Fungrim entry: 547fcd

\psi^{(m)}\!\left(z\right) = -\int_{0}^{1} \frac{{t}^{z - 1}}{1 - t} \log^{m}\!\left(t\right) \, dt

Assumptions:

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \operatorname{Re}(z) > 0 \;\mathbin{\operatorname{and}}\; m \in \mathbb{Z}_{\ge 1}

TeX:

\psi^{(m)}\!\left(z\right) = -\int_{0}^{1} \frac{{t}^{z - 1}}{1 - t} \log^{m}\!\left(t\right) \, dt

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \operatorname{Re}(z) > 0 \;\mathbin{\operatorname{and}}\; m \in \mathbb{Z}_{\ge 1}

Definitions:

Fungrim symbol	Notation	Short description
`DigammaFunction`	$\psi\!\left(z\right)$	Digamma function
`Integral`	$\int_{a}^{b} f(x) \, dx$	Integral
`Pow`	${a}^{b}$	Power
`Log`	$\log(z)$	Natural logarithm
`CC`	$\mathbb{C}$	Complex numbers
`Re`	$\operatorname{Re}(z)$	Real part
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("547fcd"),
    Formula(Equal(DigammaFunction(z, m), Neg(Integral(Mul(Div(Pow(t, Sub(z, 1)), Sub(1, t)), Pow(Log(t), m)), For(t, 0, 1))))),
    Variables(z, m),
    Assumptions(And(Element(z, CC), Greater(Re(z), 0), Element(m, ZZGreaterEqual(1)))))

Topics using this entry

Digamma function