Difference between revisions of "Clausen sine"

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Let $s \in \mathbb{C}$. The Clausen sine function $\mathrm{Cl}_s \colon \mathbb{C} \rightarrow \mathbb{C}$ is defined by the formula
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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
 
$$\mathrm{Cl}_s(z)=\displaystyle\sum_{k=1}^{\infty} \dfrac{\sin(kz)}{k^s},$$
 
$$\mathrm{Cl}_s(z)=\displaystyle\sum_{k=1}^{\infty} \dfrac{\sin(kz)}{k^s},$$
 
where $\sin$ denotes [[sine]].
 
where $\sin$ denotes [[sine]].
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<div align="center">
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<gallery>
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File:Clausensine0.5.png|Graph of $\mathrm{Cl}_{0.5}$.
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</gallery>
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</div>
  
 
=Properties=
 
=Properties=
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=See also=
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[[Clausen cosine]]<br />
  
 
=References=
 
=References=
  
 
[[Category:SpecialFunction]]
 
[[Category:SpecialFunction]]

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