Legendre transform (integral transform)

Template:About In mathematics, Legendre transform is an integral transform named after the mathematician Adrien-Marie Legendre, which uses Legendre polynomials <math>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>f(x)</math> is<ref>Template:Cite book</ref><ref>Template:Cite journal</ref><ref>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.</ref>

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

References

Template:Math-physics-stub