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Fungrim entry: 15d56a

(x(0,)  and  mZ1)        ψ(m) ⁣(x){(0,),m odd(,0),m even\left(x \in \left(0, \infty\right) \;\mathbin{\operatorname{and}}\; m \in \mathbb{Z}_{\ge 1}\right) \;\implies\; \psi^{(m)}\!\left(x\right) \in \begin{cases} \left(0, \infty\right), & m \text{ odd}\\\left(-\infty, 0\right), & m \text{ even}\\ \end{cases}
Assumptions:mZ1  and  xRm \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; x \in \mathbb{R}
\left(x \in \left(0, \infty\right) \;\mathbin{\operatorname{and}}\; m \in \mathbb{Z}_{\ge 1}\right) \;\implies\; \psi^{(m)}\!\left(x\right) \in \begin{cases} \left(0, \infty\right), & m \text{ odd}\\\left(-\infty, 0\right), & m \text{ even}\\ \end{cases}

m \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; x \in \mathbb{R}
Fungrim symbol Notation Short description
OpenInterval(a,b)\left(a, b\right) Open interval
Infinity\infty Positive infinity
ZZGreaterEqualZn\mathbb{Z}_{\ge n} Integers greater than or equal to n
DigammaFunctionψ ⁣(z)\psi\!\left(z\right) Digamma function
RRR\mathbb{R} Real numbers
Source code for this entry:
    Formula(Implies(And(Element(x, OpenInterval(0, Infinity)), Element(m, ZZGreaterEqual(1))), Element(DigammaFunction(x, m), Cases(Tuple(OpenInterval(0, Infinity), Odd(m)), Tuple(OpenInterval(Neg(Infinity), 0), Even(m)))))),
    Variables(x, m),
    Assumptions(And(Element(m, ZZGreaterEqual(1)), Element(x, RR))))

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2021-03-15 19:12:00.328586 UTC