About the Project

gamma function

AdvancedHelp

(0.018 seconds)

11—20 of 350 matching pages

11: 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 , .
12: 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.
13: 6.11 Relations to Other Functions
Incomplete Gamma Function
6.11.1 E 1 ( z ) = Γ ( 0 , z ) .
14: 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 .
15: 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 + ,
16: 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
17: Simon Ruijsenaars
18: 8.8 Recurrence Relations and Derivatives
§8.8 Recurrence Relations and Derivatives
If w ( a , z ) = γ ( a , z ) or Γ ( a , z ) , then …
8.8.4 z γ * ( a + 1 , z ) = γ * ( a , z ) 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 ) .
8.8.13 d d z γ ( a , z ) = d d z Γ ( a , z ) = z a 1 e z ,
19: 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).
20: 8.14 Integrals
§8.14 Integrals
8.14.1 0 e a x γ ( b , x ) Γ ( b ) d x = ( 1 + a ) b a , a > 0 , b > 1 ,
8.14.2 0 e a x Γ ( b , x ) d x = Γ ( b ) 1 ( 1 + a ) b a , a > 1 , b > 1 .
8.14.3 0 x a 1 γ ( b , x ) d x = Γ ( a + b ) a , a < 0 , ( a + b ) > 0 ,
8.14.4 0 x a 1 Γ ( b , x ) d x = Γ ( a + b ) a , a > 0 , ( a + b ) > 0 ,