Can anyone give a clue how the following integration can be solved by Residue theorem? $$\int^{\pi/2}_{-\pi/2}\, d\theta\,\cos^{1-\nu} \theta = \frac{\sqrt{\pi} \, \Gamma\left(1-\frac \nu 2\right)}{\Gamma\left(\frac 3 2 -\frac \nu 2\right)}$$
where $0<\nu<1$
My thought is that if I can solve the above via contour integration then it will be possible to solve the following type of integration which had been asked before: (complicated integration involving exponential and trigonometric functions) $$\int^{\pi/2}_{-\pi/2}\, d\theta\,\cos^{1-\nu} \theta \,e^{-\alpha \,sin^2\theta+\beta\,sin\theta}$$
Hint.
$$\cos^{1-\nu}\theta = \left(\frac{e^{i\theta} + e^{-i\theta}}{2}\right)^{1-\nu}$$
From here, you may try to use the Binomial theorem together with the Euler Gamma function.