3.5.1.3.40 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

Parameters:

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