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Let $f(z)$ be holomorphic in a neighborhood of the unit circle such that $|f(z)|=1$ whenever $|z|=1$. There are plenty of such functions. Examples are $z^k$ for $k\in \mathbb{Z}$ and the Möbius functions $$\frac{z-a}{1-\overline{a}\,z}$$ for $|a|\neq 1$. Then you can take products of such functions or compositions etc. The key observation is the ...

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As $\sum_{k\in\mathbb Z}\lvert a_k\rvert^2<\infty$, then this corresponds to a $2\pi-$periodic $L^2-$function $f$, with $$f(x)=\sum_{k\in\mathbb Z}a_k\,\mathrm{e}^{ikx},$$ and $\int_0^{2\pi}\lvert\,f(x)\rvert^2\,dx=2\pi\sum_{k\in\mathbb Z}\lvert a_k\rvert^2$. Now the relation $$\sum_{n\in\mathbb Z}\overline{a}_na_{n+k}=\delta_{k0}, \qquad (\star)$$ ...

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If something increases at a rate proportionate to that same something on time or angle t basis, the thing is growing exponentially. If $dr/dt$ is proportional to $r$ with a proportionality constant cot($\alpha$), then $r = {r_o} e^{ cot \alpha* t }$. To appreciate where from cot(al) came, draw the differential right angled triangle where \$ r dt / dr = ...

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The key point is that the curve will need to have self-intersections. (A simple closed curve of positive curvature must be convex). Here's an inspirational example: The parametric equations of the above curve are $$x=3\cos(-t)+\cos5 t,\qquad y = 3\sin(-t)+\sin 5t$$ Imagine a planet orbiting the origin clockwise, and its moon orbiting the planet ...

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Some smoothness is required. Read baby Rudin (3rd edition), pp. 136-137 .

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