Laplace Transform of Shifted Dirac Delta Function

Theorem

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

The Laplace transform of $\map \delta {t - a}$ is given by:

$\laptrans {\map \delta {t - a} } = e^{-a s}$

$\ds \laptrans {\map \delta {t - a} }$

$=$

$\ds \int_0^{\to +\infty} e^{-s t} \map \delta {t - a} \rd t$

Definition of Laplace Transform

$\ds $

$=$

$\ds e^{-s \times a}$

$\ds $

$=$

$\ds e^{-a s}$

$\blacksquare$

1965: Murray R. Spiegel: Theory and Problems of Laplace Transforms ... (previous) ... (next): Chapter $1$: The Laplace Transform: Laplace Transforms of Special Functions: $13$
1965: Murray R. Spiegel: Theory and Problems of Laplace Transforms ... (previous) ... (next): Appendix $\text B$: Table of Special Laplace Transforms: $113.$
1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 32$: Table of Special Laplace Transforms: $32.137$
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: $33.137.$