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inverse incomplete gamma function

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1: 8.12 Uniform Asymptotic Expansions for Large Parameter
Inverse Function
2: Bibliography T
  • N. M. Temme (1992a) Asymptotic inversion of incomplete gamma functions. Math. Comp. 58 (198), pp. 755–764.
  • 3: Bibliography D
  • 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.
  • 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.
  • 4: 19.11 Addition Theorems
    5: 8.25 Methods of Computation
    See Allasia and Besenghi (1987b) for the numerical computation of Γ ( a , z ) from (8.6.4) by means of the trapezoidal rule. … DiDonato and Morris (1986) describes an algorithm for computing P ( a , x ) and Q ( a , x ) for a 0 , x 0 , and a + x 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 ( a , x ) are specified, based on Newton’s rule (§3.8(ii)). … The computation of γ ( a , z ) and Γ ( a , z ) 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 ( a + n , x ) , Q ( a + n , x ) , or γ * ( a + n , x ) for fixed a and n = 0 , 1 , 2 , . …
    6: Bibliography O
  • 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.
  • 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.
  • A. B. Olde Daalhuis (2004a) Inverse factorial-series solutions of difference equations. Proc. Edinb. Math. Soc. (2) 47 (2), pp. 421–448.
  • 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.
  • 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.
  • 7: 6.10 Other Series Expansions
    §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
    6.10.6 Ei ( x ) = γ + ln | x | + n = 0 ( - 1 ) n ( x - a n ) ( i n ( 1 ) ( 1 2 x ) ) 2 , x 0 ,
    and ψ denotes the logarithmic derivative of the gamma function5.2(i)). …
    8: 8.18 Asymptotic Expansions of I x ( a , b )
    and Q ( a , z ) as in §8.2(i). …
    Symmetric Case
    General Case
    Let Γ ~ ( z ) denote the scaled gamma function
    Inverse Function
    9: 19.37 Tables
    Tabulated for arcsin k = 0 ( 1 ) 90 to 6D by Byrd and Friedman (1971) and to 15D by Abramowitz and Stegun (1964, Chapter 17). … Tabulated for arcsin k = 0 ( 1 ) 90 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 ϕ = 0 ( 5 ) 90 , arcsin k = 0 ( 1 ) 90 to 6D by Byrd and Friedman (1971), for ϕ = 0 ( 5 ) 90 , arcsin k = 0 ( 2 ) 90 and 5 ( 10 ) 85 to 8D by Abramowitz and Stegun (1964, Chapter 17), and for ϕ = 0 ( 10 ) 90 , arcsin k = 0 ( 5 ) 90 to 9D by Zhang and Jin (1996, pp. 674–675). … Tabulated (with different notation) for ϕ = 0 ( 15 ) 90 , α 2 = 0 ( .1 ) 1 , arcsin k = 0 ( 15 ) 90 to 5D by Abramowitz and Stegun (1964, Chapter 17), and for ϕ = 0 ( 15 ) 90 , α 2 = 0 ( .1 ) 1 , arcsin k = 0 ( 15 ) 90 to 7D by Zhang and Jin (1996, pp. 676–677). …
    10: Bibliography R
  • H. E. Rauch and A. Lebowitz (1973) Elliptic Functions, Theta Functions, and Riemann Surfaces. The Williams & Wilkins Co., Baltimore, MD.
  • I. S. Reed, D. W. Tufts, X. Yu, T. K. Truong, M. T. Shih, and X. Yin (1990) Fourier analysis and signal processing by use of the Möbius inversion formula. IEEE Trans. Acoustics, Speech, Signal Processing 38, pp. 458–470.
  • R. Reynolds and A. Stauffer (2021) Infinite Sum of the Incomplete Gamma Function Expressed in Terms of the Hurwitz Zeta Function. Mathematics 9 (16).
  • C. H. Rowe (1931) A proof of the asymptotic series for log Γ ( z ) and log Γ ( z + a ) . Ann. of Math. (2) 32 (1), pp. 10–16.
  • W. Rudin (1973) Functional Analysis. McGraw-Hill Book Co., New York.