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
|
(3)
|
The beta function can be defined over the whole complex
plane using analytic continuation,
|
(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
where
is an Euler number.
Particular values for are
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
|
(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
(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
|
(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.Referenced on Wolfram|Alpha
Dirichlet Beta Function
Cite this as:
Weisstein, Eric W. "Dirichlet Beta Function."
From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/DirichletBetaFunction.html
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