There are a family of problems that I'm trying to solve, the simplest of which looks straightforward but isn't. It's solving for this anti-derivative:
$$\operatorname{F}(\phi, a) = \int e^{ja\sin\phi}d\phi$$
The only way that I can seem to find the solution is the painful road of decomposing the exponential into it's Maclaurin series, then integrating the parts. The sine complicates this, as each term in the Maclaurin series has powers of sines, which in turn requires a polynomial expansion.
I fully expect a messy equation with Bessel functions. Indeed this is one definition of the bessel function: $$J_0(a) = \int_{-\pi}^{\pi} e^{ja\sin\phi}d\phi$$
Is there a more straightforward way to do this than working with the Maclaurin series of $e^{ja\sin(\phi)}$?
