Dirichlet beta function

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.



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},

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:

\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 OEISA195103
1 0.7853981633974483096156608 OEISA003881
2 0.9159655941772190150546035 OEISA006752
3 0.9689461462593693804836348 OEISA153071
4 0.9889445517411053361084226 OEISA175572
5 0.9961578280770880640063194 OEISA175571
6 0.9986852222184381354416008 OEISA175570
7 0.9995545078905399094963465
8 0.9998499902468296563380671
9 0.9999496841872200898213589
10 0.9999831640261968774055407

See also


Wikimedia Foundation. 2010.

Игры ⚽ Нужен реферат?

Look at other dictionaries:

  • Beta function — This article is about Euler beta function. For other uses, see Beta function (disambiguation). In mathematics, the beta function, also called the Euler integral of the first kind, is a special function defined by for The beta function was studied …   Wikipedia

  • Dirichlet character — In number theory, Dirichlet characters are certain arithmetic functions which arise from completely multiplicative characters on the units of . Dirichlet characters are used to define Dirichlet L functions, which are meromorphic functions with a… …   Wikipedia

  • Dirichlet distribution — Several images of the probability density of the Dirichlet distribution when K=3 for various parameter vectors α. Clockwise from top left: α=(6, 2, 2), (3, 7, 5), (6, 2, 6), (2, 3, 4). In probability and… …   Wikipedia

  • Beta distribution — Probability distribution name =Beta| type =density pdf cdf parameters =alpha > 0 shape (real) eta > 0 shape (real) support =x in [0; 1] ! pdf =frac{x^{alpha 1}(1 x)^{eta 1 {mathrm{B}(alpha,eta)}! cdf =I x(alpha,eta)! mean… …   Wikipedia

  • Función beta de Dirichlet — Este artículo trata sobre función beta de Dirichlet. Para otras funciones beta, véase Función beta (desambiguación). En matemática, la función beta de Dirichlet (también conocida como la función beta de Catalan) es una función especial,… …   Wikipedia Español

  • Dirichletsche Beta-Funktion — Die dirichletsche Beta Funktion, geschrieben β(s), ist eine spezielle Funktion; sie ist verwandt mit der riemannschen Zeta Funktion. Benannt wurde sie nach dem deutschen Mathematiker Peter Gustav Lejeune Dirichlet (1805−1859). Inhaltsverzeichnis… …   Deutsch Wikipedia

  • Dirichlet process — In probability theory, a Dirichlet process is a stochastic process that can be thought of as a probability distribution whose domain is itself a random distribution. That is, given a Dirichlet process , where H (the base distribution) is an… …   Wikipedia

  • Beta-binomial model — In empirical Bayes methods, the Beta binomial model is an analytic model where the likelihood function L(x| heta) is specifed by a binomial distribution:L(x| heta) = operatorname{Bin}(x, heta),::: = {nchoose x} heta^x(1 heta)^{n x},and the… …   Wikipedia

  • Generalized Dirichlet distribution — In statistics, the generalized Dirichlet distribution (GD) is a generalization of the Dirichlet distribution with a more general covariance structure and twice the number of parameters. Random variables with a GD distribution are neutral [R. J.… …   Wikipedia

  • Loi bêta — Beta Densité de probabilité / Fonction de masse Fonction de répartition …   Wikipédia en Français

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”