# Definition:Legendre Transform

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## Definition

Let $\map f x$ be a strictly convex real function.

Let $p = \map {f'} x$.

Let $\map {f^*} p = - \map f{\map x p} + p \map x p$.

The **Legendre Transform on $x$ and $f$** is the mapping of the variable and function pair:

- $\paren{x, \map f x} \to \paren{p, \map {f^*} p}$

## Source of Name

This entry was named for Adrien-Marie Legendre.

## Sources

- 1963: I.M. Gelfand and S.V. Fomin:
*Calculus of Variations*... (previous) ... (next): $\S 4.18$: The Legendre Tranformation