Evaluate $\displaystyle \int_{0}^{\infty} cos(x^2) dx$.
What complex integral and branch cut should I use??
I'd evaluate it using the fact that the residue of $\displaystyle \cos(z^2)=\frac{e^{iz^2}+e^{-iz^2}}{2}$ is 0.
Using that, we can set our integral's contour, C, over the real line from 0 to infinity and arc around back.
Thus, we'll get $\displaystyle \oint\limits_{C}\cos(z^2)\,dz = 0 = \int\limits_{[0,\infty]}cos(z^2) \, dz - \frac{in}{2} \lim_{n\to\infty^+}\int\limits_{[0, \pi]}\cos(\frac{n^2}{4}(1+e^{i\theta})^2)e^{i\theta}\, d\theta.$. Make sense?