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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.

Definition

The Dirichlet beta function is defined as

$$\beta(s) = \sum_{n=0}^\infty \frac{(-1)^n} {(2n+1)^s},$$

or, equivalently,

$$\beta(s) = \frac{1}{\Gamma(s)}\int_0^{\infty}\frac{x^{s-1}e^{-x}}{1 + e^{-2x}}\,dx.$$

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:

$$\beta(s) = 4^{-s} \left( \zeta\left(s,{1 \over 4}\right)-\zeta\left( s, {3 \over 4}\right) \right).$$

Another equivalent definition, in terms of the Lerch transcendent, is:

$$\beta(s) = 2^{-s} \Phi\left(-1,s,{{1} \over {2}}\right),$$

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

$$\beta(s)=\left(\frac{\pi}{2}\right)^{s-1} \Gamma(1-s) \cos \frac{\pi s}{2}\,\beta(1-s)$$

where Γ(s) is the gamma function.
Special values

Some special values include:

$$\beta(0)= \frac{1}{2},$$

$$\beta(1)\;=\;\tan^{-1}(1)\;=\;\frac{\pi}{4},$$

$$\beta(2)\;=\;G,$$

where G represents Catalan's constant, and

$$\beta(3)\;=\;\frac{\pi^3}{32},$$

$$\beta(4)\;=\;\frac{1}{768}(\psi_3(\frac{1}{4})-8\pi^4),$$

$$\beta(5)\;=\;\frac{5\pi^5}{1536},$$

$$\beta(7)\;=\;\frac{61\pi^7}{184320},$$

where $$\psi_3(1/4)$$ in the above is an example of the polygamma function. More generally, for any positive integer k:

$$\beta(2k+1)={{{({-1})^k}{E_{2k}}{\pi^{2k+1}} \over {4^{k+1}}(2k)!}},$$

where $$\!\ E_{n}$$ represent the Euler numbers. For integer k ≥ 0, this extends to:

$$\beta(-k)={{E_{k}} \over {2}}.$$

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

Hurwitz zeta function

References

Glasser, M. L. (1972). "The evaluation of lattice sums. I. Analytic procedures". J. Math. Phys. 14: 409. doi:10.1063/1.1666331.
J. Spanier and K. B. Oldham, An Atlas of Functions, (1987) Hemisphere, New York.
Weisstein, Eric W., "Dirichlet Beta Function" from MathWorld.