Integral/infinite sum related to Bessels which pop up in optical coherence theory In propagating partially coherent optical fields, the following integral pops up:
$$I_1=\int_0^{2\pi} e^{i(a\cos[\theta]+b\cos^2[\theta])}d\theta,$$
where $a$ and $b$ are real numbers. If we consider reducing the power on the cosine we find a related integral:
$$I_2=\int_0^{2\pi} e^{i(a\cos[\theta]+b\cos[2\theta])}d\theta.$$
If we use the Jacobi-Anger expansion we can instead consider an infinite sum:
$$I_2=2\pi\sum_{m=-\infty}^{\infty}i^{-m}J_{2m}(a)J_m(b)$$
However, in either case I have been unable to find a closed form solution for $I_2$. It would be very helpful to find a closed form solution in order to reduce computation time. Any thoughts out there?
 A: One interesting thing about $I_1$ is that it satisfies a PDE. Specifically, it satisfies a version of Schroedinger's equation:: $$\partial_a^2 I_1 =-\int_0^{2\pi} \cos^2 \theta\;e^{i a\cos[\theta]+i b\cos^2[\theta])}d\theta=i\partial_b I_1.$$ Similar remarks apply to $I_2$. Since the $a=0$ and $b=0$ cases are both reducible to a zeroth-order Bessel function, this suggests that an attack along PDE lines (e.g. separation of variables, method of characteristics) may be more fruitful than direct integration.
A: In terms of the modified generalized Bessel functions introduced in [1] and [2],
$$
\int_0^{2\pi} \mathrm{d}\theta\; e^{i(a \cos\theta \,+\, b \cos 2\theta)} = 2\pi \sum_{m=-\infty}^\infty i^{-m} J_{2m}(a)\,J_m(b) = 2\pi J_0(a,b;-i) = 2\pi I_0(a,-ib)
$$
In [1], [2], and other publications of the authors, they discuss the numerical evaluation of these functions, as well as many examples of their usefulness in quantum mechanics.
[1] G Dattoli et al, Theory of generalized Bessel functions
[2] G Dattoli et al, Theory of generalized Bessel functions II
