DirichletEta

The Dirichlet eta function , the alternating zeta — entire, with . Provided by `@enumeratio/analytic`.

DirichletEta(s)the Dirichlet eta function .

Domain: Special functions

Details

: the alternating series converges for , and the factor cancels 's pole, so is entire with .
Integer values follow from Zeta's: , , , and stays in terms of .
Zeros: the nontrivial zeros of , plus those of on the line .
Plain evaluation reduces exact integer through ; other exact stays symbolic (as in Wolfram) until N() or a floating-point argument. Numerically, complex is supported; within of the alternating series is summed directly (Euler transform, via LerchPhi) so no pole is cancelled.

Examples

referenceenginenotatio

The defining identity — except at s = 1, where the native kernel sums the alternating series instead.

nativeenginetypescriptpackages/analytic/src/dirichlet.ts

mappedexternalwolfram / mpmathpackages/oracle/src/mappings.ts

DirichletEta[s]; mpmath.altzeta(s).