# Fungrim entry: f50c74

$G'(n) = \begin{cases} 0, & n < 0\\1, & n = 0\\\frac{1}{2} \left(\log\!\left(2 \pi\right) - 1\right), & n = 1\\G(n) \left(\frac{1}{2} \log\!\left(2 \pi\right) + \left(n - 1\right) \left(H_{n - 2} - \gamma - 1\right) + \frac{1}{2}\right), & n \ge 2\\ \end{cases}$
Assumptions:$n \in \mathbb{Z}$
TeX:
G'(n) = \begin{cases} 0, & n < 0\\1, & n = 0\\\frac{1}{2} \left(\log\!\left(2 \pi\right) - 1\right), & n = 1\\G(n) \left(\frac{1}{2} \log\!\left(2 \pi\right) + \left(n - 1\right) \left(H_{n - 2} - \gamma - 1\right) + \frac{1}{2}\right), & n \ge 2\\ \end{cases}

n \in \mathbb{Z}
Definitions:
Fungrim symbol Notation Short description
ComplexDerivative$\frac{d}{d z}\, f\!\left(z\right)$ Complex derivative
BarnesG$G(z)$ Barnes G-function
Log$\log(z)$ Natural logarithm
Pi$\pi$ The constant pi (3.14...)
ConstGamma$\gamma$ The constant gamma (0.577...)
ZZ$\mathbb{Z}$ Integers
Source code for this entry:
Entry(ID("f50c74"),
Formula(Equal(ComplexDerivative(BarnesG(z), For(z, n)), Cases(Tuple(0, Less(n, 0)), Tuple(1, Equal(n, 0)), Tuple(Mul(Div(1, 2), Sub(Log(Mul(2, Pi)), 1)), Equal(n, 1)), Tuple(Mul(BarnesG(n), Add(Add(Mul(Div(1, 2), Log(Mul(2, Pi))), Mul(Sub(n, 1), Sub(Sub(HarmonicNumber(Sub(n, 2)), ConstGamma), 1))), Div(1, 2))), GreaterEqual(n, 2))))),
Variables(n),
Assumptions(Element(n, ZZ)))

## 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