inverse incomplete gamma function
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1: 8.12 Uniform Asymptotic Expansions for Large Parameter
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Inverse Function
…2: Bibliography T
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Asymptotic inversion of incomplete gamma functions.
Math. Comp. 58 (198), pp. 755–764.
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3: Bibliography D
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Computation of the incomplete gamma function ratios and their inverses.
ACM Trans. Math. Software 12 (4), pp. 377–393.
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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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4: 19.11 Addition Theorems
5: 8.25 Methods of Computation
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►See Allasia and Besenghi (1987b) for the numerical computation of from (8.6.4) by means of the trapezoidal rule.
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►DiDonato and Morris (1986) describes an algorithm for computing and for , , and from the uniform expansions in §8.12.
…A numerical inversion procedure is also given for calculating the value of (with 10S accuracy), when and are specified, based on Newton’s rule (§3.8(ii)).
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►The computation of and by means of continued fractions is described in Jones and Thron (1985) and Gautschi (1979b, §§4.3, 5).
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►Expansions involving incomplete gamma functions often require the generation of sequences , , or for fixed and .
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6: Bibliography O
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Asymptotic expansions for -gamma, -exponential, and -Bessel functions.
J. Math. Anal. Appl. 186 (3), pp. 896–913.
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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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Inverse factorial-series solutions of difference equations.
Proc. Edinb. Math. Soc. (2) 47 (2), pp. 421–448.
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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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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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7: 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
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►and denotes the logarithmic derivative of the gamma function (§5.2(i)).
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8: 8.18 Asymptotic Expansions of
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►and as in §8.2(i).
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Symmetric Case
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►Let denote the scaled gamma function … ►Inverse Function
…9: 19.37 Tables
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►Tabulated for to 6D by Byrd and Friedman (1971) and to 15D by Abramowitz and Stegun (1964, Chapter 17).
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►Tabulated for to 6D by Byrd and Friedman (1971) and to 15D by Abramowitz and Stegun (1964, Chapter 17).
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§19.37(iii) Legendre’s Incomplete Integrals
… ►Tabulated for , to 6D by Byrd and Friedman (1971), for , and to 8D by Abramowitz and Stegun (1964, Chapter 17), and for , to 9D by Zhang and Jin (1996, pp. 674–675). … ►Tabulated (with different notation) for , , to 5D by Abramowitz and Stegun (1964, Chapter 17), and for , , to 7D by Zhang and Jin (1996, pp. 676–677). …10: Bibliography R
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Elliptic Functions, Theta Functions, and Riemann Surfaces.
The Williams & Wilkins Co., Baltimore, MD.
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Fourier analysis and signal processing by use of the Möbius inversion formula.
IEEE Trans. Acoustics, Speech, Signal Processing 38, pp. 458–470.
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Infinite Sum of the Incomplete Gamma Function Expressed in Terms of the Hurwitz Zeta Function.
Mathematics 9 (16).
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A proof of the asymptotic series for log and log
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Ann. of Math. (2) 32 (1), pp. 10–16.
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Functional Analysis.
McGraw-Hill Book Co., New York.
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