incomplete beta functions

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11: Bibliography T

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Asymptotic inversion of the incomplete beta function. J. Comput. Appl. Math. 41 (1-2), pp. 145–157.

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Notes:

Asymptotic methods in analysis and combinatorics

External Links:

ISSN 0377-0427, Document, MathReview (Dumitru Acu), ZentralBlatt

Referenced by:

§8.18(ii).

Encodings:

BibTeX

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12: 3.10 Continued Fractions

… ►For special functions see §5.10 (gamma function), §7.9 (error function), §8.9 (incomplete gamma functions), §8.17(v) (incomplete beta function), §8.19(vii) (generalized exponential integral), §§10.10 and 10.33 (quotients of Bessel functions), §13.6 (quotients of confluent hypergeometric functions), §13.19 (quotients of Whittaker functions), and §15.7 (quotients of hypergeometric functions). …

13: Bibliography G

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Algorithm 222: Incomplete beta function ratios. Comm. ACM 7 (3), pp. 143–144.

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Notes:

Variable-precision Algol program. For a Certification see same journal 7 (1964), p. 244.

External Links:

ISSN 0001-0782, Document, Certification

Referenced by:

§8.28(iv).

Encodings:

BibTeX

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14: Bibliography K

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Comment on “The incomplete beta function law for parallel tempering sampling of classical canonical systems” [J. Chem. Phys. 120, 4119 (2004)]. J. Chem. Phys. 121 (2), pp. 1167.

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External Links:

Document

Referenced by:

§8.24(ii).

Encodings:

BibTeX

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15: 10.46 Generalized and Incomplete Bessel Functions; Mittag-Leffler Function

§10.46 Generalized and Incomplete Bessel Functions; Mittag-Leffler Function

►The function $\varphi (\rho ,\beta ;z)$ is defined by … ►The Laplace transform of $\varphi (\rho ,\beta ;z)$ can be expressed in terms of the Mittag-Leffler function: … ►For incomplete modified Bessel functions and Hankel functions, including applications, see Cicchetti and Faraone (2004).

16: 8.10 Inequalities

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8.10.11 $${(1-{\mathrm{e}}^{-{\alpha }_{a}x})}^a\le P(a,x)\le {(1-{\mathrm{e}}^{-{\beta }_{a}x})}^a,$$ $x\ge 0$, $a>0$,

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Symbols:

$\mathrm{e}$: base of natural logarithm, $P(a,z)$: normalized incomplete gamma function, $x$: real variable, $a$: parameter, ${\alpha }_a$ and ${\beta }_a$

Referenced by:

(7.8.8), §8.10

Permalink:

http://dlmf.nist.gov/8.10.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §8.10, §8.10 and Ch.8

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17: 19.5 Maclaurin and Related Expansions

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19.5.4_1 $$F(\varphi ,k)=\sum _{m=0}^{\mathrm{\infty }}\frac{{\left(\frac{1}{2}\right)}_m{\sin }^{2m+1}\varphi }{(2m+1)m!}{}_2{}^{}F_1^{}(\genfrac{}{}{0pt}{}{m+\frac{1}{2},\frac{1}{2}}{m+\frac{3}{2}};{\sin }^2\varphi )k^{2m}=\sin \varphi F_1(\frac{1}{2};\frac{1}{2},\frac{1}{2};\frac{3}{2};{\sin }^2\varphi ,k^2{\sin }^2\varphi ),$$

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Symbols:

$F_1(\alpha ;\beta ,{\beta }^{\prime };\gamma ;x,y)$: first Appell function, ${}_2{}^{}F_1^{}(a,b;c;z)$: $=F(a,b;c;z)$ notation for Gauss’ hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $F(\varphi ,k)$: Legendre’s incomplete elliptic integral of the first kind, $!$: factorial (as in $n!$), $\sin z$: sine function, $m$: nonnegative integer, $\varphi$: real or complex argument and $k$: real or complex modulus

Source:

Carlson (1961b, (2.8))

Referenced by:

§19.5, Erratum (V1.1.2) for Additions

Permalink:

http://dlmf.nist.gov/19.5.E4_1

Encodings:

pMML, png, TeX

Addition (effective with 1.1.2):

This equation was added.

See also:

Annotations for §19.5 and Ch.19

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19.5.4_2 $$E(\varphi ,k)=\sum _{m=0}^{\mathrm{\infty }}\frac{{\left(-\frac{1}{2}\right)}_m{\sin }^{2m+1}\varphi }{(2m+1)m!}{}_2{}^{}F_1^{}(\genfrac{}{}{0pt}{}{m+\frac{1}{2},\frac{1}{2}}{m+\frac{3}{2}};{\sin }^2\varphi )k^{2m}=\sin \varphi F_1(\frac{1}{2};\frac{1}{2},-\frac{1}{2};\frac{3}{2};{\sin }^2\varphi ,k^2{\sin }^2\varphi ),$$

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Symbols:

$F_1(\alpha ;\beta ,{\beta }^{\prime };\gamma ;x,y)$: first Appell function, ${}_2{}^{}F_1^{}(a,b;c;z)$: $=F(a,b;c;z)$ notation for Gauss’ hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $E(\varphi ,k)$: Legendre’s incomplete elliptic integral of the second kind, $!$: factorial (as in $n!$), $\sin z$: sine function, $m$: nonnegative integer, $\varphi$: real or complex argument and $k$: real or complex modulus

Source:

Carlson (1961b, (2.9))

Referenced by:

§19.5, Erratum (V1.1.2) for Additions

Permalink:

http://dlmf.nist.gov/19.5.E4_2

Encodings:

pMML, png, TeX

Addition (effective with 1.1.2):

This equation was added.

See also:

Annotations for §19.5 and Ch.19

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19.5.4_3 $$\mathrm{\Pi }(\varphi ,{\alpha }^2,k)=\sum _{m=0}^{\mathrm{\infty }}\frac{{\left(\frac{1}{2}\right)}_m{\sin }^{2m+1}\varphi }{(2m+1)m!}{F}_1(m+\frac{1}{2};\frac{1}{2},1;m+\frac{3}{2};{\sin }^2\varphi ,{\alpha }^2{\sin }^2\varphi )k^{2m},$$

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Symbols:

$F_1(\alpha ;\beta ,{\beta }^{\prime };\gamma ;x,y)$: first Appell function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\mathrm{\Pi }(\varphi ,{\alpha }^2,k)$: Legendre’s incomplete elliptic integral of the third kind, $!$: factorial (as in $n!$), $\sin z$: sine function, $m$: nonnegative integer, $\varphi$: real or complex argument, $k$: real or complex modulus and ${\alpha }^2$: real or complex parameter

Source:

Carlson (1961b, (2.11))

Referenced by:

§19.5, Erratum (V1.1.2) for Additions

Permalink:

http://dlmf.nist.gov/19.5.E4_3

Encodings:

pMML, png, TeX

Addition (effective with 1.1.2):

This equation was added.

See also:

Annotations for §19.5 and Ch.19

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18: 19.7 Connection Formulas

… ►Provided the functions in these identities are correctly analytically continued in the complex $\beta$-plane, then the identities will also hold in the complex $\beta$-plane. … ►There are three relations connecting $\mathrm{\Pi }(\varphi ,{\alpha }^2,k)$ and $\mathrm{\Pi }(\varphi ,{\omega }^2,k)$, where ${\omega }^2$ is a rational function of ${\alpha }^2$. … ►

19.7.8 $$\mathrm{\Pi }(\varphi ,{\alpha }^2,k)+\mathrm{\Pi }(\varphi ,{\omega }^2,k)=F(\varphi ,k)+\sqrt{c}{R}_C((c-1)(c-k^2),(c-{\alpha }^2)(c-{\omega }^2)),$$ ${\alpha }^2{\omega }^2=k^2$.

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Defines:

${\omega }^2$: change of variable (locally)

Symbols:

$R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions, $F(\varphi ,k)$: Legendre’s incomplete elliptic integral of the first kind, $\mathrm{\Pi }(\varphi ,{\alpha }^2,k)$: Legendre’s incomplete elliptic integral of the third kind, $\varphi$: real or complex argument, $k$: real or complex modulus, ${\alpha }^2$: real or complex parameter and $c$: change of variable

Referenced by:

§19.25(i), §19.6(i)

Permalink:

http://dlmf.nist.gov/19.7.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §19.7(iii), §19.7 and Ch.19

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19.7.9 $$(k^2-{\alpha }^2)\mathrm{\Pi }(\varphi ,{\alpha }^2,k)+(k^2-{\omega }^2)\mathrm{\Pi }(\varphi ,{\omega }^2,k)=k^{2}F(\varphi ,k)-{\alpha }^2{\omega }^2\sqrt{c-1}{R}_C(c(c-k^2),(c-{\alpha }^2)(c-{\omega }^2)),$$ $(1-{\alpha }^2)(1-{\omega }^2)=1-k^2$.

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Defines:

${\omega }^2$: change of variable (locally)

Symbols:

$R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions, $F(\varphi ,k)$: Legendre’s incomplete elliptic integral of the first kind, $\mathrm{\Pi }(\varphi ,{\alpha }^2,k)$: Legendre’s incomplete elliptic integral of the third kind, $\varphi$: real or complex argument, $k$: real or complex modulus, ${\alpha }^2$: real or complex parameter and $c$: change of variable

Permalink:

http://dlmf.nist.gov/19.7.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §19.7(iii), §19.7 and Ch.19

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19.7.10 $$(1-{\alpha }^2)\mathrm{\Pi }(\varphi ,{\alpha }^2,k)+(1-{\omega }^2)\mathrm{\Pi }(\varphi ,{\omega }^2,k)=F(\varphi ,k)+(1-{\alpha }^2-{\omega }^2)\sqrt{c-k^2}{R}_C(c(c-1),(c-{\alpha }^2)(c-{\omega }^2)),$$ $(k^2-{\alpha }^2)(k^2-{\omega }^2)=k^2(k^2-1)$.

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Defines:

${\omega }^2$: change of variable (locally)

Symbols:

$R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions, $F(\varphi ,k)$: Legendre’s incomplete elliptic integral of the first kind, $\mathrm{\Pi }(\varphi ,{\alpha }^2,k)$: Legendre’s incomplete elliptic integral of the third kind, $\varphi$: real or complex argument, $k$: real or complex modulus, ${\alpha }^2$: real or complex parameter and $c$: change of variable

Permalink:

http://dlmf.nist.gov/19.7.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §19.7(iii), §19.7 and Ch.19

19: Bibliography M

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Hypergeometric Functions.

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Notes:

Unpublished Lecture Notes, University of London. An online version exists.

External Links:

FullText

Referenced by:

§35.7(ii), §35.8(iii).

Encodings:

BibTeX

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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

External Links:

FullText, Code

Referenced by:

4th item.

Encodings:

BibTeX

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

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Referenced by:

§25.21(ii).

Encodings:

BibTeX

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Uniform computation of the error function and other related functions. Math. Comp. 25 (114), pp. 339–344.

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External Links:

FullText, MathReview, ZentralBlatt

Referenced by:

§7.22(i).

Encodings:

BibTeX

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Algorithm AS 187. Derivatives of the incomplete gamma integral. Appl. Statist. 31 (3), pp. 330–335.

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Notes:

Single-precision Fortran.

External Links:

FullText, Code, Document, ZentralBlatt

Referenced by:

4th item.

Encodings:

BibTeX

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20: Software Index

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	Open Source													With Book						Commercial
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8 Incomplete Gamma and Related 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. …