Digital Library of Mathematical Functions
The Gamma Function -- Definitions

### Definitions

###### Contents

- Gamma and Psi Functions
- Euler's Constant
- Pochhammer's Symbol

#### Gamma and Psi Functions

##### Euler's Integral

Γ ( z ) = ∫ 0 ∞ ⅇ - t t z - 1 ⅆ t
ℜ z > 0 .
When ℜ z ≤ 0 , Γ ( z ) is defined by analytic continuation. It is a meromorphic function with no zeros, and with simple poles of residue ( - 1 ) n n ! at z = - n . 1 Γ ( z ) is entire, with simple zeros at z = - n .

ψ ( z ) = Γ ′ ( z ) Γ ( z )

z ≠ 0 , - 1 , - 2 , …

ψ ( z ) is meromorphic with simple poles of residue - 1 at z = - n .

#### Euler's Constant

γ = lim n → ∞ ( 1 + 1 2 + 1 3 + … + 1 n - ln n ) = 0.57721 56649 01532 86060 …

#### Pochhammer's Symbol

( a ) 0 = 1 ( a ) n = a ( a + 1 ) ( a + 2 ) ⋯ ( a + n - 1 )
( a ) n = Γ ( a + n ) Γ ( a )
a ≠ - n , - n - 1 , - n - 2 , …