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incomplete gamma functions

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1: 8.23 Statistical Applications
§8.23 Statistical Applications
The functions P ( a , x ) and Q ( a , x ) 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 Q ( a , x ) ; 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 γ ( a , x ) 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
These functions can be expressed in terms of the incomplete gamma function γ ( a , z ) 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 γ and Γ have their general values. The function γ * ( a , z ) is entire in z and a . …
§8.2(iii) Differential Equations
6: 6.11 Relations to Other Functions
Incomplete Gamma Function
6.11.1 E 1 ( z ) = Γ ( 0 , z ) .
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 Γ ( a , z ) , with | ph a | 1 2 π and ph z = 1 2 π , has an intimate connection with the Riemann zeta function ζ ( s ) 25.2(i)) on the critical line s = 1 2 . See Paris and Cang (1997). …
8: 8.15 Sums
§8.15 Sums
8.15.1 γ ( a , λ x ) = λ a k = 0 γ ( a + k , x ) ( 1 - λ ) k k ! .
For sums of infinite series whose terms include incomplete gamma functions, see Prudnikov et al. (1986b, §5.2).
9: 8.7 Series Expansions
§8.7 Series Expansions
8.7.2 γ ( a , x + y ) - γ ( a , x ) = Γ ( a , x ) - Γ ( a , x + y ) = e - x x a - 1 n = 0 ( 1 - a ) n ( - x ) n ( 1 - e - y e n ( y ) ) , | y | < | x | .
8.7.4 γ ( a , x ) = Γ ( a ) x 1 2 a e - x n = 0 e n ( - 1 ) x 1 2 n I n + a ( 2 x 1 / 2 ) , a 0 , - 1 , - 2 , .
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).
10: 8.3 Graphics
§8.3(i) Real Variables
Some monotonicity properties of γ * ( a , x ) and Γ ( a , x ) in the four quadrants of the ( a , x )-plane in Figure 8.3.6 are given in Erdélyi et al. (1953b, §9.6).
See accompanying text
Figure 8.3.7: x - a - γ * ( a , x ) (= x - a Q ( a , x ) ), 0 x 4 , - 5 a 5 . Magnify 3D Help
§8.3(ii) Complex Argument
See accompanying text
Figure 8.3.16: γ * ( 2.5 , x + i y ) , - 3 x 3 , - 3 y 3 . Magnify 3D Help