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Fungrim entry: 37a95a

logΓ ⁣(z)=(z12)log ⁣(z)z+log ⁣(2π)2+k=1n1B2k2k(2k1)z2k1+Rn ⁣(z)\log \Gamma\!\left(z\right) = \left(z - \frac{1}{2}\right) \log\!\left(z\right) - z + \frac{\log\!\left(2 \pi\right)}{2} + \sum_{k=1}^{n - 1} \frac{B_{2 k}}{2 k \left(2 k - 1\right) {z}^{2 k - 1}} + R_{n}\!\left(z\right)
Assumptions:zCandz(,0]andnZ1z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, z \notin \left(-\infty, 0\right] \,\mathbin{\operatorname{and}}\, n \in \mathbb{Z}_{\ge 1}
TeX:
\log \Gamma\!\left(z\right) = \left(z - \frac{1}{2}\right) \log\!\left(z\right) - z + \frac{\log\!\left(2 \pi\right)}{2} + \sum_{k=1}^{n - 1} \frac{B_{2 k}}{2 k \left(2 k - 1\right) {z}^{2 k - 1}} + R_{n}\!\left(z\right)

z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, z \notin \left(-\infty, 0\right] \,\mathbin{\operatorname{and}}\, n \in \mathbb{Z}_{\ge 1}
Definitions:
Fungrim symbol Notation Short description
LogGammalogΓ ⁣(z)\log \Gamma\!\left(z\right) Logarithmic gamma function
Loglog ⁣(z)\log\!\left(z\right) Natural logarithm
ConstPiπ\pi The constant pi (3.14...)
Sumnf ⁣(n)\sum_{n} f\!\left(n\right) Sum
BernoulliBBnB_{n} Bernoulli number
Powab{a}^{b} Power
StirlingSeriesRemainderRn ⁣(z)R_{n}\!\left(z\right) Remainder term in the Stirling series for the logarithmic gamma function
CCC\mathbb{C} Complex numbers
OpenClosedInterval(a,b]\left(a, b\right] Open-closed interval
Infinity\infty Positive infinity
ZZGreaterEqualZn\mathbb{Z}_{\ge n} Integers greater than or equal to n
Source code for this entry:
Entry(ID("37a95a"),
    Formula(Equal(LogGamma(z), Add(Add(Add(Sub(Mul(Sub(z, Div(1, 2)), Log(z)), z), Div(Log(Mul(2, ConstPi)), 2)), Sum(Div(BernoulliB(Mul(2, k)), Mul(Mul(Mul(2, k), Sub(Mul(2, k), 1)), Pow(z, Sub(Mul(2, k), 1)))), Tuple(k, 1, Sub(n, 1)))), StirlingSeriesRemainder(n, z)))),
    Variables(z, n),
    Assumptions(And(Element(z, CC), NotElement(z, OpenClosedInterval(Neg(Infinity), 0)), Element(n, ZZGreaterEqual(1)))))

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

2019-08-21 11:44:15.926409 UTC