The Dirichlet beta function is defined by the sum
where is the Lerch transcendent. The beta function can be written in terms of the Hurwitz zeta function by
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(3) |
The beta function can be defined over the whole complex plane using analytic continuation,
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(4) |
where is the gamma function.
The Dirichlet beta function is implemented in the Wolfram Language as DirichletBeta[x].
The beta function can be evaluated directly special forms of arguments as
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(5) | |||
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(6) | |||
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(7) |
where is an Euler number.
Particular values for are
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(8) | |||
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(9) | |||
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(10) | |||
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(11) |
where is Catalan's constant and is the polygamma function. For , 3, 5, ..., , where the multiples are 1/4, 1/32, 5/1536, 61/184320, ... (OEIS A046976 and A053005).
It is involved in the integral
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(12) |
(Guillera and Sondow 2005).
Rivoal and Zudilin (2003) proved that at least one of the seven numbers , , , , , , and is irrational.
The derivative can also be computed analytically at a number of integer values of including
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(13) | |||
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(14) | |||
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(15) | |||
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(16) | |||
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(17) | |||
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(18) | |||
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(19) |
(OEIS A133922, A113847, and A078127), where is Catalan's constant, is the gamma function, and is the Euler-Mascheroni constant.
A nice sum involving is given by
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(20) |
for a positive integer.
See also
Catalan's Constant, Dirichlet Eta Function, Dirichlet Lambda Function, Hurwitz Zeta Function, Legendre's Chi-Function, Lerch Transcendent, Riemann Zeta Function, Sierpiński Constant, Zeta Function
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References
Abramowitz, M. and Stegun, I. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 807-808, 1972.Borwein, J. M. and Borwein, P. B. Pi & the AGM: A Study in Analytic Number Theory and Computational Complexity. New York: Wiley, p. 384, 1987.Comtet, L. Problem 37 in Advanced Combinatorics: The Art of Finite and Infinite Expansions, rev. enl. ed. Dordrecht, Netherlands: Reidel, p. 89, 1974.Guillera, J. and Sondow, J. "Double Integrals and Infinite Products for Some Classical Constants Via Analytic Continuations of Lerch's Transcendent." 16 June 2005 http://arxiv.org/abs/math.NT/0506319.Rivoal, T. and Zudilin, W. "Diophantine Properties of Numbers Related to Catalan's Constant." Math. Ann. 326, 705-721, 2003. http://www.mi.uni-koeln.de/~wzudilin/beta.pdf.Sloane, N. J. A. Sequences A046976, A053005, A078127, A113847, and A133922 in "The On-Line Encyclopedia of Integer Sequences."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.Mathews, J. and Walker, R. L. Mathematical Methods of Physics, 2nd ed. Reading, MA: W. A. Benjamin/Addison-Wesley, p. 57, 1970.
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Cite this as:
Weisstein, Eric W. "Dirichlet Beta Function." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/DirichletBetaFunction.html