Dirichlet beta function

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.

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 ψ₃(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

See also

• Hurwitz zeta function

References

• J. Spanier and K. B. Oldham, An Atlas of Functions, (1987) Hemisphere, New York.
• Weisstein, Eric W., "Dirichlet Beta Function" from MathWorld.