# Laplace–Carson transform

In mathematics, the Laplace–Carson transform, named after Pierre Simon Laplace and John Renshaw Carson, is an integral transform with significant applications in the field of physics and engineering, particularly in the field of railway engineering.

## Definition

Let $V(j,t)$ be a function and $p$ a complex variable. The Laplace–Carson transform is defined as:[1]

$V^\ast(j,p) = p\int^{\infty}_0 V(j,t) e^{-pt} \, dt$

The inverse Laplace–Carson transform is:

$V(j,t) = \frac{1}{2\pi i} \int^{a_0+i\infty}_{a_0-i\infty} e^{tp} \frac{V^\ast(j,p)}{p} \, dp$

where $a_0$ is a real-valued constant, $i\infty$ refers to the imaginary axis, which indicates the integral is carried out along a straight line parallel to the imaginary axis lying to the right of all the singularities of the following expression:

$e^{tp}\frac{V(j,t)}{p}$