Q6Advanced Engineering Mathematics

To find the Fourier transform of the rectangular pulse function $f(x)=1$ for $|x|<1$, we integrate $e^{isx}$ across the narrow limits from -1 to 1. The result of this definite integration is ${\left[\frac{e^{isx}}{is}\right]}_{-1}^1=\frac{e^{is}-e^{-is}}{is}$. By expertly applying Euler's identity, this complex exponential expression simplifies into the well-known real-valued sinc-like function $\frac{2\sin s}{s}$.