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Let $S^1=\mathbb R/\mathbb Z,$ I was wondering how to calculate the integral of a function over $S^1$ and why. Like, $\int_{S^1}1 dx=?$ Given an "appropriate" function $f$, what is $\int_{S^1}f(x)dx?$

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    Is the function over $S^1$ of period 1? Why?2012-02-04
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    The topology of $S^1$ identifies 0 and 1 so it forces functions to be periodic when viewed over $\mathbb{R}$.2012-02-04
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    @Chris The period is uncertain? It depends on the context?2012-02-04
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    Well I mean in theory you could have a function with period less than one (take a function with period $\frac{1}{2}$ over $S^1$ then extend it over $\mathbb{R}$ to get a function with period $\frac{1}{2}$) The definition of the group $\mathbb{R}/ \mathbb{Z}$ says that if you take any $a \in [0,1)$ and add $n \in \mathbb{Z}$ to it, you stay in equivalence class of $a$. That says that $f(a + n) = f(a)$. Thus when viewed over $\mathbb{R}$ the function must have period 1. Again though it could also have period less than 1.2012-02-04
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    Maybe you should think through what your real question is? To me $S^1$ is the circle group (however since I do not know an other common notation is $\mathbb{T}$, which I prefer as a 1-dimensional *T*orus2012-02-04

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