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^[1][2][3]

${\displaystyle {\mathcal {J}}_{n}\{f(x)\}={\tilde {f}}(n)=\int _{-1}^{1}P_{n}(x)\ f(x)\ dx}$

The inverse Legendre transform is given by

${\displaystyle {\mathcal {J}}_{n}^{-1}\{{\tilde {f}}(n)\}=f(x)=\sum _{n=0}^{\infty }{\frac {2n+1}{2}}{\tilde {f}}(n)P_{n}(x)}$

Associated Legendre transform

Associated Legendre transform is defined as

${\displaystyle {\mathcal {J}}_{n,m}\{f(x)\}={\tilde {f}}(n,m)=\int _{-1}^{1}(1-x^{2})^{-m/2}P_{n}^{m}(x)\ f(x)\ dx}$

The inverse Legendre transform is given by

${\displaystyle {\mathcal {J}}_{n,m}^{-1}\{{\tilde {f}}(n,m)\}=f(x)=\sum _{n=0}^{\infty }{\frac {2n+1}{2}}{\frac {(n-m)!}{(n+m)!}}{\tilde {f}}(n,m)(1-x^{2})^{m/2}P_{n}^{m}(x)}$

Some Legendre transform pairs

${\displaystyle f(x)\,}$	${\displaystyle {\tilde {f}}(n)\,}$
${\displaystyle x^{n}\,}$	${\displaystyle {\frac {2^{n+1}(n!)^{2}}{(2n+1)!}}}$
${\displaystyle e^{ax}\,}$	${\displaystyle {\sqrt {\frac {2\pi }{a}}}I_{n+1/2}(a)}$
${\displaystyle e^{iax}\,}$	${\displaystyle {\sqrt {\frac {2\pi }{a}}}i^{n}J_{n+1/2}(a)}$
${\displaystyle xf(x)\,}$	${\displaystyle {\frac {1}{2n+1}}[(n+1){\tilde {f}}(n+1)+n{\tilde {f}}(n-1)]}$
${\displaystyle (1-x^{2})^{-1/2}\,}$	${\displaystyle \pi P_{n}^{2}(0)}$
${\displaystyle [2(a-x)]^{-1}\,}$	${\displaystyle Q_{n}(a)}$
${\displaystyle (1-2ax+a^{2})^{-1/2},\ |a|<1\,}$	${\displaystyle 2a^{n}(2n+1)^{-1}}$
${\displaystyle (1-2ax+a^{2})^{-3/2},\ |a|<1\,}$	${\displaystyle 2a^{n}(1-a^{2})^{-1}}$
${\displaystyle \int _{0}^{a}{\frac {t^{b-1}\,dt}{(1-2xt+t^{2})^{1/2}}},\ |a|<1\ b>0\,}$	${\displaystyle {\frac {2a^{n+b}}{(2n+1)(n+b)}}}$
${\displaystyle {\frac {d}{dx}}\left[(1-x^{2}){\frac {d}{dx}}\right]f(x)\,}$	${\displaystyle -n(n+1){\tilde {f}}(n)}$
${\displaystyle \left\{{\frac {d}{dx}}\left[(1-x^{2}){\frac {d}{dx}}\right]\right\}^{k}f(x)\,}$	${\displaystyle (-1)^{k}n^{k}(n+1)^{k}{\tilde {f}}(n)}$
${\displaystyle {\frac {f(x)}{4}}-{\frac {d}{dx}}\left[(1-x^{2}){\frac {d}{dx}}\right]f(x)\,}$	${\displaystyle \left(n+{\frac {1}{2}}\right)^{2}{\tilde {f}}(n)}$
${\displaystyle \ln(1-x)\,}$	${\displaystyle {\begin{cases}2(\ln 2-1),&n=0\\-{\frac {2}{n(n+1)}},&n>0\end{cases}}\,}$
${\displaystyle f(x)*g(x)\,}$	${\displaystyle {\tilde {f}}(n){\tilde {g}}(n)}$
${\displaystyle \int _{-1}^{x}f(t)\,dt\,}$	${\displaystyle {\begin{cases}{\tilde {f}}(0)-{\tilde {f}}(1),&n=0\\{\frac {{\tilde {f}}(n-1)-{\tilde {f}}(n+1)}{2n+1}},&n>1\end{cases}}\,}$
${\displaystyle {\frac {d}{dx}}g(x),\ g(x)=\int _{-1}^{x}f(t)\,dt}$	${\displaystyle g(1)-\int _{-1}^{1}g(x){\frac {d}{dx}}P_{n}(x)\,dx}$

References