2011-04-04, 10:23 AM
I've found the Wikipedia description quite clear on the way that the Legendre's transform works.
On the unrelated question:
The effect of large order derivatives on the original function decreases on 1/n! (coming from the taylor expansion), so even if your derivatives are increasing on w^n that "loses" to the factorial and thus it converges. Deriving infinite times has no meaning as well, it has about the same meaning as saying "if we take an infinitely large number for x, does x^2 diverge?"
On the unrelated question:
The effect of large order derivatives on the original function decreases on 1/n! (coming from the taylor expansion), so even if your derivatives are increasing on w^n that "loses" to the factorial and thus it converges. Deriving infinite times has no meaning as well, it has about the same meaning as saying "if we take an infinitely large number for x, does x^2 diverge?"