Question related to fourier transform?


I just have a simple question

Can someone prove the fourier transform of 1. it is 2*pi*delta(omega)

but i am not able to prove it without using the properties just prove from scratch (fourier transform formula)

any link would also be useful.

Best Answer

By definition of the impulse distribution, we have

$$\int_{-\infty}^{\infty}g(\omega)\delta(\omega)d\omega = g(0)$$

The Fourier synthesis equation is

$$f(t) = \frac{1}{2\pi}\int_{-\infty}^{\infty}F(\omega)e^{j\omega t}d\omega$$

These are all you need to get the result.