Is there any good rational approximation of the Legendre transform of the function $$f(x)=\log\left(\frac{e^x-1}{x}\right)$$ I tried to approximate the derivate of $f$, using Taylor, however, the function I got was not a good approximation, since it wasn't always positive and having infite limits at $0$ and at $1$.
2026-02-23 20:35:19.1771878919
Legendre transform approximation
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