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Values of the Gamma function at rational numbers
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Numbers
$s$ 
$\Gamma(s)$
INPUT{numbers.yaml} (not shown in preview)
Definition
The list contains values of the Gamma function $\Gamma(s)$ at rational number $s = a/b$.
For $\Re(s) > 0$, $\Gamma(s)$ is defined as the Mellin transform of $e^{-t}$, that is, $\Gamma(s) = \int_0^\infty t^s e^{-t}\, (dt/t)$. $\Gamma$ extends to a meromorphic function on $\mathbb{C}$ via analytic continuation.
Parameters
$s$
—   rational number ($s \not \in \mathbb{Z}_{\leq 0}$)
Formulas
(1)
$\Gamma(s+1) = s\Gamma(s)$ for all $s\in\mathbb{C}$.
(2)
$\Gamma(n) = (n-1)!$ for positive integers $n$.
(3)
$\Gamma(s)\Gamma(1-s) = \pi / \sin(\pi s)$ (Euler's reflection formula).
Comments
(4)
The analytic continuation of $\Gamma$ can be deduced from (1).
Programs
(P1)
Sage
numbers = {a/b: gamma(RIF(a/b)) 
                for b,a in cartesian_product(([1..30],[-30..30])) 
                if gcd(a,b) == 1}
Links
Data properties
Entries are of type: real number
Table is complete: no