不完全Γ函數

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数学中,上不完全 \Gamma函数下不完全 \Gamma函数 \Gamma函数的推广。它们的定义分别如下:

 \Gamma(s,x) = \int_x^{\infty} t^{s-1}\,e^{-t}\,{\rm d}t .\,\! \qquad \gamma(s,x) = \int_0^x t^{s-1}\,e^{-t}\,{\rm d}t .\,\!

基本性质[编辑]

在定义中,s都是实部为正值的复变量。通过分部积分,可以计算得递归关系:

\Gamma(s,x)= (s-1)\Gamma(s-1,x) + x^{s-1} e^{-x}

以及反过来:

 \gamma(s,x) =(s-1)\gamma(s-1,x) - x^{s-1} e^{-x}

因为正常的 \Gamma函数定义为:

 \Gamma(s) = \int_0^{\infty} t^{s-1}\,e^{-t}\,{\rm d}t

我们有:

 \Gamma(s) = \Gamma(s,0)

以及

 \gamma(s,x) + \Gamma(s,x) = \Gamma(s).

解析开拓[编辑]

特殊值[编辑]

导数[编辑]

从特殊情况下的的“Meijer”G功能[1]:

  •  \frac{\partial \Gamma (a,x) }{\partial x} = - \frac{x^{a-1}}{e^x}
T(m,a,x) = G_{m-1,\,m}^{\,m,\,0} \!\left( \left. \begin{matrix} 0, 0, \dots, 0 \\ a-1, -1, \dots, -1 \end{matrix} \; \right| \, x \right)
T(m,a,z) = - \frac{(-1)^{m-1} }{(m-2)! } \frac{{\rm d}^{m-2} }{{\rm d}t^{m-2} } \left[\Gamma (a-t) z^{t-1}\right]\Big|_{t=0} + \sum_{n=0}^{\infty} \frac{(-1)^n z^{a-1+n}}{n! (-a-n)^{m-1} } 何時 |z| < 1

外部链接[编辑]

  1. ^ K.O. Geddes, M.L. Glasser, R.A. Moore and T.C. Scott, Evaluation of Classes of Definite Integrals Involving Elementary Functions via Differentiation of Special Functions, AAECC (Applicable Algebra in Engineering, Communication and Computing), vol. 1, (1990), pp. 149-165, [1]

积分方程[编辑]

参考资料[编辑]

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  • G. Arfken and H. Weber. Mathematical Methods for Physicists. Harcourt/Academic Press, 2000. (See Chapter 10.)
  • DiDonato, Armido R.; Morris, Jr., Alfred H. Computation of the incomplete gamma function ratios and their inverse. ACM Transactions on Mathematical Software (TOMS). 1986.Dec., 12 (4): 377–393. doi:10.1145/22721.23109. 
  • Barakat, Richard. Evaluation of the Incomplete Gamma Function of Imaginary Argument by Chebyshev Polynomials. Math. Comp. 1961, 15 (73): 7–11. MR 0128058. 
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  • DiDonato, Armido R.; Morris, Jr., Alfred H. ALGORITHM 654: FORTRAN subroutines for computing the incomplete gamma function ratios and their inverse. ACM Transactions on Mathematical Software (TOMS). 1987.Sept., 13 (3): 318–319. doi:10.1145/29380.214348.  (See also www.netlib.org/toms/654).
  • Früchtl, H.; Otto, P. A new algorithm for the evaluation of the incomplete Gamma Function on vector computers. ACM Trans. Math. Softw. 1994, 20 (4): 436–446. 
  • Gautschi, Walter. The incomplete gamma function since Tricomi. Atti Convegni Lincei. 1998, 147: 203–237. MR 1737497. 
  • Gautschi, Walter. A Note on the recursive calculation of Incomplete Gamma Functions. ACM Trans. Math. Softw. 1999, 25 (1): 101–107. MR 1697463. 
  • Gradshteyn, I.S.; Ryzhik, I.M. Tables of Integrals, Series, and Products 4th. New York: Academic Press. 1980. ISBN 0-12-294760-6.  (See Chapter 8.35.)
  • Jones, William B.; Thron, W. J. On the computation of incomplete gamma functions in the complex domain. 12-13. 401–417. 1985. MR 0793971. 
  • Miller, Allen R.; Moskowitz, Ira S. On certain Generalized incomplete Gamma functions. J. Comput. Appl. Math. 1998, 91 (2): 179–190. 
  • Takenaga, Roy. On the Evaluation of the Incomplete Gamma Function. Math. Comp. 1966, 20 (96): 606–610. MR 0203911. 
  • Temme, Nico. Uniform Asymptotic Expansions of the Incomplete Gamma Functions and the Incomplete Beta Function. Math. Comp. 1975, 29 (132): 1109–1114. MR 0387674. 
  • Terras, Riho. The determination of incomplete Gamma Functions through analytic integration. J. Comp. Phys. 1979, 31. MR 0531128.  已忽略文本“ pages-146-151” (帮助)
  • Tricomi, F. G. Asympotitsche Eigenschaften der unvollst. Gammafunktion. Math. Zeitsch. 1950, 53 (2): 136–148. MR 0045253. 
  • van Deun, Joris; Cools, Ronald. A stable recurrence for the incomplete gamma function with imaginary second argument. Numer. Math. 2006, 104: 445–456. doi:10.1007/s00211-006-0026-1. MR 2249673. 
  • Winitzki, Serge. Computing the incomplete gamma function to arbitrary precision. Lect. Not. Comp. Sci. 2003, 2667: 790–798. MR 2110953. 


参见[编辑]

外部链接[编辑]