Definition
The Dirichlet beta function is defined as
or, equivalently,
In each case, it is assumed that Re(s) > 0.
Alternatively, the following definition, in terms of the Hurwitz zeta function, is valid in the whole complex s-plane:
- proof
Another equivalent definition, in terms of the Lerch transcendent, is:
which is once again valid for all complex values of s.
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