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I have function $f(t)$ for which both $h=\mathcal{L}(f)$ and $g=\mathcal{L}^{-1}[f]$ have nice closed forms, and I want to approximate $f$ with a nice expression which has correct $t\to \infty$ asymptotics but is also accurate for $t=10,20,\ldots,100$

Any tips?

g = (4 Sqrt[2])/(
   Sqrt[x] (8 + \[Pi]^2 x + 
      4 Sqrt[2] Sqrt[x] ArcTanh[(2 Sqrt[2] Sqrt[x])/(2 + x)] + 
      x ArcTanh[(2 Sqrt[2] Sqrt[x])/(2 + x)]^2)) Boole[0 < x < 2];
h = (Sqrt[2] (\[Pi] - 2 ArcTan[Sqrt[s]/Sqrt[2]]))/(
  2 Sqrt[s] + Sqrt[2] s ArcTan[Sqrt[2]/Sqrt[s]]);
DiscretePlot[{LaplaceTransform[g, x, t], 
  InverseLaplaceTransform[h, s, t]}, {t, 10., 100, 10.}, 
 Joined -> {True, False}, PlotLegends -> {"L[g]", "invL[h]"}, 
 PlotLabel -> f, AxesLabel -> {"t"}]

enter image description here

(This is a solution of differential equation here, a continuous mean-field model)

Yaroslav Bulatov
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0 Answers0