Dirichlet Beta Function

 (1) (2)

where is the Lerch Transcendent. The beta function can be written in terms of the Hurwitz Zeta Function by
 (3)

The beta function can be evaluated directly for Positive Odd as
 (4)

where is an Euler Number. The beta function can be defined over the whole Complex Plane using Analytic Continuation,
 (5)

where is the Gamma Function.

Particular values for are

 (6) (7) (8)

where 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.