Fungrim entry: c76eaf

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

Assumptions:

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

TeX:

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

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

Definitions:

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

Source code for this entry:

Entry(ID("c76eaf"),
    Formula(Equal(DigammaFunction(Add(1, z)), Add(Neg(ConstGamma), Sum(Mul(Mul(Pow(-1, Add(n, 1)), RiemannZeta(Add(n, 1))), Pow(z, n)), For(n, 1, Infinity))))),
    Variables(z),
    Assumptions(And(Element(z, CC), Less(Abs(z), 1))))

Topics using this entry

Digamma function