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Fungrim entry: 547fcd

ψ(m) ⁣(z)=01tz11tlogm ⁣(t)dt\psi^{(m)}\!\left(z\right) = -\int_{0}^{1} \frac{{t}^{z - 1}}{1 - t} \log^{m}\!\left(t\right) \, dt
Assumptions:zC  and  Re(z)>0  and  mZ1z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \operatorname{Re}(z) > 0 \;\mathbin{\operatorname{and}}\; m \in \mathbb{Z}_{\ge 1}
\psi^{(m)}\!\left(z\right) = -\int_{0}^{1} \frac{{t}^{z - 1}}{1 - t} \log^{m}\!\left(t\right) \, dt

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \operatorname{Re}(z) > 0 \;\mathbin{\operatorname{and}}\; m \in \mathbb{Z}_{\ge 1}
Fungrim symbol Notation Short description
DigammaFunctionψ ⁣(z)\psi\!\left(z\right) Digamma function
Integralabf(x)dx\int_{a}^{b} f(x) \, dx Integral
Powab{a}^{b} Power
Loglog(z)\log(z) Natural logarithm
CCC\mathbb{C} Complex numbers
ReRe(z)\operatorname{Re}(z) Real part
ZZGreaterEqualZn\mathbb{Z}_{\ge n} Integers greater than or equal to n
Source code for this entry:
    Formula(Equal(DigammaFunction(z, m), Neg(Integral(Mul(Div(Pow(t, Sub(z, 1)), Sub(1, t)), Pow(Log(t), m)), For(t, 0, 1))))),
    Variables(z, m),
    Assumptions(And(Element(z, CC), Greater(Re(z), 0), Element(m, ZZGreaterEqual(1)))))

Topics using this entry

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