UnkleRhaukus
  • UnkleRhaukus
\[\mathcal L\left\{e^{-t^2}\right\}=\]
Mathematics
chestercat
  • chestercat
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UnkleRhaukus
  • UnkleRhaukus
\[\begin{align*} \mathcal L\left\{e^{-t^2}\right\}&= \int\limits_0^\infty e^{-t^2}e^{-pt}\cdot\text dt\\ &=\int\limits_0^\infty e^{-t(p+t)}\cdot\text dt\\ \end{align*}\]
anonymous
  • anonymous
I would think that any base, including e, when raised to a negative exponent, becomes a decay problem that approaches 0.
UnkleRhaukus
  • UnkleRhaukus
hmm?

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AccessDenied
  • AccessDenied
Maybe completing the square would be helpful? I did a similar problem (it was the only 'question' i asked) and it seemed to work out well for that one. http://puu.sh/1r1nO
UnkleRhaukus
  • UnkleRhaukus
\[\begin{align*} \mathcal L\left\{e^{-t^2}\right\}&= \int\limits_0^\infty e^{-t^2}e^{-pt}\cdot\text dt\\ &=\int\limits_0^\infty e^{-(t^2+pt)}\cdot\text dt\\ &=\int\limits_0^\infty e^{-\left(t^2+pt+\left(\tfrac p2\right)^2\right)+\left(\tfrac p2\right)^2}\cdot\text dt\\ &=e^{\left(\tfrac p2\right)^2}\int\limits_0^\infty e^{-\left(t+\tfrac p2\right)^2}\cdot\text dt\\ &=e^{\tfrac {p^2}4}\int\limits_0^\infty e^{-\left(t+\tfrac p2\right)^2}\cdot\text dt\\ \text{let }t+\tfrac p2=w\\ \text dt=\text dw\\ t=0\rightarrow w=\tfrac p2\\ t=\infty\rightarrow w=\infty\\ &=e^{\tfrac {p^2}4}\int\limits_{\tfrac p2}^\infty e^{-w^2}\cdot\text dw\\ &=e^{\tfrac {p^2}4}\left(\frac{\sqrt\pi}{2}\text{erfc}\left(\tfrac p2\right)\right)\\ &=\frac{\sqrt\pi}{2}e^{\tfrac {p^2}4}\text{erfc}\left(\tfrac p2\right)\color{red}\checkmark\\ \end{align*}\]
UnkleRhaukus
  • UnkleRhaukus
thankyou @AccessDenied
AccessDenied
  • AccessDenied
You're welcome! :)

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