Contour integration of cosine of a complex number I am trying to find the value of 
$$ -\frac{1}{\pi}\int_{-\pi/2}^{\pi/2} \cos\left(be^{i\theta}\right) \mathrm{d}\theta,$$
where $b$ is a real number.
Any helps will be appreciated!

$$\cos(be^{i\theta})=\cos(b\cos\theta+ib\sin\theta)=-\cos (b\cos \theta)\cosh(b\sin \theta)+i\sin(b\cos\theta)\sinh(b\sin \theta)$$ Thus your integral reduces to $$I=\frac{2}{\pi}\int_{0}^{\pi/2}\cos (b\cos \theta)\cosh(b\sin \theta)d\theta$$ I think you have to use  Bessel's function hereon to simplify this, though I am not quite sure.
