---
tag: AVS6G54711
kind: concept
name: Clausen function
msc: ["33B30", "33E20"]
sources:
  - url: https://en.wikipedia.org/wiki/Clausen_function
same_as:
  wikidata: Q823290
  wikipedia: Clausen function
---

## Summary

The Clausen function $\operatorname{Cl}_2$ is the real function $\theta \mapsto -\int_0^\theta \log \abs{2 \sin(t/2)} \, dt$, equal to the [[Fourier series|G12KXDHPPS]] $\sum_{k \ge 1} \sin(k\theta) / k^2$.
It is the imaginary part of the [[dilogarithm|G0X0NWSDB7]] on the unit circle.
It appears in evaluations of log-sine integrals, in values such as Catalan's constant, and in volumes of hyperbolic polyhedra.

## Explanations

### L1 human

Take the dilogarithm series $\sum z^k / k^2$ and put $z$ on the unit circle, $z = e^{i\theta}$.
The real part is a quadratic polynomial in $\theta$ on $[0, 2\pi]$, but the imaginary part, $\sum \sin(k\theta)/k^2$, is a new function: the Clausen function.
Its graph looks like a tilted sine wave, odd and periodic with period $2\pi$, with its maximum at $\theta = \pi/3$.
At $\theta = \pi/2$ the series is $1 - \frac{1}{9} + \frac{1}{25} - \cdots$, which is Catalan's constant $G \approx 0.91597$.