Gamma Functions#
The gamma function is a special function extending the factorial to real numbers.
- gamma(x)#
Gamma function.
- Parameters:
x (Real) – The value to take the gamma function of.
- Definition:
\(\Gamma(x) = \displaystyle\int_0^\infty t^{x-1} e^{-t} \, dt\) for \(x > 0\), extended to the rest of the domain by analytic continuation.
- Domain:
\(x \in \mathbb{R}\), excluding \(0, -1, -2, \ldots\)
- Range:
\((-\infty, \infty)\), excluding \(0\)
- Returns:
The value of \(\Gamma(x)\).
- Return type:
Real
- lgamma(x)#
Natural logarithm of the absolute value of the gamma function.
- Parameters:
x (Real) – The value to take the function of.
- Definition:
\(\mathrm{lgamma}(x) = \log{|\Gamma(x)|}\)
- Domain:
\(x \in \mathbb{R}\), excluding \(0, -1, -2, \ldots\)
- Range:
\((-\infty, \infty)\)
- Returns:
The value of \(\log{|\Gamma(x)|}\).
- Return type:
Real