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.
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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:
Another equivalent definition, in terms of the Lerch transcendent, is:
which is once again valid for all complex values of s.
Functional equation
The functional equation extends the beta function to the left side of the complex plane Re(s)<0. It is given by
where Γ(s) is the gamma function.
Special values
Some special values include:
where G represents Catalan's constant, and
where ψ3(1 / 4) 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.
See also
References
- J. Spanier and K. B. Oldham, An Atlas of Functions, (1987) Hemisphere, New York.
- Weisstein, Eric W., "Dirichlet Beta Function" from MathWorld.
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