error-control function

SearchAdvancedHelp

(0.002 seconds)

41—50 of 957 matching pages

41: 35.8 Generalized Hypergeometric Functions of Matrix Argument

§35.8 Generalized Hypergeometric Functions of Matrix Argument

►

§35.8(i) Definition

… ►

Convergence Properties

… ►

§35.8(ii) Relations to Other Functions

… ►

Confluence

…

42: 7.18 Repeated Integrals of the Complementary Error Function

§7.18 Repeated Integrals of the Complementary Error Function

… ►

§7.18(iv) Relations to Other Functions

… ►

Confluent Hypergeometric Functions

… ►

Parabolic Cylinder Functions

… ►

Probability Functions

…

43: 7.2 Definitions

… ►

§7.2(i) Error Functions

… ► $\mathrm{erf}z$, $\mathrm{erfc}z$, and $w\left(z\right)$ are entire functions of $z$, as is $F\left(z\right)$ in the next subsection. ►

Values at Infinity

… ► $ℱ\left(z\right)$, $C\left(z\right)$, and $S\left(z\right)$ are entire functions of $z$, as are $\mathrm{f}\left(z\right)$ and $\mathrm{g}\left(z\right)$ in the next subsection. … ►

§7.2(iv) Auxiliary Functions

…

44: 28.20 Definitions and Basic Properties

… ►

§28.20(ii) Solutions ${\mathrm{Ce}}_{\nu }$, ${\mathrm{Se}}_{\nu }$, ${\mathrm{Me}}_{\nu }$, ${\mathrm{Fe}}_n$, ${\mathrm{Ge}}_n$

… ►

§28.20(iv) Radial Mathieu Functions ${\mathrm{Mc}}_n^{(j)}$, ${\mathrm{Ms}}_n^{(j)}$

… ►

§28.20(vi) Wronskians

… ►

§28.20(vii) Shift of Variable

… ►And for the corresponding identities for the radial functions use (28.20.15) and (28.20.16).

45: 19.16 Definitions

… ►

§19.16(ii) $R_{-a}(𝐛;𝐳)$

… ►The $R$-function is often used to make a unified statement of a property of several elliptic integrals. …where $\mathrm{B}(x,y)$ is the beta function (§5.12) and … ►For generalizations and further information, especially representation of the $R$-function as a Dirichlet average, see Carlson (1977b). …

46: 28.2 Definitions and Basic Properties

… ►Since (28.2.1) has no finite singularities its solutions are entire functions of $z$. … ►

§28.2(vi) Eigenfunctions

… ► … ► ►The functions are orthogonal, that is, …

47: 2.7 Differential Equations

… ►such that ►

2.7.23 $$|{ϵ}_j(x)|,\frac{1}{2}{f}^{-1/2}(x)|{ϵ}_j^{\prime }(x)|\le \exp \left(\frac{1}{2}{𝒱}_{a_j,x}\left(F\right)\right)-1,$$ $j=1,2$,

ⓘ

Symbols:

$\exp z$: exponential function, ${𝒱}_{a,b}\left(f\right)$: total variation, $(a_1,a_2)$: interval, ${ϵ}_j(x)$: function and $F$: error-control function

Permalink:

http://dlmf.nist.gov/2.7.E23

Encodings:

pMML, png, TeX

See also:

Annotations for §2.7(iii), §2.7(iii), §2.7 and Ch.2

… ►Here $F(x)$ is the error-control function ►

2.7.24 $$F(x)=\int \left(\frac{1}{f^{1/4}}\frac{d^2}{{dx}^2}\left(\frac{1}{f^{1/4}}\right)-\frac{g}{f^{1/2}}\right)dx,$$

ⓘ

Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $dx$: differential of $x$ , $\int$: integral and $F$: error-control function

Permalink:

http://dlmf.nist.gov/2.7.E24

Encodings:

pMML, png, TeX

See also:

Annotations for §2.7(iii), §2.7(iii), §2.7 and Ch.2

… ►

2.7.25 $${𝒱}_{a_j,x}\left(F\right)=\left|{\int }_{a_j}^x\left|\frac{1}{f^{1/4}(t)}\frac{d^2}{{dt}^2}\left(\frac{1}{f^{1/4}(t)}\right)-\frac{g(t)}{f^{1/2}(t)}\right|dt\right|.$$

ⓘ

Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $dx$: differential of $x$ , $\int$: integral, ${𝒱}_{a,b}\left(f\right)$: total variation, $(a_1,a_2)$: interval and $F$: error-control function

Referenced by:

Erratum (V1.1.2) for Equation (2.7.25)

Permalink:

http://dlmf.nist.gov/2.7.E25

Encodings:

pMML, png, TeX

Correction (effective with 1.1.2):

The integrand was corrected so that the absolute value does not include the differential. Also an absolute value was introduced on the right-hand side to ensure a non-negative value.

See also:

Annotations for §2.7(iii), §2.7(iii), §2.7 and Ch.2

…

48: 4.13 Lambert $W$-Function

§4.13 Lambert $W$-Function

►The Lambert $W$-function $W\left(z\right)$ is the solution of the equation … ►and has several advantages over the Lambert $W$-function (see Lawrence et al. (2012)), and the tree $T$-function $T\left(z\right)=-W\left(-z\right)$, which is a solution of … ►Properties include: … ►For these and other integral representations of the Lambert $W$-function see Kheyfits (2004), Kalugin et al. (2012) and Mező (2020). …

49: 22.16 Related Functions

§22.16 Related Functions

►

§22.16(i) Jacobi’s Amplitude ($\mathrm{am}$) Function

… ►

§22.16(ii) Jacobi’s Epsilon Function

… ►

Relation to Theta Functions

… ►

§22.16(iii) Jacobi’s Zeta Function

…

50: Bibliography K

… ►

On the evaluation of the Gauss hypergeometric function. C. R. Acad. Bulgare Sci. 45 (6), pp. 35–36.

ⓘ

External Links:

ISSN 0861-1459, MathReview, ZentralBlatt

Referenced by:

§15.15, §15.19(i).

Encodings:

BibTeX

… ►

Generalized functions. Mathematics in Science and Engineering, Vol. 171, Academic Press, Inc., Orlando, FL.

ⓘ

Notes:

Theory and technique

External Links:

ISBN 0-12-396560-8, MathReview (Qing Yi Chen), ZentralBlatt

Referenced by:

§1.16(viii).

Encodings:

BibTeX

… ►

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

ⓘ

External Links:

FullText, MathReview, ZentralBlatt

Referenced by:

§8.13(iii), §8.22(ii).

Encodings:

BibTeX

… ►

Programs for computing the logarithm of the gamma function, and the digamma function, for complex argument. Comput. Phys. Comm. 4, pp. 221–226.

ⓘ

Notes:

Single-precision Fortran, accuracy 12-14S

External Links:

Document, Code

Referenced by:

1st item.

Encodings:

BibTeX

… ►

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

ⓘ

External Links:

ISSN 0001-0782, Document

Referenced by:

§5.24(iv).

Encodings:

BibTeX

…