Fourier Transform

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

$F$ is a continuous function of $\omega$, and is called the spectrum of $f(t)$.

Always done using numerical methods. Used for non-repetitive signals. Used in telecommunications, and in transient analysis, especially where the circuit parameters are frequency dependent.

Fourier inverse transform

$$F(t) = \int_{-\infty}^{\infty} F(\omega) e^{j\omega t}\,\text{d}\omega$$