Fungrim entry: af31ae

\frac{d}{d z}\, \left[\log G(z)\right] = \left(z - 1\right) \psi\!\left(z\right) - z + \frac{\log\!\left(2 \pi\right) + 1}{2}

Assumptions:

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \notin \{0, -1, \ldots\}

TeX:

\frac{d}{d z}\, \left[\log G(z)\right] = \left(z - 1\right) \psi\!\left(z\right) - z + \frac{\log\!\left(2 \pi\right) + 1}{2}

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \notin \{0, -1, \ldots\}

Definitions:

Fungrim symbol	Notation	Short description
`ComplexBranchDerivative`	$\frac{d}{d z}\, f\!\left(z\right)$	Complex derivative, allowing branch cuts
`LogBarnesG`	$\log G(z)$	Logarithmic Barnes G-function
`DigammaFunction`	$\psi\!\left(z\right)$	Digamma function
`Log`	$\log(z)$	Natural logarithm
`Pi`	$\pi$	The constant pi (3.14...)
`CC`	$\mathbb{C}$	Complex numbers
`ZZLessEqual`	$\mathbb{Z}_{\le n}$	Integers less than or equal to n

Source code for this entry:

Entry(ID("af31ae"),
    Formula(Equal(ComplexBranchDerivative(Brackets(LogBarnesG(z)), For(z, z)), Add(Sub(Mul(Sub(z, 1), DigammaFunction(z)), z), Div(Add(Log(Mul(2, Pi)), 1), 2)))),
    Variables(z),
    Assumptions(And(Element(z, CC), NotElement(z, ZZLessEqual(0)))))

Topics using this entry

Barnes G-function