Has someone implemented an inverse Stieltjes Transform in Mathematica? In the following example, I'm hoping to recover Beta PDF from its Stieltjes Transform in a robust way.
dist = BetaDistribution[1/2, 1];
stieltjes = Expectation[1/(s - y), y \[Distributed] dist]
Because Stieltjes transform is equivalent to Laplace Transform performed twice, this would give an alternative way of recovering density from MGF (Mathematica's InverseLaplaceTransform gets stuck inverting Beta mgf directly)
mgf = MomentGeneratingFunction[BetaDistribution[1/2, 1], t];
doubleLaplace = LaplaceTransform[mgf, t, s] // FullSimplify;
doubleLaplace == stieltjes (*True if s>1*)
Cross-posted on mathematica.SE
