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Fungrim entry: af31ae

ddz[logG(z)]=(z1)ψ ⁣(z)z+log ⁣(2π)+12\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:zC  and  z{0,1,}z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \notin \{0, -1, \ldots\}
\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\}
Fungrim symbol Notation Short description
ComplexBranchDerivativeddzf ⁣(z)\frac{d}{d z}\, f\!\left(z\right) Complex derivative, allowing branch cuts
LogBarnesGlogG(z)\log G(z) Logarithmic Barnes G-function
DigammaFunctionψ ⁣(z)\psi\!\left(z\right) Digamma function
Loglog(z)\log(z) Natural logarithm
Piπ\pi The constant pi (3.14...)
CCC\mathbb{C} Complex numbers
ZZLessEqualZn\mathbb{Z}_{\le n} Integers less than or equal to n
Source code for this entry:
    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)))),
    Assumptions(And(Element(z, CC), NotElement(z, ZZLessEqual(0)))))

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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