inverse incomplete beta function

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1: 8.18 Asymptotic Expansions of $I_{x}\left(a,b\right)$

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Inverse Function

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2: Bibliography T

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N. M. Temme (1992b) 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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3: Bibliography D

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A. R. DiDonato and A. H. Morris (1986) Computation of the incomplete gamma function ratios and their inverses. ACM Trans. Math. Software 12 (4), pp. 377–393.

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External Links:: ISSN 0098-3500, Document, ZentralBlatt
Referenced by:: §8.12, §8.25(iii).
Encodings:: BibTeX

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A. R. DiDonato and A. H. Morris (1987) Algorithm 654: Fortran subroutines for computing the incomplete gamma function ratios and their inverses. ACM Trans. Math. Software 13 (3), pp. 318–319.

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Notes:: Single-precision Fortran, maximum accuracy 14D.
External Links:: ISSN 0098-3500, Document, GAMS (module 654; package TOMS), ZentralBlatt
Referenced by:: 1st item.
Encodings:: BibTeX

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A. R. DiDonato and A. H. Morris (1992) Algorithm 708: Significant digit computation of the incomplete beta function ratios. ACM Trans. Math. Software 18 (3), pp. 360–373.

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Notes:: Single-precision Fortran, maximum accuracy 14D.
External Links:: ISSN 0098-3500, Document, GAMS (module 708; package TOMS), ZentralBlatt
Referenced by:: 3rd item.
Encodings:: BibTeX

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T. M. Dunster (2006) Uniform asymptotic approximations for incomplete Riemann zeta functions. J. Comput. Appl. Math. 190 (1-2), pp. 339–353.

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External Links:: ISSN 0377-0427, Document, MathReview Entry, ZentralBlatt
Referenced by:: §8.22(ii).
Encodings:: BibTeX

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J. Dutka (1981) The incomplete beta function—a historical profile. Arch. Hist. Exact Sci. 24 (1), pp. 11–29.

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External Links:: ISSN 0003-9519, MathReview (Willard Parker), ZentralBlatt
Referenced by:: §8.17(i).
Encodings:: BibTeX

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… ►(For other notation see Notation for the Special Functions.) … ►All derivatives are denoted by differentials, not by primes. … ►of the first, second, and third kinds, respectively, and Legendre’s incomplete integrals … ►In Abramowitz and Stegun (1964, Chapter 17) the functions (19.1.1) and (19.1.2) are denoted, in order, by

K(\alpha)

E(\alpha)

\Pi(n\backslash\alpha)

F(\phi\backslash\alpha)

E(\phi\backslash\alpha)

, and

\Pi(n;\phi\backslash\alpha)

, where

\alpha=\operatorname{arcsin}k

and

n

is the

\alpha^{2}

(not related to

k

) in (19.1.1) and (19.1.2). … ►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.

5: 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

\Pi\left(\phi,\alpha^{2},k\right)

and

\Pi\left(\phi,\omega^{2},k\right)

, where

\omega^{2}

is a rational function of

\alpha^{2}

. … ►

19.7.8

\Pi\left(\phi,\alpha^{2},k\right)+\Pi\left(\phi,\omega^{2},k\right)=F\left(% \phi,k\right)+\sqrt{c}R_{C}\left((c-1)(c-k^{2}),(c-\alpha^{2})(c-\omega^{2})% \right),

\alpha^{2}\omega^{2}=k^{2}

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Defines:: $\omega^{2}$ : change of variable (locally)
Symbols:: $R_{C}\left(\NVar{x},\NVar{y}\right)$ : Carlson’s combination of inverse circular and inverse hyperbolic functions, $F\left(\NVar{\phi},\NVar{k}\right)$ : Legendre’s incomplete elliptic integral of the first kind, $\Pi\left(\NVar{\phi},\NVar{\alpha}^{2},\NVar{k}\right)$ : Legendre’s incomplete elliptic integral of the third kind, $\phi$ : 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})\Pi\left(\phi,\alpha^{2},k\right)+(k^{2}-\omega^{2})\Pi\left% (\phi,\omega^{2},k\right)=k^{2}F\left(\phi,k\right)-\alpha^{2}\omega^{2}\sqrt{% c-1}R_{C}\left(c(c-k^{2}),(c-\alpha^{2})(c-\omega^{2})\right),

(1-\alpha^{2})(1-\omega^{2})=1-k^{2}

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Defines:: $\omega^{2}$ : change of variable (locally)
Symbols:: $R_{C}\left(\NVar{x},\NVar{y}\right)$ : Carlson’s combination of inverse circular and inverse hyperbolic functions, $F\left(\NVar{\phi},\NVar{k}\right)$ : Legendre’s incomplete elliptic integral of the first kind, $\Pi\left(\NVar{\phi},\NVar{\alpha}^{2},\NVar{k}\right)$ : Legendre’s incomplete elliptic integral of the third kind, $\phi$ : 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})\Pi\left(\phi,\alpha^{2},k\right)+(1-\omega^{2})\Pi\left(\phi,% \omega^{2},k\right)=F\left(\phi,k\right)+(1-\alpha^{2}-\omega^{2})\sqrt{c-k^{2% }}\*R_{C}\left(c(c-1),(c-\alpha^{2})(c-\omega^{2})\right),

(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:: $R_{C}\left(\NVar{x},\NVar{y}\right)$ : Carlson’s combination of inverse circular and inverse hyperbolic functions, $F\left(\NVar{\phi},\NVar{k}\right)$ : Legendre’s incomplete elliptic integral of the first kind, $\Pi\left(\NVar{\phi},\NVar{\alpha}^{2},\NVar{k}\right)$ : Legendre’s incomplete elliptic integral of the third kind, $\phi$ : 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

6: 19.16 Definitions

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§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: …

7: Bibliography K

… ►

S. H. Khamis (1965) Tables of the Incomplete Gamma Function Ratio: The Chi-square Integral, the Poisson Distribution. Justus von Liebig Verlag, Darmstadt (German, English).

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Notes:: In cooperation with W. Rudert
External Links:: MathReview, ZentralBlatt
Referenced by:: 1st item.
Encodings:: BibTeX

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D. A. Kofke (2004) 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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K. S. Kölbig (1970) Complex zeros of an incomplete Riemann zeta function and of the incomplete gamma function. Math. Comp. 24 (111), pp. 679–696.

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External Links:: FullText, MathReview, ZentralBlatt
Referenced by:: §8.13(iii), §8.22(ii).
Encodings:: BibTeX

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K. S. Kölbig (1972a) Complex zeros of two incomplete Riemann zeta functions. Math. Comp. 26 (118), pp. 551–565.

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External Links:: FullText, MathReview, ZentralBlatt
Referenced by:: §8.22(ii).
Encodings:: BibTeX

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K. S. Kölbig (1972b) On the zeros of the incomplete gamma function. Math. Comp. 26 (119), pp. 751–755.

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External Links:: FullText, MathReview, ZentralBlatt
Referenced by:: §8.13(i), §8.13(iii), §8.13(iii).
Encodings:: BibTeX

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8: Bibliography C

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L. Carlitz (1963) The inverse of the error function. Pacific J. Math. 13 (2), pp. 459–470.

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External Links:: MathReview (D. P. Banerjee), ZentralBlatt
Referenced by:: §7.17(ii).
Encodings:: BibTeX

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B. C. Carlson (2008) Power series for inverse Jacobian elliptic functions. Math. Comp. 77 (263), pp. 1615–1621.

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External Links:: ISSN 0025-5718, Document, MathReview (Shaun Cooper), ZentralBlatt
Referenced by:: §19.25(v), §22.15(ii).
Encodings:: BibTeX

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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

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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

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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

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9: 3.10 Continued Fractions

… ►For example, by converting the Maclaurin expansion of

\operatorname{arctan}z

(4.24.3), we obtain a continued fraction with the same region of convergence (

\left|z\right|\leq 1

z\neq\pm\mathrm{i}

), whereas the continued fraction (4.25.4) converges for all

z\in\mathbb{C}

except on the branch cuts from

i

i\infty

and

-i

-i\infty

. … ►For applications to Bessel functions and Whittaker functions (Chapters 10 and 13), see Gargantini and Henrici (1967). … ►For elementary functions, see §§ 4.9 and 4.35. ►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). … ►In Gautschi (1979c) the forward series algorithm is used for the evaluation of a continued fraction of an incomplete gamma function (see §8.9). …