Laplace Transform of Dirac Delta Function/Proof 1

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

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

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


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