Fungrim entry: 37a95a

\log \Gamma(z) = \left(z - \frac{1}{2}\right) \log(z) - 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:

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \notin \left(-\infty, 0\right] \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z}_{\ge 1}

TeX:

\log \Gamma(z) = \left(z - \frac{1}{2}\right) \log(z) - 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
`LogGamma`	$\log \Gamma(z)$	Logarithmic gamma function
`Log`	$\log(z)$	Natural logarithm
`Pi`	$\pi$	The constant pi (3.14...)
`Sum`	$\sum_{n} f(n)$	Sum
`BernoulliB`	$B_{n}$	Bernoulli number
`Pow`	${a}^{b}$	Power
`StirlingSeriesRemainder`	$R_{n}\!\left(z\right)$	Remainder term in the Stirling series for the logarithmic gamma function
`CC`	$\mathbb{C}$	Complex numbers
`OpenClosedInterval`	$\left(a, b\right]$	Open-closed interval
`Infinity`	$\infty$	Positive infinity
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	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, Pi)), 2)), Sum(Div(BernoulliB(Mul(2, k)), Mul(Mul(Mul(2, k), Sub(Mul(2, k), 1)), Pow(z, Sub(Mul(2, k), 1)))), For(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)))))

Topics using this entry

Gamma function