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.
 
  
[[File:Complex Dirichlet eta function.jpg|500px]]
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<div align="center">
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<gallery>
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File:Dirichletetaplot.png|Graph of $\eta$.
 +
File:Complexdirichletetaplot.png|[[Domain coloring]] of $\eta$.
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</gallery>
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</div>
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=See Also=
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[[Riemann zeta]]<br />
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[[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