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As Jacobi-Anger expansion suggest:

$e^{i z \mathrm{cos}(\theta)} = \sum_{n=-\infty}^{\infty} i^n J_n(z) e^{i n \theta}$

What if $\mathrm{cos}(\theta)$ from the expression above would be replaced with some other function $f(\theta)$. Is there a method to derive a generalized expansion of the function $e^{i z f(\theta)}$ ?

To be more specific, the expansion of $e^{i k \pi \frac{\mathrm{cos}(\omega t)}{1+\mathrm{cos}(\omega t) + \mathrm{cos}(2 \omega t)}}$ and $e^{i k \pi \frac{1}{1+\mathrm{cos}(\omega t) + \mathrm{cos}(2 \omega t)}}$ is what I need.

Thank you!

slm992
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1 Answers1

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This is a particular case of complex Fourier series expansion.

You can expand any periodic function that way, the formulas for coefficients are well known:

$$f(\theta) = \sum_{n=-\infty}^\infty A_n e^{i n \theta}$$

$$A_n= \frac{1}{2\pi} \int_{-\pi}^\pi f(\theta) e^{- i n \theta} d \theta$$

Here we assume $f(\theta+2 \pi)=f (\theta)$, since this is the case in the OP.

We can take $\theta=\omega t$ in the expression above, so we have:

$$A_n= \frac{1}{2\pi} \int_{-\pi}^\pi \exp \left( \frac{ik}{1+ \cos \theta+ \cos 2 \theta}- i n \theta \right) d \theta$$

Which I don't think has a simple explicit form.

Yuriy S
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