Evaluate $\displaystyle \int_{0}^{\infty} cos(x^2) dx$.What complex integral and branch cut should I use??

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- Apr 30th 2013, 12:16 AMCuriosityCabinetDefinite Integration using Residue Theorem
**Evaluate $\displaystyle \int_{0}^{\infty} cos(x^2) dx$.**What complex integral and branch cut should I use??

- Jun 14th 2013, 12:30 PMPhantasmaRe: Definite Integration using Residue Theorem
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?