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

:$\beta(s) = \sum_^\infty \frac ,$

or, equivalently,

:$\beta(s) = \frac\int_0^\frac\,dx.$

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:Dirichlet Beta – Hurwitz zeta relation
Engineering Mathematics

:$\beta(s) = 4^ \left( \zeta\left(s,\right)-\zeta\left( s, \right) \right).$

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

:$\beta(s) = 2^ \Phi\left(-1,s,\right),$

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

The Dirichlet beta function can also be written in terms of the Polylogarithm function:

:$\beta(s) = \frac \left(\text_s(-i)-\text_s(i)\right).$

Also the series representation of Dirichlet beta function can be formed in terms of the polygamma function

:\beta(s) =\frac \sum_^\infty\frac=\frac1\left psi^\left(\frac\right)-\psi^\left(\frac\right)\right/math>

but this formula is only valid at positive integer values of $s$.

Euler product formula

It is also the simplest example of a series non-directly related to $\zeta(s)$ which can also be factorized as an Euler product, thus leading to the idea of Dirichlet character defining the exact set of Dirichlet series having a factorization over the prime numbers.

At least for Re(''s'') ≥ 1:

:$\beta(s) = \prod_ \frac \prod_ \frac$

where are the primes of the form (5,13,17,...) and are the primes of the form (3,7,11,...). This can be written compactly as

:$\beta(s) = \prod_ \frac.$

Functional equation

The functional equation extends the beta function to the left side of the complex plane Re(''s'') ≤ 0. It is given by
:$\beta(1-s)=\left(\frac\right)^\sin\left(\fracs\right)\Gamma(s)\beta(s)$
where Γ(''s'') is the gamma function.

Special values

Some special values include:

:$\beta(0)= \frac,$

:$\beta(1)\;=\;\arctan(1)\;=\;\frac,$

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

where ''G'' represents Catalan's constant, and

:$\beta(3)\;=\;\frac,$

:$\beta(4)\;=\;\frac\left(\psi_3\left(\frac\right)-8\pi^4\right),$

:$\beta(5)\;=\;\frac,$

:$\beta(7)\;=\;\frac,$

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

:$\beta(2k+1)=,$

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

:$\beta(-k)=.$

Hence, the function vanishes for all odd negative integral values of the argument.

For every positive integer ''k'':
:$\beta(2k)=\frac\sum_^\infty\left(\left(\sum_^\binom\frac\right)-\frac\right)\frac^,$
where $A_$ is the Euler zigzag number.

Also it was derived by Malmsten in 1842 that

:$\beta'(1)=\sum_^(-1)^\frac \,=\,\frac\big(\gamma-\ln\pi) +\pi\ln\Gamma\left(\frac\right)$

There are zeros at -1; -3; -5; -7 etc.

See also

* Hurwitz zeta function
*Dirichlet eta function
*Polylogarithm

References

*
* J. Spanier and K. B. Oldham, ''An Atlas of Functions'', (1987) Hemisphere, New York.
* {{MathWorld, title=Dirichlet Beta Function, urlname=DirichletBetaFunction

Zeta and L-functions