# Laplace Transform of Null Function

## Theorem

Let $\mathcal N: \R \to \R$ be a null function.

The Laplace transform of $\map {\mathcal N} t$ is given by:

$\laptrans {\map {\mathcal N} t} = 0$

## Proof

 $\displaystyle \laptrans {\map {\mathcal N} t}$ $=$ $\displaystyle \int_0^{\to +\infty} e^{-s t} \map {\mathcal N} t \rd t$ Definition of Laplace Transform