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:

$\beta (s)=4^{-s}\left(\zeta (s,{{1} \over {4}})-\zeta (s,{{3} \over {4}})\right)$.

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.

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:

$\beta (0)={\frac {1}{2}}$,

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

$\beta (2)\;=\;K$,

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 $\psi _{3}(1/4)$ 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.

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". MathWorld.