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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: 5 Gamma Function
Chapter 5 Gamma Function
5: 5.3 Graphics
§5.3 Graphics
See accompanying text
Figure 5.3.1: Γ ( x ) and 1 / Γ ( x ) . … Magnify
See accompanying text
Figure 5.3.2: ln Γ ( x ) . … Magnify
See accompanying text
Figure 5.3.5: 1 / | Γ ( x + i y ) | . Magnify 3D Help
See accompanying text
Figure 5.3.6: | ψ ( x + i y ) | . Magnify 3D Help
6: 8.2 Definitions and Basic Properties
§8.2 Definitions and Basic Properties
The general values of the incomplete gamma functions γ ( a , z ) and Γ ( a , z ) are defined by …
§8.2(ii) Analytic Continuation
In this subsection the functions γ and Γ have their general values. …
§8.2(iii) Differential Equations
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: 5.1 Special Notation
j , m , n

nonnegative integers.

The main functions treated in this chapter are the gamma function Γ ( z ) , the psi function (or digamma function) ψ ( z ) , the beta function B ( a , b ) , and the q -gamma function Γ q ( z ) . … Alternative notations for this function are: Π ( z - 1 ) (Gauss) and ( z - 1 ) ! . …
9: 5.5 Functional Relations
§5.5(i) Recurrence
§5.5(ii) Reflection
§5.5(iii) Multiplication
Duplication Formula
§5.5(iv) Bohr–Mollerup Theorem
10: 8.7 Series Expansions
§8.7 Series Expansions
8.7.1 γ * ( a , z ) = e - z k = 0 z k Γ ( a + k + 1 ) = 1 Γ ( a ) k = 0 ( - z ) k k ! ( a + k ) .
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 , .
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).