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How would I manage $\scr L \{e^{t^2}\}$? Does it even make sense to ask? Is it just a given that there are diverging Laplace functions that can't be handled?

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You should always be able to use first principals (the formal definition of the Laplace Transform) and test to see if the result is valid. For example, see Example 41.1 d. I would recommend that you work problems a. through d. and make sure you understand why these are true. Does that make sense? Regards – Amzoti Dec 22 '12 at 4:23
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It doesn't make sense. The domain of the Laplace transform is by definition the set of functions $f(t)$ for which $\int_0^\infty f(t)e^{-st}\,dt$ converges. (The set of all $s\in\mathbb{C}$ for which the integral converges is called the region of convergence of the transform.)

As such, the function you named simply is not in the domain of the Laplace transform operator.

It's like "needing" the integral of an function that is not integrable, or the derivative of a function that is not differentiable: oh well.

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