PlanetMath: Laplace transform of a Gaussian function

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Laplace transform of a Gaussian function

(Application)

We evaluate the Laplace transform ¹

$\displaystyle \mathcal{L}\{e^{-t^2}\}=\int_0^\infty e^{-st}e^{-t^2}\,dt=F(s).$

(1)

In fact,

$\displaystyle \mathcal{L}\{e^{-t^2}\}= \int_0^\infty e^{-(t^2+2\frac{s}{2}t+\fr... ...rac{s^2}{4})}\,dt= e^\frac{s^2}{4}\!\!\int_0^\infty e^{-(t+\frac{s}{2})^2}\,dt.$

By making the change of variable $t+\frac{s}{2}=u$ , we have (by the second equality in (1), the variable on operator's argument is immaterial)

$\displaystyle \mathcal{L}\{e^{-t^2}\}= e^\frac{s^2}{4}\!\!\int_{\frac{s}{2}}^\infty e^{-u^2}\,du.$

That is,

$\displaystyle \mathcal{L}\{e^{-t^2}\}=F(s)= \frac{\sqrt{\pi}}{2}e^\frac{s^2}{4}\mathrm{erfc}\Big(\frac{s}{2}\Big),$

where $\mathrm{erfc}(\cdot)$ is the complementary error function. Its path of integration is subject to the restriction $\arg{(u)}\to\theta$ , with $|\theta|\leq\pi/4$ as $u\to\infty$ along the path, with equality only if $\Re{(u^2)}$ remains bounded to the left.

Footnotes

...¹: cf. $\emph{Gaussian function}$ , wikipedia.org

"Laplace transform of a Gaussian function" is owned by perucho.

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Cross-references: bounded, restriction, path, complementary error function, argument, operator's, equality, variable, Wikipedia, Laplace transform
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This is version 2 of Laplace transform of a Gaussian function, born on 2006-06-30, modified 2006-06-30.
Object id is 8108, canonical name is LaplaceTransformOfAGaussianFunction.
Accessed 6174 times total.

Classification:

AMS MSC:

42-01 (Fourier analysis :: Instructional exposition )

Pending Errata and Addenda

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