Inverse Stieltjes transform in Wolfram Language?

Questions
1

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

2

Mariusz Iwaniuk was quite helpful in finding techniques that often succeed. I’m using this post as a discussion placeholder, aiming to post lessons as a compressed version here when it’s ready

https://mathematica.stackexchange.com/questions/307837/robust-computation-of-inverse-stieltjes-transform

3

Mariusz Iwaniuk’s answer can be found here: https://community.wolfram.com/groups/-/m/t/3302428