- Dirichlet beta function
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This article is about the Dirichlet beta function. For other beta functions, see Beta function (disambiguation).
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.
Contents
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.
s approximate value β(s) OEIS 1/5 0.5737108471859466493572665 1/4 0.5907230564424947318659591 1/3 0.6178550888488520660725389 1/2 0.6676914571896091766586909 A195103
1 0.7853981633974483096156608 A003881
2 0.9159655941772190150546035 A006752
3 0.9689461462593693804836348 A153071
4 0.9889445517411053361084226 A175572
5 0.9961578280770880640063194 A175571
6 0.9986852222184381354416008 A175570
7 0.9995545078905399094963465 8 0.9998499902468296563380671 9 0.9999496841872200898213589 10 0.9999831640261968774055407 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.
Categories:- Zeta and L-functions
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