Fungrim entry: 15d56a

\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:

m \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; x \in \mathbb{R}

TeX:

\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}

Definitions:

Fungrim symbol	Notation	Short description
`OpenInterval`	$\left(a, b\right)$	Open interval
`Infinity`	$\infty$	Positive infinity
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`DigammaFunction`	$\psi\!\left(z\right)$	Digamma function
`RR`	$\mathbb{R}$	Real numbers

Source code for this entry:

Entry(ID("15d56a"),
    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))))

Topics using this entry

Digamma function