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11: 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).
12: 5.2 Definitions
§5.2(i) Gamma and Psi Functions
Euler’s Integral
5.2.1 Γ ( z ) = 0 e - t t z - 1 d t , z > 0 .
5.2.2 ψ ( z ) = Γ ( z ) / Γ ( z ) , z 0 , - 1 , - 2 , .
5.2.5 ( a ) n = Γ ( a + n ) / Γ ( a ) , a 0 , - 1 , - 2 , .
13: 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.
14: 6.11 Relations to Other Functions
Incomplete Gamma Function
6.11.1 E 1 ( z ) = Γ ( 0 , z ) .
15: 5.17 Barnes’ G -Function (Double Gamma Function)
§5.17 Barnes’ G -Function (Double Gamma Function)
5.17.2 G ( n ) = ( n - 2 ) ! ( n - 3 ) ! 1 ! , n = 2 , 3 , .
5.17.3 G ( z + 1 ) = ( 2 π ) z / 2 exp ( - 1 2 z ( z + 1 ) - 1 2 γ z 2 ) k = 1 ( ( 1 + z k ) k exp ( - z + z 2 2 k ) ) .
5.17.4 Ln G ( z + 1 ) = 1 2 z ln ( 2 π ) - 1 2 z ( z + 1 ) + z Ln Γ ( z + 1 ) - 0 z Ln Γ ( t + 1 ) d t .
5.17.5 Ln G ( z + 1 ) 1 4 z 2 + z Ln Γ ( z + 1 ) - ( 1 2 z ( z + 1 ) + 1 12 ) Ln z - ln A + k = 1 B 2 k + 2 2 k ( 2 k + 1 ) ( 2 k + 2 ) z 2 k .
16: 5.10 Continued Fractions
§5.10 Continued Fractions
5.10.1 Ln Γ ( z ) + z - ( z - 1 2 ) ln z - 1 2 ln ( 2 π ) = a 0 z + a 1 z + a 2 z + a 3 z + a 4 z + a 5 z + ,
17: 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
18: Simon Ruijsenaars
19: 8.8 Recurrence Relations and Derivatives
§8.8 Recurrence Relations and Derivatives
If w ( a , z ) = γ ( a , z ) or Γ ( a , z ) , then …
8.8.5 P ( a + 1 , z ) = P ( a , z ) - z a e - z Γ ( a + 1 ) ,
8.8.6 Q ( a + 1 , z ) = Q ( a , z ) + z a e - z Γ ( a + 1 ) .
8.8.12 Q ( a + n , z ) = Q ( a , z ) + z a e - z k = 0 n - 1 z k Γ ( a + k + 1 ) .
20: 5.21 Methods of Computation
§5.21 Methods of Computation
Similarly for ln Γ ( z ) , ψ ( z ) , and the polygamma functions. … For the computation of the q -gamma and q -beta functions see Gabutti and Allasia (2008).