Difference between revisions of "Clausen cosine"

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Let $s \in \mathbb{C}$. The Clausen cosine function $\tilde{\mathrm{Cl}}_s \colon \mathbb{C} \rightarrow \mathbb{C}$ is defined by
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Let $s \in \mathbb{C}$. The Clausen cosine function $\tilde{\mathrm{Cl}}_s \colon \mathbb{C} \rightarrow \mathbb{C}$ is defined as the [[analytic continuation]] of the series
 
$$\tilde{\mathrm{Cl}}_s(z)=\displaystyle\sum_{k=1}^{\infty} \dfrac{\cos(kz)}{k^s},$$
 
$$\tilde{\mathrm{Cl}}_s(z)=\displaystyle\sum_{k=1}^{\infty} \dfrac{\cos(kz)}{k^s},$$
 
where $\cos$ denotes [[cosine]].
 
where $\cos$ denotes [[cosine]].
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<div align="center">
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<gallery>
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File:Clausencosine0.5plot.png|Graph of $\tilde{\mathrm{Cl}}_{0.5}$.
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</gallery>
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</div>
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Clausenplot.png
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=Properties=
 
=Properties=

Latest revision as of 18:56, 7 September 2020

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

Clausenplot.png


Properties

See also

Clausen sine

References