“Dirichlet Beta Function”的概念、定义、习题解题方法及翻译-数学辞典

单词

Dirichlet Beta Function

释义

Dirichlet Beta Function

			(1)
			(2)

where

is the Lerch Transcendent. The beta function can be written in terms of the Hurwitz ZetaFunction

(3)

The beta function can be evaluated directly for Positive Odd

(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 LambdaFunction, 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.

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