Legendre transform

In mathematics, Legendre transform is an integral transform named after the mathematician Adrien-Marie Legendre, which uses Legendre polynomials $P_{n}(x)$ as kernels of the transform. Legendre transform is a special case of Jacobi transform.

The Legendre transform of a function $f(x)$ is[1][2][3]

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

The inverse Legendre transform is given by

{\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

{\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

{\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)

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

Debnath, Lokenath, and Dambaru Bhatta. Integral transforms and their applications. CRC press, 2014.
Churchill, R. V. "The operational calculus of Legendre transforms." Studies in Applied Mathematics 33.1–4 (1954): 165–178.
Churchill, R. V., and C. L. Dolph. "Inverse transforms of products of Legendre transforms." Proceedings of the American Mathematical Society 5.1 (1954): 93–100.

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