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Fungrim entry: 9f32fe

ψ ⁣(z+n)=ψ ⁣(z)+k=0n11z+k\psi\!\left(z + n\right) = \psi\!\left(z\right) + \sum_{k=0}^{n - 1} \frac{1}{z + k}
Assumptions:zC  and  z{0,1,}  and  nZ0z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \notin \{0, -1, \ldots\} \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z}_{\ge 0}
\psi\!\left(z + n\right) = \psi\!\left(z\right) + \sum_{k=0}^{n - 1} \frac{1}{z + k}

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \notin \{0, -1, \ldots\} \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z}_{\ge 0}
Fungrim symbol Notation Short description
DigammaFunctionψ ⁣(z)\psi\!\left(z\right) Digamma function
Sumnf(n)\sum_{n} f(n) Sum
CCC\mathbb{C} Complex numbers
ZZLessEqualZn\mathbb{Z}_{\le n} Integers less than or equal to n
ZZGreaterEqualZn\mathbb{Z}_{\ge n} Integers greater than or equal to n
Source code for this entry:
    Formula(Equal(DigammaFunction(Add(z, n)), Add(DigammaFunction(z), Sum(Div(1, Add(z, k)), For(k, 0, Sub(n, 1)))))),
    Variables(z, n),
    Assumptions(And(Element(z, CC), NotElement(z, ZZLessEqual(0)), Element(n, ZZGreaterEqual(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