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Consider the real valued function $f(x):=\cos{(x^2)}$. How can we calculate its Fourier transform?

In other words, I have to calculate $ \hat{f}(\omega):=\frac{1}{2\pi}\int_{\mathbb R}\cos{(x^2)}e^{-i\omega x}dx. $ Any ideas? I'm sincerely stuck... I tried to calculate $ \int_{\mathbb R}e^{ix^2-ikx}dx $ in order to get the Fourier transforms of both $\cos x^2$ and $\sin x^2$ but I do not know how to begin. Mathematica says that the Fourier transform of $f$ has this simple expression: $ \frac{1}{2} \left[\cos\left(\frac{\omega^2}{4}\right)+\sin\left(\frac{\omega^2}{4}\right)\right] $ Thanks in advance.

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You almost finished. You just need to complete the square in the exponential term, and use a Gaussian integral $ \int_{-\infty}^{\infty}e^{ix^2-ikx}dx=e^{\frac{(-ik)^2}{4i}}\sqrt{\frac{\pi}{-i}}=e^{\frac{-ik^2}{4}}\sqrt{i\pi} $

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    Eventually, I've come to a solution: we have to use Fresnel integrals. With them we conclude immediately. Thanks for your help.2012-12-03