Legendre transform

In mathematics, Legendre transform is an integral transform named after the mathematician Adrien-Marie Legendre, which uses Legendre polynomials ${\displaystyle P_{n}(x)}$ as kernels of the transform. Legendre transform is a special case of Jacobi transform.

The Legendre transform of a function ${\displaystyle f(x)}$ is

The inverse Legendre transform is given by

Associated Legendre transform is defined as

The inverse Legendre transform is given by

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