Difference between revisions of "Clausen sine"

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$$\mathrm{Cl}_s(z)=\displaystyle\sum_{k=1}^{\infty} \dfrac{\sin(kz)}{k^s},$$
 
$$\mathrm{Cl}_s(z)=\displaystyle\sum_{k=1}^{\infty} \dfrac{\sin(kz)}{k^s},$$
 
where $\sin$ denotes [[sine]].
 
where $\sin$ denotes [[sine]].
 
 
  
 
<div align="center">
 
<div align="center">
 
<gallery>
 
<gallery>
File:Clausensine0plots.png|Plot of $\mathrm{Cl}_0$ on $[-20,20]$.
+
File:Clausensine0.5.png|Graph of $\mathrm{Cl}_{0.5}$.
 
</gallery>
 
</gallery>
 
</div>
 
</div>

Latest revision as of 19:44, 7 September 2020

Let $s \in \mathbb{C}$. The Clausen sine function $\mathrm{Cl}_s \colon \mathbb{C} \rightarrow \mathbb{C}$ is defined as the analytic continuation of the series $$\mathrm{Cl}_s(z)=\displaystyle\sum_{k=1}^{\infty} \dfrac{\sin(kz)}{k^s},$$ where $\sin$ denotes sine.

Properties

See also

Clausen cosine

References