I am trying to find the inverse Laplace transform of the Gaussian function $$ G(s)=e^{-cs^2}$$
$L^{-1}\left\{G(s)\right\}= ?$ Where $c> 0$.
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Welcome to MSE. Your question is phrased as an isolated problem, without any further information or context. This does not match many users' quality standards, so it may attract downvotes, or be put on hold. To prevent that, please [edit] the question. This will help you recognise and resolve the issues. Concretely: please provide context, and include your work and thoughts on the problem. These changes can help in formulating more appropriate answers. – José Carlos Santos Aug 26 '19 at 10:11
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1You mean $c < 0$ then its bilateral inverse Laplace transform is $\sqrt{\pi/2}(-2c)^{-1/2}e^{-t^2/(-4c)}$. For $c> 0$ its growth on vertical lines is way too large to be a Laplace transform. – reuns Aug 26 '19 at 10:35