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If we say $f(x)$ has periods $T$,in other works, the following equation holds $f(x+T)=f(x)$. today I encounter with the notion about two variables period function. it says: $f(x,y)$has periods $T$. and without any explicit and detailed explanation. so what does it mean in general?

  • $f(x+T,y+T)=f(x,y)$
  • $f(x+T,y)=f(x,y)$
  • $f(x,y+T)=f(x,y)$

    which one is the proper definition?

    thanks very much

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    up vote 2 down vote accepted

    A function $f:\ {\mathbb R}^2\to{\mathbb R}$ is doubly periodic if there are two linearly independent vectors ${\bf p}_1$, ${\bf p}_2\in {\mathbb R}^2$ such that $$f({\bf z}+{\bf p_1})=f({\bf z}), \quad f({\bf z}+{\bf p_2})=f({\bf z})\qquad\forall{\bf z}\in{\mathbb R}^2\ .$$ The set $$\Lambda:=\{ m{\bf p}_1+n{\bf p}_2\ |\ m, n\in{\mathbb Z}\}$$ is then called the period lattice of $f$.

    If ${\bf p}_1=(a,0)$ with $a>0$ and ${\bf p}_2=(0,b)$ with $b>0$ this lattice is orthogonal, and if $a=b=T$ for some $T>0$ it is a square lattice.

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