Integrating an exponential containing a cosine function $\int_0^{2\pi} e^{i \nu \theta \pm ix\cos(\theta-\theta_0)} d \theta$

Question

I'm trying to integrate $\int_0^{2\pi} e^{i \nu \theta \pm ix\cos(\theta-\theta_0)} d \theta$. According to a reference from the Journal of Mathematical Physics (https://doi.org/10.1063/1.5108599), the result is: $$ 2\pi e^{i\nu(\theta_0 \pm \pi/2 )}J_{\nu}(x). $$ I've tried searching for this integral in multiple reference tables, including the one they've cited but I can't find it. I'd like to verify this is the correct answer since I can't find it listed in any other reference tables and I suspect there are other factors/terms when you get beyond the case of $\nu = 0$. I did find something close that I think may help me do this: $$ \int_0^{\pi} e^{\pm i (\nu x - \beta\sin(x))} dx = \pi[J_\nu(\beta) \pm iE_\nu(\beta)]. $$ From Bateman, Harry (1953) Higher Transcendental Functions Volume II (page 35, equation 32). Here $J_\nu(\beta)$ is an Anger function and $E_\nu(\beta)$ is a Weber function. I really have no experience with Anger or Weber functions. I'd like to start by changing $\theta$ to change my sine to a cosine and dealing with the integration bounds but I don't know how this would affect those functions. Any help figuring this out or even other references to find my original integral will be appreciated.