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
J n { f ( x ) } = f ~ ( n ) = ∫ − 1 1 P n ( x ) f ( x ) d x {\displaystyle {\mathcal {J}}_{n}\{f(x)\}={\tilde {f}}(n)=\int _{-1}^{1}P_{n}(x)\ f(x)\ dx}
The inverse Legendre transform is given by
J n − 1 { f ~ ( n ) } = f ( x ) = ∑ n = 0 ∞ 2 n + 1 2 f ~ ( n ) P n ( x ) {\displaystyle {\mathcal {J}}_{n}^{-1}\{{\tilde {f}}(n)\}=f(x)=\sum _{n=0}^{\infty }{\frac {2n+1}{2}}{\tilde {f}}(n)P_{n}(x)}
Associated Legendre transform Associated Legendre transform is defined as
J n , m { f ( x ) } = f ~ ( n , m ) = ∫ − 1 1 ( 1 − x 2 ) − m / 2 P n m ( x ) f ( x ) d x {\displaystyle {\mathcal {J}}_{n,m}\{f(x)\}={\tilde {f}}(n,m)=\int _{-1}^{1}(1-x^{2})^{-m/2}P_{n}^{m}(x)\ f(x)\ dx}
The inverse Legendre transform is given by
J n , m − 1 { f ~ ( n , m ) } = f ( x ) = ∑ n = 0 ∞ 2 n + 1 2 ( n − m ) ! ( n + m ) ! f ~ ( n , m ) ( 1 − x 2 ) m / 2 P n m ( x ) {\displaystyle {\mathcal {J}}_{n,m}^{-1}\{{\tilde {f}}(n,m)\}=f(x)=\sum _{n=0}^{\infty }{\frac {2n+1}{2}}{\frac {(n-m)!}{(n+m)!}}{\tilde {f}}(n,m)(1-x^{2})^{m/2}P_{n}^{m}(x)}
Some Legendre transform pairs
References