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- Homework Statement
- Find the inverse Laplace transform of 1/s^2 * e^(-sx^2/2)

- Relevant Equations
- Laplace convolution theorem equation.

My attempt at finding this was via convolution theorem, where we take F(s) = 1/s^2 and G(s) = e^(-sx^2/2). Then to use convolution we need to find the inverses of those transforms. From a table of Laplace transforms we know that f(t) = t. But I am sort of struggling with e^(-sx^2/2). My 'guess' is that the inverse Laplace transform of e^(-sx^2/2) is δ(t - (x^2)/2). This is from the fact that the inverse Laplace transform of e^sc is δ(t+c). But then integrating this in the convolution integral proves difficult.

Am I on the right track or should I use another method of finding the inverse Laplace transform of 1/s^2 * e^(-sx^2/2)?

Note: Here I have used * to denote multiplication and not convolution.

Am I on the right track or should I use another method of finding the inverse Laplace transform of 1/s^2 * e^(-sx^2/2)?

Note: Here I have used * to denote multiplication and not convolution.