Fungrim entry: 4e3853

\sum_{n=1}^{\infty} \psi\!\left(n\right) {z}^{n} = \frac{z \left(\gamma + \log\!\left(1 - z\right)\right)}{z - 1}

Assumptions:

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \left|z\right| < 1

TeX:

\sum_{n=1}^{\infty} \psi\!\left(n\right) {z}^{n} = \frac{z \left(\gamma + \log\!\left(1 - z\right)\right)}{z - 1}

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \left|z\right| < 1

Definitions:

Fungrim symbol	Notation	Short description
`Sum`	$\sum_{n} f(n)$	Sum
`DigammaFunction`	$\psi\!\left(z\right)$	Digamma function
`Pow`	${a}^{b}$	Power
`Infinity`	$\infty$	Positive infinity
`ConstGamma`	$\gamma$	The constant gamma (0.577...)
`Log`	$\log(z)$	Natural logarithm
`CC`	$\mathbb{C}$	Complex numbers
`Abs`	$\left\|z\right\|$	Absolute value

Source code for this entry:

Entry(ID("4e3853"),
    Formula(Equal(Sum(Mul(DigammaFunction(n), Pow(z, n)), For(n, 1, Infinity)), Div(Mul(z, Add(ConstGamma, Log(Sub(1, z)))), Sub(z, 1)))),
    Variables(z),
    Assumptions(And(Element(z, CC), Less(Abs(z), 1))))

Topics using this entry

Digamma function