incomplete beta functions

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21: 19.1 Special Notation

… ►(For other notation see Notation for the Special Functions.) … ►All derivatives are denoted by differentials, not by primes. ►The first set of main functions treated in this chapter are Legendre’s complete integrals …of the first, second, and third kinds, respectively, and Legendre’s incomplete integrals … ►The first three functions are incomplete integrals of the first, second, and third kinds, and the

\operatorname{cel}

function includes complete integrals of all three kinds.

22: Bibliography C

… ►

M. A. Chaudhry, N. M. Temme, and E. J. M. Veling (1996) Asymptotics and closed form of a generalized incomplete gamma function. J. Comput. Appl. Math. 67 (2), pp. 371–379.

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External Links:: ISSN 0377-0427, Document, MathReview, ZentralBlatt
Referenced by:: §8.16.
Encodings:: BibTeX

►

M. A. Chaudhry and S. M. Zubair (1994) Generalized incomplete gamma functions with applications. J. Comput. Appl. Math. 55 (1), pp. 99–124.

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External Links:: ISSN 0377-0427, Document, MathReview, ZentralBlatt
Referenced by:: §8.16.
Encodings:: BibTeX

►

M. A. Chaudhry and S. M. Zubair (2001) On a Class of Incomplete Gamma Functions with Applications. Chapman & Hall/CRC, Boca Raton, FL.

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External Links:: ISBN 1-58488-143-7, MathReview (R. K. Raina), ZentralBlatt
Referenced by:: §8.16.
Encodings:: BibTeX

… ►

R. Cicchetti and A. Faraone (2004) Incomplete Hankel and modified Bessel functions: A class of special functions for electromagnetics. IEEE Trans. Antennas and Propagation 52 (12), pp. 3373–3389.

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External Links:: ISSN 0018-926X, Document, MathReview (T. Erber), ZentralBlatt
Referenced by:: §10.46.
Encodings:: BibTeX

… ►

E. D. Constantinides and R. J. Marhefka (1993) Efficient and accurate computation of the incomplete Airy functions. Radio Science 28 (4), pp. 441–457.

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External Links:: ISSN 0048-6604, Document
Referenced by:: §9.14.
Encodings:: BibTeX

…

23: 19.16 Definitions

… ►

§19.16(ii) $R_{-a}\left(\mathbf{b};\mathbf{z}\right)$

… ►The

R

-function is often used to make a unified statement of a property of several elliptic integrals. …where

\mathrm{B}\left(x,y\right)

is the beta function (§5.12) and … ► … ►Each of the four complete integrals (19.16.20)–(19.16.23) can be integrated to recover the incomplete integral: …

24: 19.14 Reduction of General Elliptic Integrals

… ►

19.14.1

\int_{1}^{x}\frac{\,\mathrm{d}t}{\sqrt{t^{3}-1}}=3^{-1/4}F\left(\phi,k\right),

\cos\phi=\dfrac{\sqrt{3}+1-x}{\sqrt{3}-1+x}

k^{2}=\dfrac{2-\sqrt{3}}{4}

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Symbols:: $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $F\left(\NVar{\phi},\NVar{k}\right)$ : Legendre’s incomplete elliptic integral of the first kind, $\int$ : integral, $\phi$ : real or complex argument and $k$ : real or complex modulus
Referenced by:: §19.14(i)
Permalink:: http://dlmf.nist.gov/19.14.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §19.14(i), §19.14 and Ch.19

►

19.14.2

\int_{x}^{1}\frac{\,\mathrm{d}t}{\sqrt{1-t^{3}}}=3^{-1/4}F\left(\phi,k\right),

\cos\phi=\dfrac{\sqrt{3}-1+x}{\sqrt{3}+1-x}

k^{2}=\dfrac{2+\sqrt{3}}{4}

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Symbols:: $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $F\left(\NVar{\phi},\NVar{k}\right)$ : Legendre’s incomplete elliptic integral of the first kind, $\int$ : integral, $\phi$ : real or complex argument and $k$ : real or complex modulus
Permalink:: http://dlmf.nist.gov/19.14.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §19.14(i), §19.14 and Ch.19

… ►

19.14.4

{\int_{y}^{x}\frac{\,\mathrm{d}t}{\sqrt{(a_{1}+b_{1}t^{2})(a_{2}+b_{2}t^{2})}}% =\frac{1}{\sqrt{\gamma-\alpha}}F\left(\phi,k\right)},

k^{2}=\ifrac{(\gamma-\beta)}{(\gamma-\alpha)}

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Symbols:: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $F\left(\NVar{\phi},\NVar{k}\right)$ : Legendre’s incomplete elliptic integral of the first kind, $\int$ : integral, $\phi$ : real or complex argument and $k$ : real or complex modulus
Referenced by:: §19.14(i), §19.14(i), §19.14(i), §19.15, §19.29(iii)
Permalink:: http://dlmf.nist.gov/19.14.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §19.14(i), §19.14 and Ch.19

►In (19.14.4)

0\leq y<x

, each quadratic polynomial is positive on the interval

(y,x)

, and

\alpha,\beta,\gamma

is a permutation of

0,a_{1}b_{2},a_{2}b_{1}

(not all 0 by assumption) such that

\alpha\leq\beta\leq\gamma

. … ►Legendre (1825–1832) showed that every elliptic integral can be expressed in terms of the three integrals in (19.1.2) supplemented by algebraic, logarithmic, and trigonometric functions. …

25: Errata

… ►

Equation (18.12.2)

18.12.2

{{}_{0}{\mathbf{F}}_{1}}\left({-\atop\alpha+1};\frac{(x-1)z}{2}\right)\*{{}_{0% }{\mathbf{F}}_{1}}\left({-\atop\beta+1};\frac{(x+1)z}{2}\right)=\left(\tfrac{1% }{2}(1-x)z\right)^{-\frac{1}{2}\alpha}J_{\alpha}\left(\sqrt{2(1-x)z}\right)\*% \left(\tfrac{1}{2}(1+x)z\right)^{-\frac{1}{2}\beta}I_{\beta}\left(\sqrt{2(1+x)% z}\right)=\sum_{n=0}^{\infty}\frac{P^{(\alpha,\beta)}_{n}\left(x\right)}{% \Gamma\left(n+\alpha+1\right)\Gamma\left(n+\beta+1\right)}z^{n}

This equation was updated to include on the left-hand side, its definition in terms of a product of two ${{}_{0}{\mathbf{F}}_{1}}$ functions.

… ►

Equations (31.3.10), (31.3.11)

31.3.10

z^{-\alpha}\mathit{H\!\ell}\left(\frac{1}{a},\frac{q}{a}-\alpha\left(\beta-% \epsilon\right)-\frac{\alpha}{a}\left(\beta-\delta\right);\alpha,\alpha-\gamma% +1,\alpha-\beta+1,\delta;\frac{1}{z}\right)

31.3.11

z^{-\beta}\mathit{H\!\ell}\left(\frac{1}{a},\frac{q}{a}-\beta\left(\alpha-% \epsilon\right)-\frac{\beta}{a}\left(\alpha-\delta\right);\beta,\beta-\gamma+1% ,\beta-\alpha+1,\delta;\frac{1}{z}\right)

In both equations, the second entry in the $\mathit{H\!\ell}$ has been corrected with an extra minus sign.

… ►

Subsection 20.10(ii)

… ►

Paragraph Mellin–Barnes Integrals (in §8.6(ii))

The descriptions for the paths of integration of the Mellin-Barnes integrals (8.6.10)–(8.6.12) have been updated. The description for (8.6.11) now states that the path of integration is to the right of all poles. Previously it stated incorrectly that the path of integration had to separate the poles of the gamma function from the pole at $s=0$ . The paths of integration for (8.6.10) and (8.6.12) have been clarified. In the case of (8.6.10), it separates the poles of the gamma function from the pole at $s=a$ for $\gamma\left(a,z\right)$ . In the case of (8.6.12), it separates the poles of the gamma function from the poles at $s=0,1,2,\ldots$ .

Reported 2017-07-10 by Kurt Fischer.

… ►

Equation (8.12.5)

To be consistent with the notation used in (8.12.16), Equation (8.12.5) was changed to read

8.12.5

\frac{{\mathrm{e}}^{\pm\pi\mathrm{i}a}}{2\mathrm{i}\sin(\pi a)}Q\left(-a,z{% \mathrm{e}}^{\pm\pi\mathrm{i}}\right)=\pm\tfrac{1}{2}\operatorname{erfc}\left(% \pm\mathrm{i}\eta\sqrt{a/2}\right)-\mathrm{i}T(a,\eta)

…

26: 13.7 Asymptotic Expansions for Large Argument

§13.7 Asymptotic Expansions for Large Argument

… ►

§13.7(ii) Error Bounds

… ►

§13.7(iii) Exponentially-Improved Expansion

… ►

13.7.12

G_{p}(z)=\frac{e^{z}}{2\pi}\Gamma\left(p\right)\Gamma\left(1-p,z\right).

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Defines:: $G_{p}(z)$ : expansion (locally)
Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\Gamma\left(\NVar{a},\NVar{z}\right)$ : incomplete gamma function and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/13.7.E12
Encodings:: pMML, png, TeX
See also:: Annotations for §13.7(iii), §13.7 and Ch.13

… ►For extensions to hyperasymptotic expansions see Olde Daalhuis and Olver (1995a).

27: Bibliography

… ►

M. M. Agrest and M. S. Maksimov (1971) Theory of Incomplete Cylindrical Functions and Their Applications. Springer-Verlag, Berlin.

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Notes:: Translated from the Russian by H. E. Fettis, J. W. Goresh and D. A. Lee, Die Grundlehren der mathematischen Wissenschaften, Band 160
External Links:: MathReview, ZentralBlatt
Referenced by:: 1st item.
Encodings:: BibTeX

… ►

W. A. Al-Salam and M. E. H. Ismail (1994) A

q

-beta integral on the unit circle and some biorthogonal rational functions. Proc. Amer. Math. Soc. 121 (2), pp. 553–561.

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External Links:: ISSN 0002-9939, FullText, MathReview (S. P. Goyal), ZentralBlatt
Referenced by:: §18.33(v).
Encodings:: BibTeX

… ►

G. Allasia and R. Besenghi (1987b) Numerical calculation of incomplete gamma functions by the trapezoidal rule. Numer. Math. 50 (4), pp. 419–428.

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External Links:: ISSN 0029-599X, Document, MathReview (F. W. J. Olver), ZentralBlatt
Referenced by:: §8.25(ii).
Encodings:: BibTeX

… ►

H. Alzer (1997b) On some inequalities for the incomplete gamma function. Math. Comp. 66 (218), pp. 771–778.

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External Links:: ISSN 0025-5718, Document, MathReview, ZentralBlatt
Referenced by:: §8.10.
Encodings:: BibTeX

… ►

H. Appel (1968) Numerical Tables for Angular Correlation Computations in

\alpha

\beta

- and

\gamma

-Spectroscopy:

3j

6j

9j

-Symbols, F- and

\Gamma

-Coefficients. Landolt-Börnstein Numerical Data and Functional Relationships in Science and Technology, Springer-Verlag.

ⓘ

External Links:: ISBN 978-3-540-04218-1
Referenced by:: §34.14.
Encodings:: BibTeX

…

28: 19.33 Triaxial Ellipsoids

… ►

19.33.2

\frac{S}{2\pi}=c^{2}+\frac{ab}{\sin\phi}\left(E\left(\phi,k\right){\sin}^{2}% \phi+F\left(\phi,k\right){\cos}^{2}\phi\right),

a\geq b\geq c

ⓘ

Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $F\left(\NVar{\phi},\NVar{k}\right)$ : Legendre’s incomplete elliptic integral of the first kind, $E\left(\NVar{\phi},\NVar{k}\right)$ : Legendre’s incomplete elliptic integral of the second kind, $\sin\NVar{z}$ : sine function, $\phi$ : real or complex argument, $k$ : real or complex modulus and $S$ : area
Permalink:: http://dlmf.nist.gov/19.33.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §19.33(i), §19.33 and Ch.19

… ►where

\alpha,\beta,\gamma

are dimensionless positive constants. The contours of constant density are a family of similar, rather than confocal, ellipsoids. … ►

19.33.11

U=\tfrac{1}{2}(\alpha\beta\gamma)^{2}R_{F}\left(\alpha^{2},\beta^{2},\gamma^{2% }\right)\int_{0}^{\infty}(g(r))^{2}\,\mathrm{d}r,

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Symbols:: $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of first kind, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\alpha$ : positive constant, $\beta$ : positive constant, $\gamma$ : positive constant, $U$ : energy and $g(r)$ : function
Permalink:: http://dlmf.nist.gov/19.33.E11
Encodings:: pMML, png, TeX
See also:: Annotations for §19.33(iv), §19.33 and Ch.19

►where …