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expansions in series of incomplete gamma functions

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1: 8.7 Series Expansions
§8.7 Series Expansions
8.7.6 Γ ( a , x ) = x a e - x n = 0 L n ( a ) ( x ) n + 1 , x > 0 .
For an expansion for γ ( a , i x ) in series of Bessel functions J n ( x ) that converges rapidly when a > 0 and x ( 0 ) is small or moderate in magnitude see Barakat (1961).
2: 8.22 Mathematical Applications
See Paris and Cang (1997). …
3: 5.11 Asymptotic Expansions
For re-expansions of the remainder terms in (5.11.1) and (5.11.3) in series of incomplete gamma functions with exponential improvement (§2.11(iii)) in the asymptotic expansions, see Berry (1991), Boyd (1994), and Paris and Kaminski (2001, §6.4). …
4: 6.10 Other Series Expansions
§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
and ψ denotes the logarithmic derivative of the gamma function5.2(i)). …
5: Bibliography G
  • W. Gautschi (1979a) Algorithm 542: Incomplete gamma functions. ACM Trans. Math. Software 5 (4), pp. 482–489.
  • W. Gautschi (1959b) Some elementary inequalities relating to the gamma and incomplete gamma function. J. Math. Phys. 38 (1), pp. 77–81.
  • W. Gautschi (1979b) A computational procedure for incomplete gamma functions. ACM Trans. Math. Software 5 (4), pp. 466–481.
  • W. Gautschi (1998) The incomplete gamma functions since Tricomi. In Tricomi’s Ideas and Contemporary Applied Mathematics (Rome/Turin, 1997), Atti Convegni Lincei, Vol. 147, pp. 203–237.
  • A. Guthmann (1991) Asymptotische Entwicklungen für unvollständige Gammafunktionen. Forum Math. 3 (2), pp. 105–141 (German).
  • 6: Bibliography V
  • H. Van de Vel (1969) On the series expansion method for computing incomplete elliptic integrals of the first and second kinds. Math. Comp. 23 (105), pp. 61–69.
  • C. G. van der Laan and N. M. Temme (1984) Calculation of Special Functions: The Gamma Function, the Exponential Integrals and Error-Like Functions. CWI Tract, Vol. 10, Stichting Mathematisch Centrum, Centrum voor Wiskunde en Informatica, Amsterdam.
  • R. S. Varma (1941) An infinite series of Weber’s parabolic cylinder functions. Proc. Benares Math. Soc. (N.S.) 3, pp. 37.
  • H. Volkmer (1999) Expansions in products of Heine-Stieltjes polynomials. Constr. Approx. 15 (4), pp. 467–480.
  • H. Volkmer (2021) Fourier series representation of Ferrers function P .
  • 7: 8.11 Asymptotic Approximations and Expansions
    §8.11 Asymptotic Approximations and Expansions
    where δ denotes an arbitrary small positive constant. … This expansion is absolutely convergent for all finite z , and it can also be regarded as a generalized asymptotic expansion2.1(v)) of γ ( a , z ) as a in | ph a | π - δ . … This reference also contains explicit formulas for the coefficients in terms of Stirling numbers. …
    8: 8.27 Approximations
    §8.27(i) Incomplete Gamma Functions
  • DiDonato (1978) gives a simple approximation for the function F ( p , x ) = x - p e x 2 / 2 x e - t 2 / 2 t p d t (which is related to the incomplete gamma function by a change of variables) for real p and large positive x . This takes the form F ( p , x ) = 4 x / h ( p , x ) , approximately, where h ( p , x ) = 3 ( x 2 - p ) + ( x 2 - p ) 2 + 8 ( x 2 + p ) and is shown to produce an absolute error O ( x - 7 ) as x .

  • Luke (1969b, pp. 25, 40–41) gives Chebyshev-series expansions for Γ ( a , ω z ) (by specifying parameters) with 1 ω < , and γ ( a , ω z ) with 0 ω 1 ; see also Temme (1994b, §3).

  • Luke (1975, p. 103) gives Chebyshev-series expansions for E 1 ( x ) and related functions for x 5 .

  • Verbeeck (1970) gives polynomial and rational approximations for E p ( x ) = ( e - x / x ) P ( z ) , approximately, where P ( z ) denotes a quotient of polynomials of equal degree in z = x - 1 .

  • 9: Bibliography T
  • N. M. Temme (1979b) The asymptotic expansion of the incomplete gamma functions. SIAM J. Math. Anal. 10 (4), pp. 757–766.
  • N. M. Temme (1987) On the computation of the incomplete gamma functions for large values of the parameters. In Algorithms for approximation (Shrivenham, 1985), Inst. Math. Appl. Conf. Ser. New Ser., Vol. 10, pp. 479–489.
  • N. M. Temme (1992a) Asymptotic inversion of incomplete gamma functions. Math. Comp. 58 (198), pp. 755–764.
  • N. M. Temme (1994b) Computational aspects of incomplete gamma functions with large complex parameters. In Approximation and Computation. A Festschrift in Honor of Walter Gautschi, R. V. M. Zahar (Ed.), International Series of Numerical Mathematics, Vol. 119, pp. 551–562.
  • N. M. Temme (1995a) Asymptotics of zeros of incomplete gamma functions. Ann. Numer. Math. 2 (1-4), pp. 415–423.
  • 10: Bibliography F
  • C. Ferreira, J. L. López, and E. Pérez Sinusía (2005) Incomplete gamma functions for large values of their variables. Adv. in Appl. Math. 34 (3), pp. 467–485.
  • C. Ferreira and J. L. López (2001) An asymptotic expansion of the double gamma function. J. Approx. Theory 111 (2), pp. 298–314.
  • J. L. Fields and J. Wimp (1961) Expansions of hypergeometric functions in hypergeometric functions. Math. Comp. 15 (76), pp. 390–395.
  • W. B. Ford (1960) Studies on Divergent Series and Summability & The Asymptotic Developments of Functions Defined by Maclaurin Series. Chelsea Publishing Co., New York.
  • L. W. Fullerton (1972) Algorithm 435: Modified incomplete gamma function. Comm. ACM 15 (11), pp. 993–995.