# How could you identify a periodic function with a function on a circle

Guess the headline already said everything. If I have a periodic function, for example on the real line, how could it be identified with a function, say for example on the unit circle?

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Map the line onto the circle, using the map that says that $f(x) = y$, where $y$ is the remainder when $x$ is divided by the period of the function. – MJD Oct 9 '12 at 20:58

Say $f : \mathbb{R} \to \mathbb{C}$ is a $2\pi$-periodic function. Write $S^1$ is the unit circle in $\mathbb{C}$. You can identify $f$ with the function $g : S^1 \to \mathbb{C}$ defined by $g(e^{i\theta}) = f(\theta)$ (this definition is valid because if $e^{i\theta} = e^{i\theta'}$, then $\theta \equiv \theta' \mod 2\pi$ so $f(\theta) = f(\theta')$).

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If the period is $1$, identify the interval $[0,1]$ with the circle, by the map $t \to (\cos(2\pi t), \sin(2\pi t))$

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A heuristic approach is to imagine the graph of the periodic function $f\colon \mathbb R \to \mathbb R$ drawn on a sheet of paper. The graph is a subset of $\mathbb R \times \mathbb R$

If it has period $L$, roll the paper into a tube along the horizontal axis so the the circumference of the cross section is $L$. The graph over any of the intervals matches that over any other of the intervals, so the graph is now a subset of the tube, which is $S^{1} \times \mathbb R$, so it is the graph of a real valued function defined on a circle

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