I am solving a physics problem in which I've found an integral of this kind. I have seen a similar expression for the Bessel function, nevertheless the limits are different, how could I evaluate this integral?
$$ \int\limits_{-a}^{2\pi-a}e^{ixcos(u)}du\,. $$
Thank you for your help
As $\cos u$ is $2\pi$-periodic and because you integrate on an interval with $2\pi$ amplitude, you can shift the limits and integrate between 0 and $2\pi$: $$ I=\int_0^{2\pi}e^{ix\cos u}du$$ Then changing $u\to2\pi-u$ leaves the integrand inchanged, so $$I=2\int_0^{\pi}e^{ix\cos u}du$$ Using the integral representation for $J_0(z)$, $$I=2\pi J_0(x)$$