# Laplace Transform of Null Function

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## 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 |

## Sources

- 1965: Murray R. Spiegel:
*Theory and Problems of Laplace Transforms*... (previous) ... (next): Chapter $1$: The Laplace Transform: Laplace Transforms of Special Functions: $14$