I need to use Parseval's theorem to calculate the following integral:
$$\int_{-\infty}^{\infty}\left |\frac{1-e^{-iwt}}{iw} \right |^{2}dt$$
I thought to find the transform of $$f(t) = \frac{1-e^{-iwt}}{iw}$$ and then use Parseval.
I have problems finding the Fourier transform of f(t) using properties by knowing common transforms.
The definition used for the Fourier transform is:
$$\hat{f}(w) = \int_{-\infty}^{\infty} f(t) e^{-iwt}\cdot dt$$
Best Answer
Hint: I think the denominator should be $wt$ instead of $w$ as the integral would evaluate to $\infty$ otherwise. Also, $$\frac{1-e^{-iwt}}{iwt} = e^{-iwt/2}\frac{e^{iwt/2}-e^{-iwt/2}}{iwt} = e^{-iwt/2}\frac{\sin(wt/2)}{wt/2}.$$ Consequently, $$\left|\frac{1-e^{-iwt}}{iwt}\right| = \left|\frac{\sin(wt/2)}{wt/2}\right|.$$