# 双周期函数

## 定义

${\displaystyle f(z+mu+nv)=f(z),}$

## 例子

${\displaystyle f:\;\;x+yi\;\;\mapsto \;\;\sin {(2\pi x)}\cos {(2\pi y)}.}$

${\displaystyle D_{f}=\{tu+sv\;\;|\;\;0\leqslant t,s<1\))$（一个平行四边形

### 椭圆函数

${\displaystyle \wp (z)={\frac {1}{z^{2))}+\sum _{m^{2}+n^{2}\neq 0}\left$$(\frac {1}{(z-nu-mv)^{2))}-{\frac {1}{(nu+mv)^{2))}\right$$)$[1]:324

## 性质

${\displaystyle {\mathcal {B))_{f}=\{su\;|\;0\leqslant s<1\}\cup \{u+sv\;|\;0\leqslant s\leqslant 1\}\cup \{v+su\;|\;0\leqslant s<1\}\cup \{sv\;|\;0

${\displaystyle \oint _((\mathcal {B))_{f))f(z)\mathrm {d} z=0.}$

## 参考来源

1. Nico M. Temme. Special Functions: An Introduction to the Classical Functions of Mathematical Physics. John Wiley & Sons. 2011. ISBN 9781118030813 （英语）.
2. Michael T. Vaughn. Introduction to Mathematical Physics. John Wiley & Sons. 2008. ISBN 9783527618866 （英语）.
3. Gareth A. Jones. Complex Functions: An Algebraic and Geometric Viewpoint. Cambridge University Press. 1987. ISBN 9780521313667 （英语）.
4. ^ Anatolij T. Fomenko, Aleksej A. Tužilin. Elements of the Geometry and Topology of Minimal Surfaces in Three-Dimensional Space. American Mathematical Soc. 2005. ISBN 9780821898345 （英语）.