# Legendre Transform Taking Too Long

I am trying to perform Legendre transform on $x^2 \ln(1+x^{\frac{1}{3}}+x^{\frac{2}{3}})$ using this.

legendreTransform[f_, x_, y_] :=
Module[{k = Flatten[{x}], q = Flatten[{y}], h}, First[h /.
Append[k, h]], {Solve::incnst, Solve::ifun}]]]
Simplify[legendreTransform[a (Log[1 + c*x^(1/3) + d*x^(2/3)]), x, y],
Assumptions -> {Element[y, Reals]}]


And I did confirm this code gives correct Legendre transforms of $e^x , x^2$.

However, it is taking very long for my computer (2 hours past, still no response after multiple tries) to run this with the expression above. This function is convex for x>0 , and this is the interval that I am interested in. Why is this the case?

Thank you very much for reading this question.

• Likely it's the x^(1/3) causing problems, this is not a single valued function, and so it must choose which branch to calculate. Note that legendreTransform[Log[1 + x^(1/3)], x, y] is already multi-valued. – bill s Oct 2 '17 at 2:41
• @bills then is it fine to add change the last part ' Assumptions -> {x>0, Element[x, Reals], Element[x, Reals]}' to solve that problem? – Duke Smith Oct 2 '17 at 2:49
• That won't really help because the branch cuts are occurring within the legendreTransform function, while the Assumptions are part of the Simplify function. – bill s Oct 2 '17 at 3:01
• @bills Where may I find info. on choosing the branch on Mathematica? is there a built-in function for it? – Duke Smith Oct 2 '17 at 3:12
• legendreTransform[ Refine[x^2 (Log[1 + x^(1/3) + x^(2/3)]), Assumptions -> {x > 0, Element[x, Reals]}], x, y] doing this will not help either? – Duke Smith Oct 2 '17 at 3:28

You need to specify that you want Solve to work over the reals, not the complexes.

legendreTransform[f_, x_, y_] := Module[{k = Flatten[{x}], q = Flatten[{y}], h},
First[h /. Quiet[
Solve[
Append[k, h],
Reals
],
{Solve::incnst, Solve::ifun}]
]
]


This will help it choose appropriate branches, and in this case produces an answer in little over a second.

 AbsoluteTiming[Simplify[legendreTransform[a (Log[1 + c*x^(1/3) + d*x^(2/3)]), x, y],
Assumptions -> {Element[y, Reals]}];]

(* {1.33583, Null}   I suppressed output becaues the result is big and ugly *)

• Thank you, but why do I get an error (returns h$1874, with 'Solve::nsmet: This system cannot be solved with the methods available to Solve.') When I plug in a * x * (Log[1 + cx^(1/3) + dx^(2/3)]) as my function to this code? (all I did was multiplying x) – Duke Smith Oct 4 '17 at 1:42 • Unfortunately, transcendental equations are not always solvable. Apparently, the one you get for that function falls into that camp. (And you're getting h$xxxx because Solve isn't returning a rule becuase it failed, so /. never evaluates, so the First seems extracts your module variable h from the /..) You can try removing the Reals and see if that returns a result over the complexes, but it would litterally spend hours or more trying to establish a solution. That may or may not work, and you may or may not be able to simplify the result to something usable over the reals. – Itai Seggev Oct 4 '17 at 4:55