How to compute symbolically the Legendre transform of a function?

Question

Is it possible to compute symbolically the Legendre transform of a function in Mathematica?

I find no LegendreTransform function.

See:https://mathematica.stackexchange.com/questions/151512/legendre-transform-of-a-function-of-a-3-vector/151536#151536 — Mariusz Iwaniuk, Apr 20 '18 at 20:53

score 9 · Accepted Answer · answered Apr 23 '18 at 12:56

I wrote a small function to compute Legendre transform of differentiable univariate functions.

LegendreTransform[f_]:=Module[{x,y,sol,expr},
    sol = Solve[f'[x] == y, x, Reals];
    expr = FullSimplify[First[y x - f[x]/.sol], {x,y}\[Element]Reals];
    Evaluate[expr/.y->#]&]

Some examples:

LegendreTransform[-Log[#] &]
(* -1 + Log[-(1/#1)] & *)

LegendreTransform[Exp[#] &]
(* ConditionalExpression[(-1 + Log[#1]) #1, #1 > 0] & *)

It works only when $f$ is convex – tituf Mar 14 '23 at 17:18 — tituf, Mar 14 '23 at 17:18

How to compute symbolically the Legendre transform of a function?

1 Answers1