# Inverse Laplace transform

In mathematics, the inverse Laplace transform of a function F(s) is the function f(t) which has the property $\mathcal{L}\left\{ f\right\}(s) = F(s)$, or alternatively $\mathcal{L}_t\left\{ f(t)\right\}(s) = F(s)$, where $\mathcal{L}$ denotes the Laplace transform.

It can be proven, that if a function $F(s)$ has the inverse Laplace transform $f(t)$, i.e. $f$ is a piecewise-continuous and exponentially-restricted real function $f$ satisfying the condition

$\mathcal{L}_t\{f(t)\}(s) = F(s),\ \forall s \in \mathbb R$

then $f(t)$ is uniquely determined (considering functions which differ from each other only on a point set having Lebesgue measure zero as the same). This result was first proven by Mathias Lerch in 1903 and is known as Lerch's theorem.[1][2]

The Laplace transform and the inverse Laplace transform together have a number of properties that make them useful for analysing linear dynamic systems.

## Mellin's inverse formula

An integral formula for the inverse Laplace transform, called the Bromwich integral, the Fourier–Mellin integral, and Mellin's inverse formula, is given by the line integral:

$f(t) = \mathcal{L}^{-1} \{F\}(t) = \mathcal{L}^{-1}_s \{F(s)\}(t) = \frac{1}{2\pi i}\lim_{T\to\infty}\int_{\gamma-iT}^{\gamma+iT}e^{st}F(s)\,ds,$

where the integration is done along the vertical line $Re(s)=\gamma$ in the complex plane such that $\gamma$ is greater than the real part of all singularities of F(s). This ensures that the contour path is in the region of convergence. If all singularities are in the left half-plane, or F(s) is a smooth function on - ∞ < Re(s) < ∞ (i.e. no singularities), then $\gamma$ can be set to zero and the above inverse integral formula above becomes identical to the inverse Fourier transform.

In practice, computing the complex integral can be done by using the Cauchy residue theorem.

It is named after Hjalmar Mellin, Joseph Fourier and Thomas John I'Anson Bromwich.

If F(s) is the Laplace transform of the function f(t),then f(t) is called the inverse Laplace transform of F(s).