I am trying to show that $\int_0^{\infty}{\frac{\sin(x)}{x}} = \frac{\pi}{2}$ using the fourier transform of $f(x) = \sin(ax)/\pi x$. I found using symmetry formula and the transform of a rectangular pulse to find that $$\hat{f}(\omega)= \begin{cases} 1, & \text{if $|x| < a$} \\ 0, & \text{otherwise} \end{cases}$$
Setting $a$ to $1$ and multiplying my $\pi$ should do the trick for the fourier transform of $\sin(x)/x$. I figure I have to use the energy theorem or parsevals formula but I can't seem to get the correct form of the integral.
\sin(x)instead ofsin(x)to display $\sin(x)$ instead of $sin(x)$. – DMcMor Jan 23 '21 at 20:12