Legendre transform (integral transform)

In mathematics, Legendre transform is an integral transform named after the mathematician Adrien-Marie Legendre, which uses Legendre polynomials [math]\displaystyle{ P_n(x) }[/math] as kernels of the transform. Legendre transform is a special case of Jacobi transform.

The Legendre transform of a function [math]\displaystyle{ f(x) }[/math] is

[math]\displaystyle{ \mathcal{J}_n\{f(x)\} = \tilde f(n) = \int_{-1}^1 P_n(x)\ f(x) \ dx }[/math]

The inverse Legendre transform is given by

[math]\displaystyle{ \mathcal{J}_n^{-1}\{\tilde f(n)\} = f(x) = \sum_{n=0}^\infty \frac{2n+1}{2} \tilde f(n) P_n(x) }[/math]

Associated Legendre transform

Associated Legendre transform is defined as

[math]\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 }[/math]

The inverse Legendre transform is given by

[math]\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) }[/math]

Some Legendre transform pairs

[math]\displaystyle{ f(x)\, }[/math]	[math]\displaystyle{ \tilde f(n)\, }[/math]
[math]\displaystyle{ x^n \, }[/math]	[math]\displaystyle{ \frac{2^{n+1} (n!)^2}{(2n+1)!} }[/math]
[math]\displaystyle{ e^{ax} \, }[/math]	[math]\displaystyle{ \sqrt{\frac{2\pi}{a}}I_{n+1/2}(a) }[/math]
[math]\displaystyle{ e^{iax} \, }[/math]	[math]\displaystyle{ \sqrt{\frac{2\pi}{a}}i^n J_{n+1/2}(a) }[/math]
[math]\displaystyle{ xf(x) \, }[/math]	[math]\displaystyle{ \frac{1}{2n+1}[(n+1)\tilde f(n+1) + n \tilde f(n-1)] }[/math]
[math]\displaystyle{ (1-x^2)^{-1/2} \, }[/math]	[math]\displaystyle{ \pi P_n^2(0) }[/math]
[math]\displaystyle{ [2(a-x)]^{-1} \, }[/math]	[math]\displaystyle{ Q_n(a) }[/math]
[math]\displaystyle{ (1-2ax+a^2)^{-1/2}, \ |a|\lt 1 \, }[/math]	[math]\displaystyle{ 2a^n (2n+1)^{-1} }[/math]
[math]\displaystyle{ (1-2ax+a^2)^{-3/2}, \ |a|\lt 1 \, }[/math]	[math]\displaystyle{ 2a^n (1-a^2)^{-1} }[/math]
[math]\displaystyle{ \int_0^a \frac{t^{b-1} \, dt}{(1-2xt + t^2)^{1/2}}, \ |a|\lt 1 \ b\gt 0 \, }[/math]	[math]\displaystyle{ \frac{2a^{n+b}}{(2n+1)(n+b)} }[/math]
[math]\displaystyle{ \frac{d}{dx}\left[(1-x^2)\frac{d}{dx} \right] f(x)\, }[/math]	[math]\displaystyle{ -n(n+1)\tilde f(n) }[/math]
[math]\displaystyle{ \left\{\frac{d}{dx}\left[(1-x^2)\frac{d}{dx} \right]\right\}^k f(x)\, }[/math]	[math]\displaystyle{ (-1)^k n^k (n+1)^k \tilde f(n) }[/math]
[math]\displaystyle{ \frac{f(x)}{4}-\frac{d}{dx}\left[(1-x^2)\frac{d}{dx} \right] f(x)\, }[/math]	[math]\displaystyle{ \left(n+\frac{1}{2}\right)^2\tilde f(n) }[/math]
[math]\displaystyle{ \ln(1-x) \, }[/math]	[math]\displaystyle{ \begin{cases} 2(\ln 2 -1) , & n= 0\\ -\frac{2}{n(n+1)} , & n\gt 0 \end{cases}\, }[/math]
[math]\displaystyle{ f(x)*g(x)\, }[/math]	[math]\displaystyle{ \tilde f(n)\tilde g(n) }[/math]
[math]\displaystyle{ \int_{-1}^x f(t) \, dt \, }[/math]	[math]\displaystyle{ \begin{cases} \tilde f(0)-\tilde f(1) , & n= 0\\ \frac{\tilde f(n-1) - \tilde f(n+1)}{2n+1} , & n\gt 1 \end{cases}\, }[/math]
[math]\displaystyle{ \frac{d}{dx} g(x), \ g(x) = \int_{-1}^x f(t) \,dt }[/math]	[math]\displaystyle{ g(1) - \int_{-1}^1g(x) \frac{d}{dx} P_n(x) \,dx }[/math]