# 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

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

2021-03-15 19:12:00.328586 UTC