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

Barnes G-function