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Fungrim entry: f1e02b

ψ(m) ⁣(z)=dm+1dzm+1[logΓ(z)]\psi^{(m)}\!\left(z\right) = \frac{d^{m + 1}}{{d z}^{m + 1}} \left[\log \Gamma(z)\right]
Assumptions:mZ0andzCandz{0,1,}m \in \mathbb{Z}_{\ge 0} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, z \notin \{0, -1, \ldots\}
TeX:
\psi^{(m)}\!\left(z\right) = \frac{d^{m + 1}}{{d z}^{m + 1}} \left[\log \Gamma(z)\right]

m \in \mathbb{Z}_{\ge 0} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, z \notin \{0, -1, \ldots\}
Definitions:
Fungrim symbol Notation Short description
DigammaFunctionψ ⁣(z)\psi\!\left(z\right) Digamma function
ComplexBranchDerivativeddzf ⁣(z)\frac{d}{d z}\, f\!\left(z\right) Complex derivative, allowing branch cuts
LogGammalogΓ(z)\log \Gamma(z) Logarithmic gamma function
ZZGreaterEqualZn\mathbb{Z}_{\ge n} Integers greater than or equal to n
CCC\mathbb{C} Complex numbers
ZZLessEqualZn\mathbb{Z}_{\le n} Integers less than or equal to n
Source code for this entry:
Entry(ID("f1e02b"),
    Formula(Equal(DigammaFunction(z, m), ComplexBranchDerivative(Brackets(LogGamma(z)), For(z, z, Add(m, 1))))),
    Variables(m, z),
    Assumptions(And(Element(m, ZZGreaterEqual(0)), Element(z, CC), NotElement(z, ZZLessEqual(0)))))

Topics using this entry

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

2020-01-31 18:09:28.494564 UTC