Difference between revisions of "Dirichlet beta"

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__NOTOC__
 
The Dirichlet $\beta$ function is defined by
 
The Dirichlet $\beta$ function is defined by
$$\beta(x) = \displaystyle\sum_{k=0}^{\infty} (-1)^k (2k+1)^{-x} = 2^{-x} \Phi \left(-1,x,\dfrac{1}{2} \right),$$
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$$\beta(z) = \displaystyle\sum_{k=0}^{\infty} \dfrac{(-1)^k}{(2k+1)^z}.$$
where $\Phi$ denotes the [[Lerch transcendent]].
 
  
  
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=Properties=
 
=Properties=
{{:Catalan's constant using Dirichlet beta}}
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[[Catalan's constant using Dirichlet beta]]<br />
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[[Dirichlet beta in terms of Lerch transcendent]]<br />
  
 
[[Category:SpecialFunction]]
 
[[Category:SpecialFunction]]

Latest revision as of 00:54, 11 December 2016

The Dirichlet $\beta$ function is defined by $$\beta(z) = \displaystyle\sum_{k=0}^{\infty} \dfrac{(-1)^k}{(2k+1)^z}.$$


Properties

Catalan's constant using Dirichlet beta
Dirichlet beta in terms of Lerch transcendent