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I just came across this approximation that I have no idea how it was derived:

$e^{i\beta sin(\Omega t)} \approx J_0(\beta) + 2iJ_1(\beta)sin(\Omega t)$

I thought of using Moivre theorem and the fact that

$cos(\beta sin(\Omega t)) = \sum_{m=-\infty}^{m=\infty}J_mcos(m \Omega t) $

$sin(\beta sin(\Omega t)) = \sum_{m=-\infty}^{m=\infty}J_msin(m \Omega t) $

Which gets me close to the approximation but I don't know how to approximate and get rid of the summation or if I'm in the right track. Can anyone help me? The paper just pulled this approximation out of thin air and I can't find anything related to it on the web. Thanks.

Tandeitnik
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  • Recall the generating function for bessel functions \begin{eqnarray*} \operatorname{exp} \left(\frac{z}{2} \left( T- \frac{1}{T} \right) \right) = \sum_{n=-\infty}^{\infty} T^n J_{n}(z).

    \end{eqnarray*} Now $ T \rightarrow e^{i\Omega t}$ and $ z \rightarrow \beta$ starts to get close ... but there is still missing detail ... are there any other restrictions on the variables ?

     – Donald Splutterwit Feb 18 '20 at 20:46
  • So the paper says in a footnote "The reader who doesn't like Bessel functions will find that the small angle expansion $e^{i\beta sin(\Omega t)} \approx 1 + i\beta sin(\Omega t)$ works just about as well", so he's using a small angle approximation. Does this help? – Tandeitnik Feb 18 '20 at 20:58
  • I have a new clue: for n being an integer we have that $J_{-n} = (-1)^n J_n$. So in the approximation he considered only the $n = 0$ and $n = \pm 1$ terms, that's why there's a factor 2 in front of the second term (because of the relation and parity of the sine funcion). The final question is: why can he approximate by only these terms? Or, what let's he do that. – Tandeitnik Feb 19 '20 at 14:22
  • $J_n(x)$ is like $x^n$ for small $x$. So the author is neglecting quadratic & larger terms. Still does not explain why they want to approximate them with Bessel functions. http://mathworld.wolfram.com/BesselFunctionoftheFirstKind.html – Donald Splutterwit Feb 19 '20 at 14:30

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