Fungrim entry: 6d0a95

\Gamma(z) = {\left(2 \pi\right)}^{1 / 2} {z}^{z - 1 / 2} {e}^{-z} \exp\!\left(\sum_{n=1}^{\infty} \left(z + n - \frac{1}{2}\right) \log\!\left(\frac{z + n}{z + n - 1}\right) - 1\right)

Assumptions:

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \notin \left(-\infty, 0\right]

References:

B. C. Carlson (1977), Special functions of applied mathematics, Academic Press. Proposition 3.8-1.

TeX:

\Gamma(z) = {\left(2 \pi\right)}^{1 / 2} {z}^{z - 1 / 2} {e}^{-z} \exp\!\left(\sum_{n=1}^{\infty} \left(z + n - \frac{1}{2}\right) \log\!\left(\frac{z + n}{z + n - 1}\right) - 1\right)

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \notin \left(-\infty, 0\right]

Definitions:

Fungrim symbol	Notation	Short description
`Gamma`	$\Gamma(z)$	Gamma function
`Pow`	${a}^{b}$	Power
`Pi`	$\pi$	The constant pi (3.14...)
`Exp`	${e}^{z}$	Exponential function
`Sum`	$\sum_{n} f(n)$	Sum
`Log`	$\log(z)$	Natural logarithm
`Infinity`	$\infty$	Positive infinity
`CC`	$\mathbb{C}$	Complex numbers
`OpenClosedInterval`	$\left(a, b\right]$	Open-closed interval

Source code for this entry:

Entry(ID("6d0a95"),
    Formula(Equal(Gamma(z), Mul(Mul(Mul(Pow(Mul(2, Pi), Div(1, 2)), Pow(z, Sub(z, Div(1, 2)))), Exp(Neg(z))), Exp(Sum(Sub(Mul(Sub(Add(z, n), Div(1, 2)), Log(Div(Add(z, n), Sub(Add(z, n), 1)))), 1), For(n, 1, Infinity)))))),
    Variables(z),
    Assumptions(And(Element(z, CC), NotElement(z, OpenClosedInterval(Neg(Infinity), 0)))),
    References("B. C. Carlson (1977), Special functions of applied mathematics, Academic Press. Proposition 3.8-1."))

Topics using this entry

Gamma function