definition of Wikipedia
In mathematics, the Dirichlet beta function (also known as the Catalan beta function) is a special function, closely related to the Riemann zeta function. It is a particular Dirichlet L-function, the L-function for the alternating character of period four.
The Dirichlet beta function is defined as
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:
Another equivalent definition, in terms of the Lerch transcendent, is:
which is once again valid for all complex values of s.
where Γ(s) is the gamma function.
Some special values include:
where G represents Catalan's constant, and
where in the above is an example of the polygamma function. More generally, for any positive integer k:
where represent the Euler numbers. For integer k ≥ 0, this extends to:
Hence, the function vanishes for all odd negative integral values of the argument.
|s||approximate value β(s)||OEIS|
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