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In mathematics, Legendre transform is an integral transform named after the mathematician Adrien-Marie Legendre, which uses Legendre polynomials P n ( x ) {\displaystyle P_{n}(x)} as kernels of the transform. Legendre transform is a special case of Jacobi transform.
The Legendre transform of a function f ( x ) {\displaystyle f(x)} is[1][2][3]
The inverse Legendre transform is given by
Associated Legendre transform is defined as