Dirichlet beta function

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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.

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[edit] 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(s,{{1} \over {4}})-\zeta(s, {{3} \over {4}}) \right).

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

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

which is once again valid for all complex values of s.

[edit] 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.

[edit] Special values

Some special values include:

\beta(0)= \frac{1}{2} ,
\beta(1)\;=\;\tan^{-1}(1)\;=\;\frac{\pi}{4} ,
\beta(2)\;=\;K,

where K 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.

[edit] See also

[edit] References

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