Monday Math 113

The Laplace Transform
Part 14: Gaussians and Error Function

Now, let’s look at the Laplace transform of a Gaussian function, specifically the function $f(t)=e^{-t^2}$ , for t≥0. From our transform definition,
$F(s)=\mathcal{L}\left[f(t)\right](s)=\int_0^{\infty}e^{-st}e^{-t^2}\,dt$
We proceed by completing the square:
$\begin{eqnarray}F(s)&=&\int_0^{\infty}e^{-\left(t^2+st\right)}\,dt\\&=&\int_0^{\infty}e^{-\left(t^2+2\frac{s}2t+\frac{s^2}4\right)+\frac{s^2}4}\,dt\\&=&e^{\frac{s^2}4}\int_0^{\infty}e^{-\left(t+\frac{s}2\right)^2}\,dt\end{eqnarray}$
To simplify from here, we avoid complications of trajectory in the complex plane by assuming s real (the result can be extended to the region of convergence $\Re(s)\gt0$ ). Then, by substituting $u=t+\frac{s}2$ , we have
$\begin{eqnarray}F(s)&=&e^{\frac{s^2}4}\int_0^{\infty}e^{-\left(t+\frac{s}2\right)^2}\,dt\\&=&e^{\frac{s^2}4}\int_{\frac{s}2}^{\infty}e^{-u^2}\,du\end{eqnarray}$ .
In terms of the error function $\operatorname{erf}(x)=\frac2{\sqrt\pi}\int_0^xe^{-u^2}\,du$ , and using $\operatorname{erf}(\infty)=\frac2{\sqrt\pi}\int_0^{\infty}e^{-u^2}\,du=1$ , we see
$\int_{\frac{s}2}^{\infty}e^{-u^2}\,du=\frac{\sqrt\pi}2\left(1-\operatorname{erf}\left(\frac{s}2\right)\right)$ ,
and so
$F(s)=\frac{\sqrt\pi}2e^{\frac{s^2}4}\left(1-\operatorname{erf}\left(\frac{s}2\right)\right)$ .

We can also use this to find the transform of the error function, via our integration rule:
$\begin{eqnarray}\mathcal{L}\left[\operatorname{erf}(t)H(t)\right](s)&=&\frac2{\sqrt\pi}\mathcal{L}\left[H(t)\int_0^te^{-u^2}\,du\right](s)\\&=&\frac2{\sqrt\pi}\frac1s\mathcal{L}\left[H(t)e^{-u^2}\,du\right](s)\\&=&\frac2{\sqrt\pi}\frac1s\cdot\frac{\sqrt\pi}2e^{\frac{s^2}4}\left(1-\operatorname{erf}\left(\frac{s}2\right)\right)\\\mathcal{L}\left[\operatorname{erf}(t)H(t)\right](s)&=&\frac{e^{s^2/4}\left(1-\operatorname{erf}\left(\frac{s}2\right)\right)}s\end{eqnarray}$
(also with $\Re(s)\gt0$ ).

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Tags: Error Function, Gaussian, Laplace Transform, Math, Monday Math

This entry was posted on March 22, 2010 at 2:04 pm and is filed under Math/Science. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.

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Monday Math 113

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