Difference between revisions of "Clausen cosine"
From specialfunctionswiki
(Created page with "Let $s \in \mathbb{C}$. The Clausen cosine function $\tilde{\mathrm{Cl}}_s \colon \mathbb{C} \rightarrow \mathbb{C}$ is defined by $$\tilde{\mathrm{Cl}}_s(z)=\displaystyle\su...") |
|||
| (3 intermediate revisions by the same user not shown) | |||
| Line 1: | Line 1: | ||
| − | Let $s \in \mathbb{C}$. The Clausen cosine function $\tilde{\mathrm{Cl}}_s \colon \mathbb{C} \rightarrow \mathbb{C}$ is defined | + | Let $s \in \mathbb{C}$. The Clausen cosine function $\tilde{\mathrm{Cl}}_s \colon \mathbb{C} \rightarrow \mathbb{C}$ is defined as the [[analytic continuation]] of the series |
$$\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},$$ | ||
where $\cos$ denotes [[cosine]]. | where $\cos$ denotes [[cosine]]. | ||
| + | |||
| + | <div align="center"> | ||
| + | <gallery> | ||
| + | File:Clausencosine0.5plot.png|Graph of $\tilde{\mathrm{Cl}}_{0.5}$. | ||
| + | </gallery> | ||
| + | </div> | ||
| + | |||
| + | Clausenplot.png | ||
| + | |||
=Properties= | =Properties= | ||
Latest revision as of 18:56, 7 September 2020
Let $s \in \mathbb{C}$. The Clausen cosine function $\tilde{\mathrm{Cl}}_s \colon \mathbb{C} \rightarrow \mathbb{C}$ is defined as the analytic continuation of the series $$\tilde{\mathrm{Cl}}_s(z)=\displaystyle\sum_{k=1}^{\infty} \dfrac{\cos(kz)}{k^s},$$ where $\cos$ denotes cosine.
Graph of $\tilde{\mathrm{Cl}}_{0.5}$.
Clausenplot.png
