\[X(f) = \int_{-\infty}^{\infty} x(t) e^{-2 \pi j f t} dt\]
\[x(t) = \int_{-\infty}^{\infty} X(f) e^{2 \pi j f t} df\]
As a convention, we require \(x(t)\) has the form \(u(t)f(t)\) so that infinite negative tail of \(e^{-2 \pi j f t}\) is eliminated.
An alternative notation:
\[X(\omega) = \int_{-\infty}^{\infty} x(t) e^{j f t} dt\]
\[x(t) = \frac{1}{2 \pi} \int_{-\infty}^{\infty} X(f) e^{j f t} df\]
\[\text{sinc}(t) = \frac{\sin (\pi t)}{\pi t} \xrightarrow[]{CTFT} \text{rect}(f)\] \[\text{comb}_T(x(t)) \xrightarrow[]{CTFT} \frac{1}{T} \text{rep}_{1/T}(X(f))\]
\[x(t) \xrightarrow[]{CTFT} X(f)\]
\[X(t) \xrightarrow[]{CTFT} x(-f)\]
\[x(t - t_0) \xrightarrow[]{CTFT} X(f)e^{-j 2 \pi ft_0}\]
\[x\left(\frac{t}{a}\right) \xrightarrow[]{CTFT} |a| X(af)\]
\[X_d (\omega) = \sum_{n = - \infty}^\infty x[n] e^{-j \omega n}\]
\[x[n] = \frac{1}{2 \pi} \int_{-\infty}^{\infty} X_d(\omega) e^{j \omega n}\]