Examples
The exponential function ex has x ln x − x as a Legendre transform since their respective first derivatives ex and ln x are inverse to each other. This example shows that the respective domains of a function and its Legendre transform need not agree. As another easy example, for the Legendre transform is
Similarly, the quadratic form
with A a symmetric invertible n-by-n-matrix has
as a Legendre transform.
Read more about this topic: Legendre Transformation
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