Legendre transformation
（ plural ）
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(mathematics) Given a function f(x,y,z,...) which is concave up with respect to x (i.e., its second derivative with respect to x is greater than zero), an involutive procedure for replacing x with another variable, say p=∂f/∂x thus yielding another function, say F=F(p,y,z,...). This new function contains all of the information of the original f encoded, as it were, within it so that ∂F/∂p=x and applying a similar transformation to F yields the original f. The formula is: F(p,y,z,...)=p·x(p)-f(x(p),y,z,...) where x must be expressed as a function of p. (Note: The concave upwardness means that ∂f/∂x is monotonically increasing, which means that p as a function of x is invertible, so x should be expressible as a function of p.)
Legendre transformation
For a function f that is concave up in x (so ∂2f/∂x2 > 0), performing a Legendre transformation replaces x with its conjugate variable p = ∂f/∂x, yielding a new function that encodes the same information in terms of p.
For a function f that is concave up in x (so ∂2f/∂x2 > 0), performing a Legendre transformation replaces x with its conjugate variable p = ∂f/∂x, yielding a new function that encodes the same information in terms of p.
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