Fourier transform pairs and properties derivation

Fourier Transform Paris

$1\leftrightarrow 2\pi \delta (\omega )$

$$\begin{array}{c}\begin{array}{rl}X(\omega )& ={\int }_{-\infty }^{\infty }1\ast e^{-j\omega t}dt\\ & =\underset{T\to \infty }{\lim }{\int }_{-\frac{T}{2}}^{\frac{T}{2}}{e}^{-j\omega t}dt\\ & =\underset{T\to \infty }{\lim }-\frac{1}{j\omega }[e^{-j\omega t}]{|}_{t=-T/2}^{t=T/2}\end{array}\end{array}$$