Hello Math Stack Exchange community,
I have encountered a challenging integral that I am struggling to simplify and express in terms of special functions, particularly Bessel functions. The integral I am working with is:
$$ \frac{1}{\pi} \int_{0}^{2\pi} e^{-x e^{-i\pi\sin(\theta)} - y e^{-2i\pi\sin(\theta)}} d\theta $$
I have attempted to compute this integral numerically and then simplify it, but I'm having difficulty finding a closed-form expression.
Interestingly, I found a similar integral that has been simplified in terms of Bessel functions in a previous discussion on Math Stack Exchange: link to previous discussion. However, the integral in that discussion is not exactly the same as mine.
I would greatly appreciate any insights, techniques, or strategies that could help me simplify this integral and express it in terms of Bessel functions. Additionally, if anyone has encountered a similar integral or has experience working with parametric integrals involving exponentials, your expertise would be invaluable.
Thank you in advance for your assistance!