Timeline for Visualizing density given by its Stieltjes transformation?
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
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| Jun 25, 2023 at 18:19 | comment | added | Yaroslav Bulatov | I confirmed that this transformation is correct, limit + rational values seems like a useful trick to know! (background here) | |
| Jun 25, 2023 at 18:17 | vote | accept | Yaroslav Bulatov | ||
| Jun 25, 2023 at 7:01 | comment | added | Yaroslav Bulatov | Thanks for the update...this expression is also the one I hit a dead end trying to compute the inverse Laplace transform of, being able to get the explicit expression for inverse Stieltjes is promising | |
| Jun 25, 2023 at 4:44 | comment | added | JimB |
Yes but I've only got it to work for rational values of $x$: stieltjes[s_] := ArcCoth[Sqrt[s]]/Sqrt[s]; Limit[(stieltjes[x - I \[Epsilon]] - stieltjes[x + I \[Epsilon]])/(2 \[Pi] I) /. x -> 7/8, \[Epsilon] -> 0, Direction -> "FromAbove"] gives the same result as PDF[BetaDistribution[1/2, 1], 7/8].
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| Jun 25, 2023 at 4:35 | history | edited | JimB | CC BY-SA 4.0 |
Added general formula.
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| Jun 25, 2023 at 2:44 | comment | added | Yaroslav Bulatov | Does that formula work for the regular beta distribution? (Away from mathematica right now) | |
| Jun 25, 2023 at 2:29 | comment | added | Yaroslav Bulatov | Interesting!.... actually that looks close to what I expected, my expression was obtained using two Laplace transforms instead of MFG followed by Laplace, so that may explain the negative sign | |
| Jun 25, 2023 at 1:49 | history | answered | JimB | CC BY-SA 4.0 |