LerchPhi

The Lerch transcendent — a common generalization of the polylogarithm and the Hurwitz zeta. Provided by `@enumeratio/analytic`.

WikipediaLerch transcendentMathWorldLerchTranscendentWikidataQ1546155DLMF25.14.E1FungrimLerchPhiFungrim entryd6fbc8Wolfram LanguageLerchPhi!

LerchPhi(z, s, a)the Lerch transcendent .

Domain: Special functions

Details

At it is the Hurwitz zeta: , so . See HurwitzZeta.
It carries the polylogarithm: — e.g. .
At it is the Dirichlet eta: , so .
And the Dirichlet beta / Catalan's constant:
— the geometric series (and its continuation), independent of , for all .
Convergence: the series converges for (any , ), and on only for ; elsewhere it is defined by analytic continuation in .
Poles at , from the singular term, as for HurwitzZeta.
Numeric evaluation sums the series directly for (geometric convergence), routes through the Euler–Maclaurin Hurwitz kernel, and sums real (the rim included) by a van Wijngaarden Euler transform, so the alternating and Catalan cases reach machine precision. is left unevaluated except where a closed form applies (e.g. ).

Examples

referenceenginenotatio

The defining series, converging for |z| < 1 (and for |z| = 1 with Re(s) > 1); ζ(s, a) is its z = 1 edge and Li_s(z) its a = 1 slice.

nativeenginetypescriptpackages/analytic/src/lerch.ts

The direct series, which covers |z| ≤ 1; z = 1 hands off to HurwitzZeta and inherits its closed forms.

compiledgpuwgslpackages/analytic/src/shader.ts:zetaWGSL

Produces a colour per pixel — complex Φ, domain-coloured.

mappedexternalwolfram / mpmathpackages/oracle/src/mappings.ts

LerchPhi[z, s, a]; mpmath.lerchphi(z, s, a).