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!
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.