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

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+__NOTOC__
 The Dirichlet $\beta$ function is defined by
-$$\beta(x) = \displaystyle\sum_{k=0}^{\infty} (-1)^k (2k+1)^{-x} = 2^{-x} \Phi \left(-1,x,\dfrac{1}{2} \right),$$
+$$\beta(z) = \displaystyle\sum_{k=0}^{\infty} \dfrac{(-1)^k}{(2k+1)^z}.$$
-where $\Phi$ denotes the [[Lerch transcendent]].
+<div align="center">
+<gallery>
+File:Plot dirichlet beta.png|Graph of $\beta$ on $[-4,4]$.
+File:Domain coloring dirichlet beta.png|[[Domain coloring]] of [[analytic continuation]] of $\beta$.
+</gallery>
+</div>
+=Properties=
+[[Catalan's constant using Dirichlet beta]]<br />
+[[Dirichlet beta in terms of Lerch transcendent]]<br />
+[[Category:SpecialFunction]]