Fungrim entry: f1e02b

\psi^{(m)}\!\left(z\right) = \frac{d^{m + 1}}{{d z}^{m + 1}} \left[\log \Gamma(z)\right]

Assumptions:

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

TeX:

\psi^{(m)}\!\left(z\right) = \frac{d^{m + 1}}{{d z}^{m + 1}} \left[\log \Gamma(z)\right]

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
`DigammaFunction`	$\psi\!\left(z\right)$	Digamma function
`ComplexBranchDerivative`	$\frac{d}{d z}\, f\!\left(z\right)$	Complex derivative, allowing branch cuts
`LogGamma`	$\log \Gamma(z)$	Logarithmic gamma 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("f1e02b"),
    Formula(Equal(DigammaFunction(z, m), ComplexBranchDerivative(Brackets(LogGamma(z)), For(z, z, Add(m, 1))))),
    Variables(m, z),
    Assumptions(And(Element(m, ZZGreaterEqual(0)), Element(z, CC), NotElement(z, ZZLessEqual(0)))))

Topics using this entry

Digamma function