Is there a closed form for this "flowery" integral?

Question

I'm curious about shapes like this:

I think of this as the trajectory of a particle where the acceleration is perpendicular to the velocity and oscillates sinusoidally in time.

The functions I'm investigating, in the generic case, are of the form:

$$f(x) = \int e^{ai(x + b\,sin\,x)}\ dx$$

The above image is generated with $a=0.75, b=-1$.

I'm especially interested the relationship of $a$ and $b$ in the specific case:

$$0 = \int_0^{2\pi} e^{ai(x + b\,sin\,x)}\ dx$$

This corresponds to shapes like this, where the "lobes" touch at the origin:

This is generated with $a=0.75, b \approx -0.7364$.

In trying to figure out $b = g(a)$ I've tried my usual method of graphing approximate values of $b$ against $a$ and comparing it with some common functions, but I couldn't find anything that matched. I also tried entering some values of $b$ into WolframAlpha to see if it could find a closed form, but no dice.

1. Is there a closed form for the generic case?

2. What is the equation relating $a$ and $b$ in the specific case, and does it have a closed form? To me, the numbers seem magically pulled out of thin air, with no relation to any known constants.

Answer 1

Note that

\begin{align}\int_0^{2\pi}e^{aix}e^{bi\sin x}~\mathrm dx&=\frac1i\int_0^{2\pi}(e^{ix})^{a-1}e^{b((e^{ix})-1/(e^{ix}))/2}~\mathrm d(e^{ix})\\&=\frac1i\oint_{|z|=1}z^{a-1}e^{(b/2)(z-1/z)}~\mathrm dz\\&=2\pi J_{-a}(b)\end{align}

So it comes down to finding the roots of the Bessel Function of the First Kind, which are given in Bessel Function Zeros for integer $a$.