Definition A: For any x > 0 the gamma function is defined by
(Note: actually the gamma function can be defined as above for any complex number with non-negative real part.)
Definition B: For any x > 0 the lower incomplete gamma function is defined by
For any x > 0 the upper incomplete gamma function is defined by
Proof: Follows from Definitions A and B.
- Γ(1) = 1
- Γ(x + 1) = x Γ(x)
- Γ(n) = (n – 1)! For all natural numbers n = 0, 1, 2, 3, …
- Γ(½) = √π =
(3) Follows from (1) and (2) by induction
(4) The proof of the fourth assertion results from the fact (Gaussian integral) that
We won’t prove this here, but note that by using the substitution x = , we have by Definition 1
Observation: Note that the gamma function Γ(x) is only defined for x > 0. Negative values can be defined via Property 1.2, namely via Γ(x) = Γ(x+1)/x. Thus, by Property 1.4, we see that Γ(–.5) = Γ(-.5+1)/(-.5) = –2. This approach only works for non-integer values since Γ(0) = Γ(1)/0, Γ(-1) = Γ(0)/(-1), etc. are undefined.
The following formula can be used to calculate the gamma function for non-integer negative values.
Real Statistics Functions: The Real Statistic Resource Pack provides the following formulas.
XGAMMA(x) = gamma function at x even when x is negative
LowerGamma(x, a) = lower incomplete gamma function γ(x, a)
UpperGamma(x, a) = upper incomplete gamma function Γ(x, a)