DirichletBeta

The Dirichlet beta function , the L-function of the nontrivial character mod 4 — entire, with and (Catalan). Provided by `@enumeratio/analytic`.

WikipediaDirichlet beta function↗MathWorldDirichletBetaFunction↗WikidataQ1227706↗Wolfram LanguageDirichletBeta↗

DirichletBeta(s)the Dirichlet beta function .

Domain: Special functions

Details

• in terms of HurwitzZeta, which is how it is evaluated numerically (complex included).
• Odd positive integers have closed forms in and the Euler numbers: — (Leibniz), , .
• Even positive integers do not: is Catalan's constant, and stay symbolic.
• Nonpositive integers are Euler numbers: (, , ), and .
• Functional equation ; entire, no poles.

Examples

Implementation

referenceenginenotatio

nativeenginetypescriptpackages/analytic/src/dirichlet.ts

mappedexternalwolfram / mpmathpackages/oracle/src/mappings.ts

DirichletBeta[s]; mpmath.dirichlet(s, [0, 1, 0, -1]).

See also: DirichletEta, HurwitzZeta, ClausenCl, Zeta