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Fungrim entry: 6d0a95

Γ ⁣(z)=(2π)1/2zz1/2ezexp ⁣(n=1(z+n12)log ⁣(z+nz+n1)1)\Gamma\!\left(z\right) = {\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:zCandz(,0]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\!\left(z\right) = {\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
GammaFunctionΓ ⁣(z)\Gamma\!\left(z\right) Gamma function
Powab{a}^{b} Power
ConstPiπ\pi The constant pi (3.14...)
Expez{e}^{z} Exponential function
Sumnf ⁣(n)\sum_{n} f\!\left(n\right) Sum
Loglog ⁣(z)\log\!\left(z\right) Natural logarithm
Infinity\infty Positive infinity
CCC\mathbb{C} Complex numbers
OpenClosedInterval(a,b]\left(a, b\right] Open-closed interval
Source code for this entry:
Entry(ID("6d0a95"),
    Formula(Equal(GammaFunction(z), Mul(Mul(Mul(Pow(Mul(2, ConstPi), 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), Tuple(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."))

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Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2019-08-19 14:38:23.809000 UTC