I have a follow up to this question regarding the derivation of a sinc function from a complex exponential, because I haven't been able to figure out how the introduction of a scaling factor will affect the Euler's formula. Can anyone please help?
The window function is: \begin{align}g(t) = \begin{cases} 1, & \text{if |t| < $\frac{1}{2a}$} \\ 0, & \text{otherwise} \end{cases} \end{align}
So the Fourier transform is:
\begin{align} \int_{-1/2a}^{1/2a}e^{-j\omega t} dt &= \frac{1}{j\omega}[e^{j\omega/2a}-e^{-j\omega/2a}]\\ &= \frac{1}{j\omega}[\cos(\frac{\omega}{2a})+j\sin(\frac{\omega}{2a})-\cos(\frac{\omega}{2a})+j\sin(\frac{\omega}{2a})]\\ &= \frac{1}{j\omega}[2j\sin(\frac{\omega}{2a})]\\ &= \frac{\sin(\frac{\omega}{2a})}{\frac{w}{2}} \end{align}
How do I get to the sinc function?