设 $f(t)=\left\{\begin{array}{ll}1, & |t| \leq 1 \\ 0, & |t|>1\end{array}\right.$ ，求 $f(z)$ 的傅氏变换，并推证：
$$
\int_0^{+\infty} \frac{\sin \omega \cos \omega t}{\omega} d \omega= \begin{cases}\frac{\pi}{2}, & |t| < 1 \\ \frac{\pi}{4}, & |t|=1 \\ 0, & |t|>1\end{cases}
$$
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