I'm looking for a closed-form solution of the following integral in terms of standard functions such as sine, cosine, exponential, etc, as well as special functions such as Bessel functions:
$$ \int_0^{2\pi} \operatorname{sinc}(y/a) e^{ix\cos y} dy $$ where here $\operatorname{sinc}(x) = \sin(\pi x)/(\pi x)$.
I am aware that if there was no $\operatorname{sinc}$ factor, the integral would reduce to: $$ \int_0^{2\pi} e^{ix\cos y} dy = 2\pi J_0(x) $$
I've tried using a Maclaurin series expansion for $\operatorname{sinc}$: $$ \operatorname{sinc}(x) \approx 1-\frac{(\pi x)^2}{3!}+\frac{(\pi x)^4}{5!}-\ldots $$ but this requires a large number of terms, and I am unsure how to compute the integrals that result (with coefficients factored outside the integral): $$ \int_0^{2\pi} y^n e^{ix\cos y} dy, \quad n\in \{2, 4, 6, \ldots\} $$
Are either of these integrals (the one with the $\operatorname{sinc}$ factor or the one with the $y^n$ factor) integrals with known solutions, and if not, can you suggest a possible closed-form solution method?