Fungrim entry: 458a97

\sum_{k=1}^{n} \psi\!\left(\frac{k}{n}\right) {e}^{2 \pi r k i / n} = n \log\!\left(1 - {e}^{2 \pi r i / n}\right)

Assumptions:

n \in \mathbb{Z}_{\ge 2} \;\mathbin{\operatorname{and}}\; r \in \{1, 2, \ldots, n - 1\}

TeX:

\sum_{k=1}^{n} \psi\!\left(\frac{k}{n}\right) {e}^{2 \pi r k i / n} = n \log\!\left(1 - {e}^{2 \pi r i / n}\right)

n \in \mathbb{Z}_{\ge 2} \;\mathbin{\operatorname{and}}\; r \in \{1, 2, \ldots, n - 1\}

Definitions:

Fungrim symbol	Notation	Short description
`Sum`	$\sum_{n} f(n)$	Sum
`DigammaFunction`	$\psi\!\left(z\right)$	Digamma function
`Exp`	${e}^{z}$	Exponential function
`Pi`	$\pi$	The constant pi (3.14...)
`ConstI`	$i$	Imaginary unit
`Log`	$\log(z)$	Natural logarithm
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`Range`	$\{a, a + 1, \ldots, b\}$	Integers between given endpoints

Source code for this entry:

Entry(ID("458a97"),
    Formula(Equal(Sum(Mul(DigammaFunction(Div(k, n)), Exp(Div(Mul(Mul(Mul(Mul(2, Pi), r), k), ConstI), n))), For(k, 1, n)), Mul(n, Log(Sub(1, Exp(Div(Mul(Mul(Mul(2, Pi), r), ConstI), n))))))),
    Variables(r, n),
    Assumptions(And(Element(n, ZZGreaterEqual(2)), Element(r, Range(1, Sub(n, 1))))))

Topics using this entry

Digamma function