the Fourier transform of a constant

Question

How to calculate the Fourier transform of a constant without the aid of duality property? In other words, how do I calculate $$ \int_{-\infty}^{\infty}e^{-j\omega t}dt? $$

Answer 1

The Fourier transform of a constant is defined as $$ \mathcal{F}(a) = \int_{-\infty}^{\infty}ae^{-2\pi j \omega t}dt = a\delta(\omega) $$ where $\delta$ is the Dirac delta function.

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We can view the Dirac delta function as the limit of the Gaussian; that is, $$ \lim_{\sigma\to 0}\frac{1}{\sigma\sqrt{2\pi}}\exp\Bigl(-\frac{t^2}{2\sigma^2}\Bigr) $$ Then we have \begin{align} \delta(t) &= \lim_{\sigma\to 0}\frac{1}{\sigma\sqrt{2\pi}}\exp\Bigl(-\frac{t^2}{2\sigma^2}\Bigr)\\ &=\lim_{\sigma\to 0}\int_{-\infty}^{\infty}\frac{1}{2\pi}\exp\Bigl(-\frac{\omega^2\sigma^2}{2}\Bigr)e^{-i\omega t}d\omega\\ &= \frac{1}{2\pi}\int_{-\infty}^{\infty}e^{-j\omega t}d\omega \end{align}