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1: 8.17 Incomplete Beta Functions
§8.17 Incomplete Beta Functions
§8.17(iii) Integral Representation
§8.17(iv) Recurrence Relations
§8.17(v) Continued Fraction
§8.17(vi) Sums
2: 9.14 Incomplete Airy Functions
§9.14 Incomplete Airy Functions
Incomplete Airy functions are defined by the contour integral (9.5.4) when one of the integration limits is replaced by a variable real or complex parameter. …
3: 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). …The function B x ( a , b ) and its normalization I x ( a , b ) play a similar role in statistics in connection with the beta distribution; see Johnson et al. (1995, pp. 210–275). 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).
4: 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)).
§8.24(ii) Incomplete Beta Functions
5: 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). …
6: 8.22 Mathematical Applications
§8.22 Mathematical Applications
§8.22(i) Terminant Function
§8.22(ii) Riemann Zeta Function and Incomplete Riemann Zeta Function
See Paris and Cang (1997). If ζ x ( s ) denotes the incomplete Riemann zeta function defined by …
7: 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
8: 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
9: 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).
10: 8.13 Zeros
§8.13 Zeros
§8.13(i) x -Zeros of γ * ( a , x )
The function γ * ( a , x ) has no real zeros for a 0 . … Note that from (8.4.12) γ * ( - n , 0 ) = 0 , n = 1 , 2 , 3 , . …
§8.13(iii) a -Zeros of γ * ( a , x )