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Fungrim entry: c89abc

ψ(m) ⁣(z)=(1)m+10tmezt1etdt\psi^{(m)}\!\left(z\right) = {\left(-1\right)}^{m + 1} \int_{0}^{\infty} \frac{{t}^{m} {e}^{-z t}}{1 - {e}^{-t}} \, 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) = {\left(-1\right)}^{m + 1} \int_{0}^{\infty} \frac{{t}^{m} {e}^{-z t}}{1 - {e}^{-t}} \, 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
Powab{a}^{b} Power
Integralabf(x)dx\int_{a}^{b} f(x) \, dx Integral
Expez{e}^{z} Exponential function
Infinity\infty Positive infinity
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), Mul(Pow(-1, Add(m, 1)), Integral(Div(Mul(Pow(t, m), Exp(Neg(Mul(z, t)))), Sub(1, Exp(Neg(t)))), For(t, 0, Infinity))))),
    Variables(z, m),
    Assumptions(And(Element(z, CC), Greater(Re(z), 0), Element(m, ZZGreaterEqual(1)))))

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2021-03-15 19:12:00.328586 UTC