Legendre transform (integral transform)

In mathematics, Legendre transform is an integral transform named after the mathematician Adrien-Marie Legendre, which uses Legendre polynomials as kernels of the transform. Legendre transform is a special case of Jacobi transform.

The Legendre transform of a function isChurchill, R. V., and C. L. Dolph. "Inverse transforms of products of Legendre transforms." Proceedings of the American Mathematical Society 5.1 (1954): 93–100.

The inverse Legendre transform is given by

Associated Legendre transform

Associated Legendre transform is defined as

The inverse Legendre transform is given by

Some Legendre transform pairs

References