Poisson kernel

Template:Otheruses4 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

$\beta (s)=\sum _{n=0}^{\mathrm{\infty }}\frac{(-1{)}^n}{(2n+1{)}^s},$

or, equivalently,

$\beta (s)=\frac{1}{\mathrm{\Gamma }(s)}\underset{0}{\overset{\mathrm{\infty }}{\int }}\frac{x^{s-1}{e}^{-x}}{1+e^{-2x}}dx.$

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:

$\beta (s)=4^{-s}(\zeta (s,\frac{1}{4})-\zeta (s,\frac{3}{4})).$ proof

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

$\beta (s)=2^{-s}\mathrm{\Phi }(-1,s,\frac{1}{2}),$

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

$\beta (s)={\left(\frac{\pi }{2}\right)}^{s-1}\mathrm{\Gamma }(1-s)\cos \frac{\pi s}{2}\beta (1-s)$

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 ${\psi }_3(1/4)$ in the above is an example of the polygamma function. More generally, for any positive integer k:

$\beta (2k+1)=\frac{(-1{)}^k{E}_{2k}{\pi }^{2k+1}}{4^{k+1}(2k)!},$

where $E_n$ represent the Euler numbers. For integer k ≥ 0, this extends to:

$\beta (-k)=\frac{E_k}{2}.$

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

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

See also

• Hurwitz zeta function

References

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• J. Spanier and K. B. Oldham, An Atlas of Functions, (1987) Hemisphere, New York.
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