incomplete gamma functions
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§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).
§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). …
§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)). …
… ►These functions can be expressed in terms of the incomplete gamma function (§8.2(i)) by change of integration variable.
§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…
§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: 8.15 Sums
8.15.1►For sums of infinite series whose terms include incomplete gamma functions, see Prudnikov et al. (1986b, §5.2).
§8.7 Series Expansions… ►
8.7.2 .… ►
8.7.4 .… ►
8.7.6 .►For an expansion for in series of Bessel functions that converges rapidly when and () is small or moderate in magnitude see Barakat (1961).