# Fungrim entry: 739819

$\sum_{k=1}^{n} \psi\!\left(\frac{k}{n}\right) \cos\!\left(\frac{2 \pi r k}{n}\right) = n \log\!\left(2 \sin\!\left(\frac{\pi r}{n}\right)\right)$
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) \cos\!\left(\frac{2 \pi r k}{n}\right) = n \log\!\left(2 \sin\!\left(\frac{\pi r}{n}\right)\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
Cos$\cos(z)$ Cosine
Pi$\pi$ The constant pi (3.14...)
Log$\log(z)$ Natural logarithm
Sin$\sin(z)$ Sine
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("739819"),
Formula(Equal(Sum(Mul(DigammaFunction(Div(k, n)), Cos(Div(Mul(Mul(Mul(2, Pi), r), k), n))), For(k, 1, n)), Mul(n, Log(Mul(2, Sin(Div(Mul(Pi, r), n))))))),
Variables(r, n),
Assumptions(And(Element(n, ZZGreaterEqual(2)), Element(r, Range(1, Sub(n, 1))))))

## Topics using this entry

Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2021-03-15 19:12:00.328586 UTC