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

\(\ds \)

\(=\)

\(\ds e^0\)

\(\ds \)

\(=\)

\(\ds 1\)

$\blacksquare$