Difference between revisions of "Dirichlet beta"
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$$\beta(x) = \displaystyle\sum_{k=0}^{\infty} (-1)^k (2k+1)^{-x} = 2^{-x} \Phi \left(-1,x,\dfrac{1}{2} \right),$$ | $$\beta(x) = \displaystyle\sum_{k=0}^{\infty} (-1)^k (2k+1)^{-x} = 2^{-x} \Phi \left(-1,x,\dfrac{1}{2} \right),$$ | ||
where $\Phi$ denotes the [[Lerch transcendent]]. | where $\Phi$ denotes the [[Lerch transcendent]]. | ||
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| + | =Properties= | ||
| + | {{:Catalan's constant using Dirichlet beta}} | ||
Revision as of 01:14, 21 March 2015
The Dirichlet $\beta$ function is defined by $$\beta(x) = \displaystyle\sum_{k=0}^{\infty} (-1)^k (2k+1)^{-x} = 2^{-x} \Phi \left(-1,x,\dfrac{1}{2} \right),$$ where $\Phi$ denotes the Lerch transcendent.
Contents
Properties
Theorem
The following formula holds: $$K=\beta(2),$$ where $K$ is Catalan's constant and $\beta$ denotes the Dirichlet beta function.
