inverse incomplete gamma function

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1: 8.12 Uniform Asymptotic Expansions for Large Parameter

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Inverse Function

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2: Bibliography T

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N. M. Temme (1992a) Asymptotic inversion of incomplete gamma functions. Math. Comp. 58 (198), pp. 755–764.

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External Links:: ISSN 0025-5718, FullText, Document, MathReview (F. W. J. Olver), ZentralBlatt
Referenced by:: §8.12, §8.12.
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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4: 19.11 Addition Theorems

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19.11.5

\Pi\left(\theta,\alpha^{2},k\right)+\Pi\left(\phi,\alpha^{2},k\right)=\Pi\left% (\psi,\alpha^{2},k\right)-\alpha^{2}R_{C}\left(\gamma-\delta,\gamma\right),

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Symbols:: $R_{C}\left(\NVar{x},\NVar{y}\right)$ : Carlson’s combination of inverse circular and inverse hyperbolic functions, $\Pi\left(\NVar{\phi},\NVar{\alpha}^{2},\NVar{k}\right)$ : 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, $\theta$ : amplitude, $\psi$ : angle, $\gamma$ and $\delta$
Referenced by:: §19.11(i), §19.11(i), Erratum (V1.0.24) for Subsection 19.11(i)
Permalink:: http://dlmf.nist.gov/19.11.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §19.11(i), §19.11 and Ch.19

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19.11.10

\Pi\left(\phi,\alpha^{2},k\right)=\Pi\left(\alpha^{2},k\right)-\Pi\left(\theta% ,\alpha^{2},k\right)-\alpha^{2}R_{C}\left(\gamma-\delta,\gamma\right),

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Symbols:: $R_{C}\left(\NVar{x},\NVar{y}\right)$ : Carlson’s combination of inverse circular and inverse hyperbolic functions, $\Pi\left(\NVar{\alpha}^{2},\NVar{k}\right)$ : Legendre’s complete elliptic integral of the third 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, $\theta$ : amplitude, $\gamma$ and $\delta$
Permalink:: http://dlmf.nist.gov/19.11.E10
Encodings:: pMML, png, TeX
See also:: Annotations for §19.11(ii), §19.11 and Ch.19

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19.11.16

\Pi\left(\psi,\alpha^{2},k\right)=2\Pi\left(\theta,\alpha^{2},k\right)+\alpha^% {2}R_{C}\left(\gamma-\delta,\gamma\right),

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Symbols:: $R_{C}\left(\NVar{x},\NVar{y}\right)$ : Carlson’s combination of inverse circular and inverse hyperbolic functions, $\Pi\left(\NVar{\phi},\NVar{\alpha}^{2},\NVar{k}\right)$ : Legendre’s incomplete elliptic integral of the third kind, $k$ : real or complex modulus, $\alpha^{2}$ : real or complex parameter, $\theta$ : amplitude, $\delta$ , $\psi$ : angle and $\gamma$
Permalink:: http://dlmf.nist.gov/19.11.E16
Encodings:: pMML, png, TeX
See also:: Annotations for §19.11(iii), §19.11 and Ch.19

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5: 8.25 Methods of Computation

… ►See Allasia and Besenghi (1987b) for the numerical computation of

\Gamma\left(a,z\right)

from (8.6.4) by means of the trapezoidal rule. … ►DiDonato and Morris (1986) describes an algorithm for computing

P\left(a,x\right)

and

Q\left(a,x\right)

for

a\geq 0

x\geq 0

, and

a+x\neq 0

from the uniform expansions in §8.12. …A numerical inversion procedure is also given for calculating the value of

x

(with 10S accuracy), when

a

and

P\left(a,x\right)

are specified, based on Newton’s rule (§3.8(ii)). … ►The computation of

\gamma\left(a,z\right)

and

\Gamma\left(a,z\right)

by means of continued fractions is described in Jones and Thron (1985) and Gautschi (1979b, §§4.3, 5). … ►Expansions involving incomplete gamma functions often require the generation of sequences

P\left(a+n,x\right)

Q\left(a+n,x\right)

, or

\gamma^{*}\left(a+n,x\right)

for fixed

a

and

n=0,1,2,\dots

. …

L. Jacobsen, W. B. Jones, and H. Waadeland (1986) Further results on the computation of incomplete gamma functions. In Analytic Theory of Continued Fractions, II (Pitlochry/Aviemore, 1985), W. J. Thron (Ed.), Lecture Notes in Math. 1199, pp. 67–89.

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External Links:: MathReview (Marietta J. Tretter), ZentralBlatt
Referenced by:: §8.25(iv).
Encodings:: BibTeX

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D. K. Jefferson (1961) Algorithm 73: Incomplete elliptic integrals. Comm. ACM 4 (12), pp. 543.

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Notes:: Includes Algol program. For certification see same journal 6 (1963), p. 167, for remark see same journal 5 (1962), p. 514
External Links:: ISSN 0001-0782, Document
Referenced by:: §19.39(iii).
Encodings:: BibTeX

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D. J. Jeffrey, R. M. Corless, D. E. G. Hare, and D. E. Knuth (1995) Sur l’inversion de

y^{\alpha}e^{y}

au moyen des nombres de Stirling associés. C. R. Acad. Sci. Paris Sér. I Math. 320 (12), pp. 1449–1452.

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External Links:: ISSN 0764-4442, MathReview (F. Beukers)
Referenced by:: (4.13.10).
Encodings:: BibTeX

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D. J. Jeffrey and N. Murdoch (2017) Stirling Numbers, Lambert W and the Gamma Function. In Mathematical Aspects of Computer and Information Sciences, J. Blömer, I. S. Kotsireas, T. Kutsia, and D. E. Simos (Eds.), Cham, pp. 275–279.

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External Links:: ISBN 978-3-319-72453-9
Referenced by:: §4.13.
Encodings:: BibTeX

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W. B. Jones and W. J. Thron (1985) On the computation of incomplete gamma functions in the complex domain. J. Comput. Appl. Math. 12/13, pp. 401–417.

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External Links:: ISSN 0377-0427, Document, MathReview (M. Marsden), ZentralBlatt
Referenced by:: §8.25(iv), §8.9.
Encodings:: BibTeX

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7: Bibliography O

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A. B. Olde Daalhuis (1994) Asymptotic expansions for

q

-gamma,

q

-exponential, and

q

-Bessel functions. J. Math. Anal. Appl. 186 (3), pp. 896–913.

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External Links:: ISSN 0022-247X, Document, MathReview (P. K. Banerji), ZentralBlatt
Referenced by:: §5.18(ii).
Encodings:: BibTeX

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A. B. Olde Daalhuis (1998c) On the resurgence properties of the uniform asymptotic expansion of the incomplete gamma function. Methods Appl. Anal. 5 (4), pp. 425–438.

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External Links:: ISSN 1073-2772, Pdf, MathReview (Éric Delabaere), ZentralBlatt
Referenced by:: §8.12.
Encodings:: BibTeX

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A. B. Olde Daalhuis (2004a) Inverse factorial-series solutions of difference equations. Proc. Edinb. Math. Soc. (2) 47 (2), pp. 421–448.

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External Links:: ISSN 0013-0915, Document, MathReview (Niro Yanagihara), ZentralBlatt
Referenced by:: §2.9(i).
Encodings:: BibTeX

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F. W. J. Olver (1980b) Whittaker functions with both parameters large: Uniform approximations in terms of parabolic cylinder functions. Proc. Roy. Soc. Edinburgh Sect. A 86 (3-4), pp. 213–234.

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External Links:: ISSN 0308-2105, MathReview (P. F. Hsieh), ZentralBlatt
Referenced by:: §13.20(iii), §13.20(iv).
Encodings:: BibTeX

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F. W. J. Olver (1995) On an asymptotic expansion of a ratio of gamma functions. Proc. Roy. Irish Acad. Sect. A 95 (1), pp. 5–9.

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External Links:: ISSN 0035-8975, MathReview (Richard B. Paris), ZentralBlatt
Referenced by:: §5.11(iii).
Encodings:: BibTeX

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8: 6.10 Other Series Expansions

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§6.10(i) Inverse Factorial Series

… ►For a more general result (incomplete gamma function), and also for a result for the logarithmic integral, see Nielsen (1906a, p. 283: Formula (3) is incorrect). ►

§6.10(ii) Expansions in Series of Spherical Bessel Functions

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6.10.6

\operatorname{Ei}\left(x\right)=\gamma+\ln\left|x\right|+\sum_{n=0}^{\infty}(-% 1)^{n}(x-a_{n})\left({\mathsf{i}^{(1)}_{n}}\left(\tfrac{1}{2}x\right)\right)^{% 2},

x\neq 0

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Symbols:: $\gamma$ : Euler’s constant, $\operatorname{Ei}\left(\NVar{x}\right)$ : exponential integral, $\ln\NVar{z}$ : principal branch of logarithm function, ${\mathsf{i}^{(1)}_{\NVar{n}}}\left(\NVar{z}\right)$ : modified spherical Bessel function, $x$ : real variable, $n$ : nonnegative integer and $a_{n}$ : coefficients
Referenced by:: §6.18(i)
Permalink:: http://dlmf.nist.gov/6.10.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §6.10(ii), §6.10 and Ch.6

… ►and

\psi

denotes the logarithmic derivative of the gamma function (§5.2(i)). …

9: 8.18 Asymptotic Expansions of $I_{x}\left(a,b\right)$

… ►and

Q\left(a,z\right)

as in §8.2(i). … ►

Symmetric Case

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General Case

►For the scaled gamma function

\Gamma^{*}\left(z\right)

see (5.11.3). … ►

Inverse Function

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10: 19.37 Tables

… ►Tabulated for

\operatorname{arcsin}k=0(1^{\circ})90^{\circ}

to 6D by Byrd and Friedman (1971) and to 15D by Abramowitz and Stegun (1964, Chapter 17). … ►Tabulated for

\operatorname{arcsin}k=0(1^{\circ})90^{\circ}

to 6D by Byrd and Friedman (1971) and to 15D by Abramowitz and Stegun (1964, Chapter 17). … ►

§19.37(iii) Legendre’s Incomplete Integrals

… ►Tabulated for

\phi=0(5^{\circ})90^{\circ}

\operatorname{arcsin}k=0(1^{\circ})90^{\circ}

to 6D by Byrd and Friedman (1971), for

\phi=0(5^{\circ})90^{\circ}

\operatorname{arcsin}k=0(2^{\circ})90^{\circ}

and

5^{\circ}(10^{\circ})85^{\circ}

to 8D by Abramowitz and Stegun (1964, Chapter 17), and for

\phi=0(10^{\circ})90^{\circ}

\operatorname{arcsin}k=0(5^{\circ})90^{\circ}

to 9D by Zhang and Jin (1996, pp. 674–675). … ►Tabulated (with different notation) for

\phi=0(15^{\circ})90^{\circ}

\alpha^{2}=0(.1)1

\operatorname{arcsin}k=0(15^{\circ})90^{\circ}

to 5D by Abramowitz and Stegun (1964, Chapter 17), and for

\phi=0(15^{\circ})90^{\circ}

\alpha^{2}=0(.1)1

\operatorname{arcsin}k=0(15^{\circ})90^{\circ}

to 7D by Zhang and Jin (1996, pp. 676–677). …

inverse incomplete gamma function

1—10 of 14 matching pages

1: 8.12 Uniform Asymptotic Expansions for Large Parameter

Inverse Function

2: Bibliography T

3: Bibliography D

4: 19.11 Addition Theorems

5: 8.25 Methods of Computation

6: Bibliography J

7: Bibliography O

8: 6.10 Other Series Expansions

§6.10(i) Inverse Factorial Series

§6.10(ii) Expansions in Series of Spherical Bessel Functions

9: 8.18 Asymptotic Expansions of I x ⁡ ( a , b )

Symmetric Case

General Case

Inverse Function

10: 19.37 Tables

§19.37(iii) Legendre’s Incomplete Integrals

9: 8.18 Asymptotic Expansions of $I_{x}\left(a,b\right)$