# Definition:Inverse Laplace Transform

## Definition

### Definition 1

Let $f \left({s}\right) : S \to \R$ be a function, where $S \subset \R$.

Let $F$ be another function such that $F$ is the Laplace transform of $f$.

Then, $f$ is the inverse Laplace transform of $F$.

### Definition 2

Let $\map f s: S \to \R$ be a complex function, where $S \subset \R$.

The inverse Laplace transform of $f$, denoted $\map F t: \R \to S$, is defined as:

$\map F t = \dfrac 1 {2 \pi i} \PV_{c \mathop - i \, \infty}^{c \mathop + i \, \infty} e^{s t} \map f s \rd s = \frac 1 {2 \pi i} \lim_{T \mathop \to \infty} \int_{c \mathop - i \, T}^{c \mathop + i \, T} e^{s t} \map f s \rd s$

where:

$\PV$ is the Cauchy principal value of the integral
$c$ is any real constant such that all the singular points of $\map f s$ lie to the left of the line $\map \Re s = c$ in the complex $s$ plane.