ClausenCl

The Clausen functions $\mathrm{Cl}_n(\theta)$: $\sum_{k\ge1} \sin(k\theta)/k^n$ for even $n$ and $\sum_{k\ge1} \cos(k\theta)/k^n$ for odd $n$ — the imaginary or real part of $\operatorname{Li}_n(e^{i\theta})$.

WikipediaClausen function↗MathWorldClausenFunction↗WikidataQ823290↗

ClausenCl(n, θ)the Clausen function $\mathrm{Cl}_n(\theta)$ of integer order $n \ge 1$ and real $\theta$.

Typed signatures

Type	Package	Applies
(integer, real) -> number	compute-engine	always

Domain: Special functions

Details

• $\mathrm{Cl}_2(\theta) = -\int_0^\theta \ln\left|2\sin\tfrac{t}{2}\right|\,dt$ is the classical Clausen function; $\mathrm{Cl}_1(\theta) = -\ln|2\sin(\theta/2)|$, infinite at $\theta \equiv 0$.
• Parity alternates with the order (DLMF §25.12(ii), mpmath's clsin / clcos): even $n$ gives the odd, $2\pi$-periodic sine series, odd $n$ the even cosine series, so that $\mathrm{Cl}_n(\theta) = \operatorname{Im}\operatorname{Li}_n(e^{i\theta})$ or $\operatorname{Re}\operatorname{Li}_n(e^{i\theta})$ respectively. See PolyLog.
• Special values: $\mathrm{Cl}_2(\pi/2) = G$ (Catalan's constant), $\mathrm{Cl}_{2m}(0) = \mathrm{Cl}_{2m}(\pi) = 0$, $\mathrm{Cl}_{2m+1}(0) = \zeta(2m+1)$, $\mathrm{Cl}_{2m+1}(\pi) = -\eta(2m+1)$, $\mathrm{Cl}_{2m}(\pi/2) = \beta(2m)$, $\mathrm{Cl}_{2m+1}(\pi/2) = -2^{-(2m+1)}\eta(2m+1)$. See DirichletEta, DirichletBeta.
• $\mathrm{Cl}_2$ peaks at $\theta = \pi/3$ with value $1.01494\ldots$, the Gieseking constant's companion; the volume of the ideal regular tetrahedron is $3\,\mathrm{Cl}_2(\pi/3)/2$.
• Wolfram has no Clausen head: there it is spelled $\operatorname{Im}[\mathrm{PolyLog}[n, e^{i\theta}]]$, which is how the oracle checks are phrased. Numerically, the polylogarithm's expansion at the unit circle (DLMF 25.12.12) with $\theta$ reduced into $(-\pi, \pi]$.

Plot

Examples

2 more kept as data, checked against the oracles

Basic #

$\mathrm{Cl}_2(\pi/2) = G$, Catalan's constant

live ↗

In$\mathrm{ClausenCl}(2, \frac{π}{2})$Out$\operatorname{G}$

Odd orders are cosine series: $\mathrm{Cl}_3(0) = \zeta(3)$

live ↗

In$\mathrm{ClausenCl}(3, 0)$Out$ζ(3)$

$\mathrm{Cl}_2(\pi/3) = 1.01494\ldots$, the maximum of $\mathrm{Cl}_2$

live ↗

In$\mathrm{ClausenCl}(2, 1.047\,197\,551\,196\,597\,6)$Out$1.014\,941\,606\,409\,653\,3$

Properties #

$\mathrm{Cl}_2(\pi) = 0$: the sine series vanishes at multiples of $\pi$

live ↗

In$\mathrm{ClausenCl}(2, π)$Out$0$

$\mathrm{Cl}_3(\pi) = -\eta(3) = -\tfrac34\zeta(3)$

live ↗

In$\mathrm{ClausenCl}(3, π)$Out$-\frac{3ζ(3)}{4}$

$\mathrm{Cl}_4(\pi/2) = \beta(4)$, which has no closed form of its own

live ↗

In$\mathrm{ClausenCl}(4, \frac{π}{2})$Out$\mathrm{DirichletBeta}(4)$

Possible issues #

$\mathrm{Cl}_1(\theta) = -\ln|2\sin(\theta/2)|$ diverges at $\theta = 0$

live ↗

In$\mathrm{ClausenCl}(1, 0)$Out$∞$

About

Examples	9
Checked against	Wolfram Language 7 agree2 differmpmath 8 agree1 error
Crosswalk	WikipediaMathWorldWikidata

Implementation

referenceenginenotatio

Cl_n cuts Li_n(e^{iθ}) in two by parity: the even orders are the sine series (its imaginary part), the odd orders the cosine series (its real part).

nativeenginetypescriptpackages/ce-patches/src/compute-engine/numerics/clausen.ts

mappedwolframpackages/oracle/src/mappings.ts

No built-in Clausen head; the Wolfram Function Repository's own ResourceFunction["ClausenCl"] matches ours -- Catalan at (2, π/2), 0 at (2, π), -3/4 ζ(3) at (3, π), DirichletBeta[4] at (4, π/2), 1.0149416064096535 at (2, 1.047…). Needs network access to the Function Repository to run. Known quirk: at (3, 0) it returns an unevaluated If[OddQ, Zeta[3], 0] -- a bug in the repository function, not ours -- so that example's oracle verdict should read as inconclusive, not disagree.

mappedmpmathpackages/oracle/src/mappings.ts

clsin(n, θ) for even n, clcos(n, θ) for odd n. Wolfram has no head; Im/Re PolyLog[n, E^(I θ)] instead.

See also: PolyLog, DirichletBeta, DirichletEta, Zeta