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: $14$
  • 1965: Murray R. Spiegel: Theory and Problems of Laplace Transforms ... (previous) ... (next): Appendix $\text B$: Table of Special Laplace Transforms: $111.$
  • 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 32$: Table of Special Laplace Transforms: $32.135$
  • 2009: Murray R. Spiegel, Seymour Lipschutz and John Liu: Mathematical Handbook of Formulas and Tables (3rd ed.) ... (previous) ... (next): $\S 33$: Laplace Transforms: Table of Special Laplace Transforms: $33.135.$
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