Inverse Laplace transform

Definition

Let $a$ be a real number and let $F$ be a complex function on the half-plane $\{s \in \mathbb{C} : \operatorname{Re} s > a\}$. An inverse Laplace transform of $F$ is a piecewise continuous function $f : [0, \infty) \to \mathbb{C}$ with two properties. First, there are constants $M, b$ with $|f(t)| \le M e^{b t}$ for $t \ge 0$. Second, the Laplace transform of $f$ equals $F(s)$ for each $s$ with $\operatorname{Re} s > \max(a, b)$. The notation is $f = \mathcal{L}^{-1}\{F\}$.

Remarks

By Lerch's theorem, two such functions with the same transform agree at each point where both are continuous. So $\mathcal{L}^{-1}\{F\}$ is unique in practice, and unique among continuous functions. Let $\gamma$ be a real number larger than the real parts of all singularities of $F$. Under suitable conditions, the Bromwich integral recovers $f$: $f(t) = \frac{1}{2 \pi i} \lim_{T \to \infty} \int_{\gamma - iT}^{\gamma + iT} e^{s t} F(s)\, ds$. In practice, one splits $F$ into partial fractions and reads the inverse of each term from a table. For example, $\mathcal{L}^{-1}\{1/(s - a)\} = e^{a t}$.

Explanations

L1 human

The inverse transform goes back from the $s$ side to the time side. After we solve a problem in terms of $F(s)$, the inverse transform gives the solution as a function of time.