TCplx DirichletBeta(const TCplx &Z);
Dirichlet Beta function.
| # | Name | Type | Description |
|---|---|---|---|
| 1 | Z | const TCplx & | complex value at which the Dirichlet Beta function is evaluated. |
Returns: the Dirichlet Beta function evaluated at the complex point Z.
For Re(z) > 0 the Dirichlet beta function is given by the alternating series
beta(z) = sum_(n=0)^(inf)(-1)^n(2n+1)^(-z) ,
and is extended to the whole complex plane by the reflection relation
beta(1-z) = (2/pi)^zsin(1/2pi z) Gamma(z) beta(z) ,
where Gamma denotes the Math387::Gamma function. The value is computed from a 65-term Euler-sum acceleration; for Re(z) < 1/2 the reflection formula above is applied. Known values: beta(1)=pi/4, beta(2)=G (Catalan's constant), beta(3)=pi^3/32.
Domain: any complex Z. Defined behavior: at the negative odd integers z=-1,-3,-5,... (the trivial zeros) the result is 0; at z=0 it is 1/2. On the real axis the imaginary part of the result is zero.