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.