# PlanetPhysics/Laplace Transform of Diracs Delta

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A Dirac ${\displaystyle \delta }$ symbol can be interpreted as a linear functional, i.e. a linear mapping from a function space, consisting e.g. of certain real functions, to ${\displaystyle \mathbb {R} }$ (or ${\displaystyle \mathbb {C} }$), having the property


${\displaystyle \delta [f]\;=\;f(0).}$ One may think this as the inner product ${\displaystyle \langle f,\,\delta \rangle \;=\;\int _{0}^{\infty }\!f(t)\delta (t)\,dt}$ of a function ${\displaystyle f}$ and another "function" ${\displaystyle \delta }$, when the well-known formula ${\displaystyle \int _{0}^{\infty }\!f(t)\delta (t)\,dt\;=\;f(0)}$ is true.\, Applying this to\, ${\displaystyle f(t):=e^{-st}}$,\, one gets ${\displaystyle \int _{0}^{\infty }\!e^{-st}\delta (t)\,dt\;=\;e^{-0},}$

i.e. the Laplace transform

${\displaystyle {\begin{matrix}{\mathcal {L}}\{\delta (t)\}\;=\;1.\end{matrix}}}$

By the delay theorem, this result may be generalised to ${\displaystyle {\mathcal {L}}\{\delta (t\!-\!a))\}\;=\;e^{-as}.}$\\

When introducing a so-called "Dirac delta function", for example

${\displaystyle {\begin{matrix}\eta _{\varepsilon }(t)\;:=\;{\begin{cases}{\frac {1}{\varepsilon }}\quad {\mbox{for}}\;\;0\leq t\leq \varepsilon ,\\0\quad {\mbox{for}}\qquad t>\varepsilon ,\end{cases}}\end{matrix}}}$

as an "approximation" of Dirac delta, we obtain the Laplace transform ${\displaystyle {\mathcal {L}}\{\eta _{\varepsilon }(t)\}\;=\;\int _{0}^{\infty }\!e^{-st}\eta _{\varepsilon }(t)\,dt\;=\;\int _{0}^{\varepsilon }{\frac {e^{-st}}{\varepsilon }}\,dt+\int _{\varepsilon }^{\infty }\!e^{-st}\cdot 0\,dt\;=\;{\frac {1}{\varepsilon }}\int _{0}^{\varepsilon }\!e^{-st}\,dt\;=\;{\frac {1\!-\!e^{-\varepsilon s}}{\varepsilon s}}.}$ As the Taylor expansion shows, we then have ${\displaystyle \lim _{\varepsilon \to 0+}{\mathcal {L}}\{\eta _{\varepsilon }(t)\}\;=\;1,}$ according to ref.(2).

### Laplace transform of Dirac delta

The Dirac delta , ${\displaystyle \delta }$, can be correctly defined as a linear functional, i.e. a linear mapping from a function space, consisting e.g. of certain real functions, to ${\displaystyle \mathbb {R} }$ (or ${\displaystyle \mathbb {C} }$), having the property ${\displaystyle \delta [f]\;=\;f(0).}$ One may think of this as an inner product ${\displaystyle \langle f,\,\delta \rangle \;=\;\int _{0}^{\infty }\!f(t)\delta (t)\,dt}$ of a function ${\displaystyle f}$ and another "function" ${\displaystyle \delta }$, when the well-known formula ${\displaystyle \int _{0}^{\infty }\!f(t)\delta (t)\,dt\;=\;f(0)}$ holds.\, By applying this to \, ${\displaystyle f(t):=e^{-st}}$,\, one gets ${\displaystyle \int _{0}^{\infty }\!e^{-st}\delta (t)\,dt\;=\;e^{-0},}$ i.e. the Laplace transform

${\displaystyle {\begin{matrix}{\mathcal {L}}\{\delta (t)\}\;=\;1.\end{matrix}}}$

By the delay theorem, this result may be generalised to: ${\displaystyle {\mathcal {L}}\{\delta (t\!-\!a)\}\;=\;e^{-as}.}$

## References

1. Schwartz, L. (1950--1951), Th\'eorie des distributions, vols. 1--2, Hermann: Paris.
2. W. Rudin, Functional Analysis , McGraw-Hill Book Company, 1973.
3. L. H\"ormander, {\em The Analysis of Linear Partial Differential Operators I, (Distribution theory and Fourier Analysis)}, 2nd ed, Springer-Verlag, 1990.