Fungrim entry: c89abc

\psi^{(m)}\!\left(z\right) = {\left(-1\right)}^{m + 1} \int_{0}^{\infty} \frac{{t}^{m} {e}^{-z t}}{1 - {e}^{-t}} \, 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) = {\left(-1\right)}^{m + 1} \int_{0}^{\infty} \frac{{t}^{m} {e}^{-z t}}{1 - {e}^{-t}} \, 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
`Pow`	${a}^{b}$	Power
`Integral`	$\int_{a}^{b} f(x) \, dx$	Integral
`Exp`	${e}^{z}$	Exponential function
`Infinity`	$\infty$	Positive infinity
`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("c89abc"),
    Formula(Equal(DigammaFunction(z, m), Mul(Pow(-1, Add(m, 1)), Integral(Div(Mul(Pow(t, m), Exp(Neg(Mul(z, t)))), Sub(1, Exp(Neg(t)))), For(t, 0, Infinity))))),
    Variables(z, m),
    Assumptions(And(Element(z, CC), Greater(Re(z), 0), Element(m, ZZGreaterEqual(1)))))

Topics using this entry

Digamma function