Difference between revisions of "Clausen sine"
From specialfunctionswiki
| (2 intermediate revisions by the same user not shown) | |||
| Line 2: | Line 2: | ||
$$\mathrm{Cl}_s(z)=\displaystyle\sum_{k=1}^{\infty} \dfrac{\sin(kz)}{k^s},$$ | $$\mathrm{Cl}_s(z)=\displaystyle\sum_{k=1}^{\infty} \dfrac{\sin(kz)}{k^s},$$ | ||
where $\sin$ denotes [[sine]]. | where $\sin$ denotes [[sine]]. | ||
| − | |||
| − | |||
<div align="center"> | <div align="center"> | ||
<gallery> | <gallery> | ||
| − | File: | + | 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.
Graph of $\mathrm{Cl}_{0.5}$.
