incomplete gamma functions
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1: 8.23 Statistical Applications
§8.23 Statistical Applications
►The functions and are used extensively in statistics as the probability integrals of the gamma distribution; see Johnson et al. (1994, pp. 337–414). …In queueing theory the Erlang loss function is used, which can be expressed in terms of the reciprocal of ; see Jagerman (1974) and Cooper (1981, pp. 80, 316–319).2: 8.16 Generalizations
§8.16 Generalizations
►For a generalization of the incomplete gamma function, including asymptotic approximations, see Chaudhry and Zubair (1994, 2001) and Chaudhry et al. (1996). …3: 8.24 Physical Applications
§8.24 Physical Applications
►§8.24(i) Incomplete Gamma Functions
►The function appears in: discussions of power-law relaxation times in complex physical systems (Sornette (1998)); logarithmic oscillations in relaxation times for proteins (Metzler et al. (1999)); Gaussian orbitals and exponential (Slater) orbitals in quantum chemistry (Shavitt (1963), Shavitt and Karplus (1965)); population biology and ecological systems (Camacho et al. (2002)). …4: 7.16 Generalized Error Functions
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►These functions can be expressed in terms of the incomplete gamma function
(§8.2(i)) by change of integration variable.
5: 8.2 Definitions and Basic Properties
§8.2 Definitions and Basic Properties
… ►§8.2(ii) Analytic Continuation
►In this subsection the functions and have their general values. ►The function is entire in and . … ►§8.2(iii) Differential Equations
…6: 6.11 Relations to Other Functions
7: 8.22 Mathematical Applications
§8.22 Mathematical Applications
►§8.22(i) Terminant Function
… ►§8.22(ii) Riemann Zeta Function and Incomplete Riemann Zeta Function
►The function , with and , has an intimate connection with the Riemann zeta function (§25.2(i)) on the critical line . See Paris and Cang (1997). …8: 8.7 Series Expansions
§8.7 Series Expansions
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8.7.4
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8.7.6
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►For an expansion for in series of Bessel functions
that converges rapidly when and () is small or moderate in magnitude see Barakat (1961).
9: 8.3 Graphics
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§8.3(i) Real Variables
… ►Some monotonicity properties of and in the four quadrants of the ()-plane in Figure 8.3.6 are given in Erdélyi et al. (1953b, §9.6). ► ►§8.3(ii) Complex Argument
… ►10: 8.27 Approximations
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§8.27(i) Incomplete Gamma Functions
►DiDonato (1978) gives a simple approximation for the function (which is related to the incomplete gamma function by a change of variables) for real and large positive . This takes the form , approximately, where and is shown to produce an absolute error as .