It seems to me that most distributions (positive, bounded, finite integral, continuous (to some degree)) must have a polynomial Bilateral Laplace transform. How is this inverted? Most inverse Laplace transforms seem to rely on poles of function, what if it doesn't have any?
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L[Uniform distribution ,s] = Sin[s]/s, Triangular distribution (Sin[s]/s)^2 – aronp Dec 23 '15 at 21:25
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1Inverse Laplace Transform of the entire functions are often especially difficult to express in nice form, even they obviously exist. Typical examples including the inverse Laplace Transform of $e^{as^2}$ (http://math.stackexchange.com/questions/169275) . And the inverse Laplace Transform of $s^n$ for natural number $n$ should express in terms of the derivative of the Dirac delta function (http://math.stackexchange.com/questions/1010927). – doraemonpaul Dec 26 '15 at 17:42
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Thanks, so there is no 'standard' mechanism for inverting entire functions, or even polynomials. – aronp Dec 26 '15 at 17:56