Dirichlet Beta Function

$$\beta(x) \equiv \sum_{n=0}^\infty (-1)^n(2n+1)^{-x}$$ (1)

$$\beta(x) = 2^{-x}\Phi(-1,x,{\textstyle{1\over 2}}),$$ (2)

where $\Phi$ is the Lerch Transcendent. The beta function can be written in terms of the Hurwitz Zeta Function $\zeta(x,a)$ by

$$\beta(x)={1\over 4^x} [\zeta(x,{\textstyle{1\over 4}})-\zeta(x,{\textstyle{3\over 4}})].$$ (3)

The beta function can be evaluated directly for Positive Odd $x$ as

$$\beta(2k+1)={(-1)^k E_{2k}\over 2(2k)!} ({\textstyle{1\over 2}}\pi)^{2k+1},$$ (4)

where $E_n$ is an Euler Number. The beta function can be defined over the whole Complex Plane using Analytic Continuation,

$$\beta(1-z)=\left({2\over\pi}\right)^z\sin({\textstyle{1\over 2}}\pi z)\Gamma(z)\beta(z),$$ (5)

where $\Gamma(z)$ is the Gamma Function.

Particular values for $\beta$ are

$$\beta(1) = {\textstyle{1\over 4}}\pi$$ (6)

$$\beta(2) \equiv K$$ (7)

$$\beta(3) = {\textstyle{1\over 32}}\pi^3,$$ (8)

where $K$ is Catalan's Constant.

See also Catalan's Constant, Dirichlet Eta Function, Dirichlet Lambda Function, Hurwitz Zeta Function, Lerch Transcendent, Riemann Zeta Function, Zeta Function

References

Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 807-808, 1972.

Spanier, J. and Oldham, K. B. ``The Zeta Numbers and Related Functions.'' Ch. 3 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 25-33, 1987.