Laplace Transform of Null Function

Theorem

Let $\NN: \R \to \R$ be a null function.

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

$\laptrans {\map \NN t} = 0$

Proof

\(\ds \laptrans {\map \NN t}\)

\(=\)

\(\ds \int_0^{\to +\infty} e^{-s t} \map \NN t \rd t\)

Definition of Laplace Transform

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Sources

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