Gamma

Definition:

$gamma = gamma(x)$ evaluates

$\Gamma (x)=\int_0^\infty t^{x-1}e^{-t}dt$

The function is based on a Chebyshev expansion for $\Gamma$ (1+u), and uses the property $\Gamma$ (1+x) = x $\Gamma$ (x).

If x = N +1+u where N is integral and 0 ≤ u < 1 then it follows that:

for N >0 $\Gamma$ (x) = (x - 1)(x - 2) . . . (x - N) $\Gamma$ (1 + u)

for N = 0 $\Gamma$ (x) = $\Gamma$ (1+u)

for N <0 $\Gamma$ (x) = $\Gamma$ (1+u)/x(x + 1)(x + 2) . . . (x - N - 1).

For more information please review the s14aac function in the NAG document

x (input, double): The argument x of the function.; Constraint: x must not be zero or a negative integer.
$gamma$ (output, double): the value of the function $\Gamma (x)=\int_0^\infty t^{x-1}e^{-t}dt$