Difference between revisions of "Dirichlet beta"
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(Created page with "$$\beta(x) = \displaystyle\sum_{k=0}^{\infty} (-1)^k (2k+1)^{-x} = 2^{-x} \Phi \left(-1,x,\dfrac{1}{2} \right),$$ where $\Phi$ denotes the Lerch transcendent.") |
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| + | The Dirichlet $\beta$ function is defined by | ||
$$\beta(x) = \displaystyle\sum_{k=0}^{\infty} (-1)^k (2k+1)^{-x} = 2^{-x} \Phi \left(-1,x,\dfrac{1}{2} \right),$$ | $$\beta(x) = \displaystyle\sum_{k=0}^{\infty} (-1)^k (2k+1)^{-x} = 2^{-x} \Phi \left(-1,x,\dfrac{1}{2} \right),$$ | ||
where $\Phi$ denotes the [[Lerch transcendent]]. | where $\Phi$ denotes the [[Lerch transcendent]]. | ||
Revision as of 15:56, 9 October 2014
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.
