Cauchy's Cosine Integral Formula

$$\int_{-\pi/2}^{\pi/2} \cos ^{\mu+\nu-2}\theta e^{i\theta(\mu... ...(\mu+\nu-1)\over 2^{\mu+\nu-2}\Gamma(\mu+\xi)\Gamma(\nu-\xi)},$$

where $\Gamma(z)$ is the Gamma Function.