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I am looking for density distributions over a circular set (think about $[-\pi, \pi[$ or $\Delta+[0, T[$ in general $\forall T \in \mathbb{R}^{+*}, \Delta \in \mathbb{R}$). Is there a density distribution over such sets, which would be as natural as the Gaussian density?

Here is what I am thinking about:

Gaussian density:

$d : \left\{\begin{array}{l}\mathbb{R}\rightarrow\mathbb{R}^+\\t\mapsto\frac{1}{\sigma\sqrt{2\pi}}e^{-\frac{1}{2}\left(\frac{t-\mu}{\sigma}\right)^2} \end{array}\right.$

Circular Gaussian density:

$\delta : \left\{\begin{array}{l}[0,T[\rightarrow\mathbb{R}^+\\\tau\mapsto\sum_{-\infty}^{+\infty}{d(\tau+Ti)}\end{array}\right.$

Having tried to compute $\delta(\tau)\ \forall \tau \in [0,T[$, I came accross this problem so I'm stuck for now. But does this remind anyone of anything? Can one perform such winding operations over regular density functions?

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iago-lito
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1 Answers1

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It turns out that Mathematica knew the answer: this involves elliptic theta functions:

$\delta(\tau)=\frac{1 + 2 \sum _{i=1}^{+\infty } q^{i^2} \cos (2 i u)}{T}$

with $q=e^{-2 \left(\frac{ \pi \sigma }{T}\right)^2}$ and $u=\frac{\pi (\tau -\mu )}{T}$.

This is not a closed form expression though...

iago-lito
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