ClausenCl

The Clausen functions : for even and for odd — the imaginary or real part of . Provided by `@enumeratio/analytic`.

ClausenCl(n, θ)the Clausen function of integer order and real .

Domain: Special functions

Details

is the classical Clausen function; , infinite at .
Parity alternates with the order (DLMF §25.12(ii), mpmath's `clsin` / `clcos`): even gives the odd, -periodic sine series, odd the even cosine series, so that or respectively. See PolyLog.
Special values: (Catalan's constant), , , , , . See DirichletEta, DirichletBeta.
peaks at with value , the Gieseking constant's companion; the volume of the ideal regular tetrahedron is .
Wolfram has no Clausen head: there it is spelled , which is how the oracle checks are phrased. Numerically, the polylogarithm's expansion at the unit circle (DLMF 25.12.12) with reduced into .

Examples

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/analytic/src/clausen.ts

mappedexternalmpmathpackages/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.