Dirichlet beta function

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