# Laplace Transform of Dirac Delta Function/Proof 1

## Theorem

Let $\map \delta t$ denote the Dirac delta function.

The Laplace transform of $\map \delta t$ is given by:

$\laptrans {\map \delta t} = 1$

## Proof

 $\ds \laptrans {\map \delta t}$ $=$ $\ds \int_0^{\to +\infty} e^{-s t} \map \delta t \rd t$ Definition of Laplace Transform $\ds$ $=$ $\ds e^{-s \times 0}$ Integral to Infinity of Dirac Delta Function by Continuous Function $\ds$ $=$ $\ds e^0$ $\ds$ $=$ $\ds 1$

$\blacksquare$