions. For generalizations in which $z$ is also allowed to be large see Temme and Veling (2022).

20: 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

►

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

ⓘ

External Links:

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

Referenced by:

§7.22(iii).

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

…