Fungrim entry: 21f4f9

\frac{d^{n}}{{d z}^{n}} \left[\psi^{(m)}\!\left(z\right)\right] = \psi^{(m + n)}\!\left(z\right)

Assumptions:

n \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; m \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \notin \{0, -1, \ldots\}

TeX:

\frac{d^{n}}{{d z}^{n}} \left[\psi^{(m)}\!\left(z\right)\right] = \psi^{(m + n)}\!\left(z\right)

n \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; m \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \notin \{0, -1, \ldots\}

Definitions:

Fungrim symbol	Notation	Short description
`ComplexDerivative`	$\frac{d}{d z}\, f\!\left(z\right)$	Complex derivative
`DigammaFunction`	$\psi\!\left(z\right)$	Digamma function
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`CC`	$\mathbb{C}$	Complex numbers
`ZZLessEqual`	$\mathbb{Z}_{\le n}$	Integers less than or equal to n

Source code for this entry:

Entry(ID("21f4f9"),
    Formula(Equal(ComplexDerivative(Brackets(DigammaFunction(z, m)), For(z, z, n)), DigammaFunction(z, Add(m, n)))),
    Variables(n, m, z),
    Assumptions(And(Element(n, ZZGreaterEqual(0)), Element(m, ZZGreaterEqual(0)), Element(z, CC), NotElement(z, ZZLessEqual(0)))))

Topics using this entry

Digamma function