I have what is effectively a Fourier integral resulting from applying Abbe's theorem that I would like to simplify (ideally into a closed form solution): $$ f(\theta_0,\theta_1;\alpha) = \int_{\theta_0}^{\theta_1} e^{i \alpha \cos \theta} \cos \theta \,d\theta $$ Here, $i=\sqrt{-1}$ and $\alpha$ is a constant. The best case scenario is a closed form solution in terms of special functions, but that is doubtful. I'm also looking for series solutions and (rapidly converging) asymptotic expansions. Alternatively, this can be written as $$ f(u_0,u_1;\alpha) = \int_{u_0}^{u_1} e^{i \alpha u} \frac{u}{\sqrt{1-u^2}} \,du $$ or in real and imaginary parts, $$ f_{\Re,\Im}(u_0,u_1;\alpha) = \int_{u_0}^{u_1} \left\{\begin{matrix}\cos\\\sin\end{matrix}\right\}(\alpha u) \frac{u}{\sqrt{1-u^2}} \,du $$
So, can any of these integrals be written in terms of known special functions?
