Difference between revisions of "Dirichlet eta"

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Let $\mathrm{Re} \hspace{2pt} z > 0$, then define
 
Let $\mathrm{Re} \hspace{2pt} z > 0$, then define
$$\eta(z) = \displaystyle\sum_{n=1}^{\infty} \dfrac{(-1)^{n-1}}{n^s}.$$
+
$$\eta(z) = \displaystyle\sum_{n=1}^{\infty} \dfrac{(-1)^{n-1}}{n^z}.$$
This series is clearly the [[Riemann zeta function]] with alternating terms.
 
  
 
<div align="center">
 
<div align="center">
 
<gallery>
 
<gallery>
File:Complex Dirichlet eta function.jpg|[[Domain coloring]] of [[domain coloring]] of $\eta(z)$.
+
File:Dirichletetaplot.png|Graph of $\eta$.
 +
File:Complexdirichletetaplot.png|[[Domain coloring]] of $\eta$.
 
</gallery>
 
</gallery>
 
</div>
 
</div>
 +
 +
=See Also=
 +
[[Riemann zeta]]<br />
 +
 +
[[Category:SpecialFunction]]

Latest revision as of 20:58, 5 November 2017

Let $\mathrm{Re} \hspace{2pt} z > 0$, then define $$\eta(z) = \displaystyle\sum_{n=1}^{\infty} \dfrac{(-1)^{n-1}}{n^z}.$$

See Also

Riemann zeta