The inverse Laplace transform recovers a time-domain function from its transform:

$$f(t)=\mathcal L^{-1}{F(s)}.$$

In elementary applications, inversion is usually performed by recognizing known transform pairs and using algebra rather than evaluating a complex inversion integral directly.

A rational transform can often be decomposed into partial fractions. For example,

$$\frac{1}{(s-a)(s-b)}=\frac{A}{s-a}+\frac{B}{s-b}.$$

Each simpler term can then be inverted separately by linearity.

Repeated quadratic factors and irreducible quadratics lead naturally to exponentials, sines and cosines.

Partial-fraction decomposition therefore provides the algebraic bridge from a rational expression in $s$ back to a sum of recognizable time-domain modes.