Dirichlet beta function

From Wikipedia, the free encyclopedia

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.

Contents 1 Definition 2 Functional equation 3 Special values 4 See also 5 References

[edit] Definition

The Dirichlet beta function is defined as

or, equivalently,

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:

.

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

,

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

where Γ(s) is the gamma function.

[edit] Special values

Some special values include:

,

,

,

where K 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:

,

where represent the Euler numbers. For integer k ≤ 0, this extends to:

.

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

[edit] See also

• Hurwitz zeta function

[edit] References

• J. Spanier and K. B. Oldham, An Atlas of Functions, (1987) Hemisphere, New York.
• Eric W. Weisstein, Dirichlet Beta Function at MathWorld.