18 April 2023 Gamma Functions The following page includes the definitions of the gamma functions and their relations to each other.
These functions were particularly useful in the development of promethium .
Gamma Function
Γ ( z ) = ∫ 0 ∞ t z − 1 e − t d t \displaystyle \Gamma(z) = \int_{0}^{\infty} t^{z - 1} e^{-t} dt Γ ( z ) = ∫ 0 ∞ t z − 1 e − t d t
Incomplete Gamma Function
Lower Incomplete Gamma Function
γ ( s , x ) = ∫ 0 x t s − 1 e − t d t \displaystyle \gamma(s, x) = \int_{0}^{x} t^{s - 1} e^{-t} dt γ ( s , x ) = ∫ 0 x t s − 1 e − t d t
Upper Incomplete Gamma Function
Γ ( s , x ) = ∫ x ∞ t s − 1 e − t d t \displaystyle \Gamma(s, x) = \int_{x}^{\infty} t^{s - 1} e^{-t} dt Γ ( s , x ) = ∫ x ∞ t s − 1 e − t d t
Regularized Incomplete Gamma Function
Regularized Lower Incomplete Gamma Function
P ( a , x ) = 1 Γ ( a ) ∫ 0 x t a − 1 e − t d t \displaystyle P(a, x) = \frac{1}{\Gamma(a)} \int_{0}^{x} t^{a - 1} e^{-t} dt P ( a , x ) = Γ ( a ) 1 ∫ 0 x t a − 1 e − t d t
Regularized Upper Incomplete Gamma Function
Q ( a , x ) = 1 Γ ( a ) ∫ x ∞ t a − 1 e − t d t \displaystyle Q(a, x) = \frac{1}{\Gamma(a)} \int_{x}^{\infty} t^{a - 1} e^{-t} dt Q ( a , x ) = Γ ( a ) 1 ∫ x ∞ t a − 1 e − t d t
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P ( a , x ) + Q ( a , x ) = 1 \displaystyle P(a, x) + Q(a, x) = 1 P ( a , x ) + Q ( a , x ) = 1