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 is
The inverse Legendre transform is given by

Associated Legendre transform

Associated Legendre transform is defined as
The inverse Legendre transform is given by