Legendre transform (integral transform) explained

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]} l{J} _{n\{f(x)\}} = \tilde f(n) = \int ^{1} _{-1} P _{n(x)} f(x) dx

The inverse Legendre transform is given by

l{J} ^{-1} _{n} \{\tilde f(n)\} = f(x) = \sum ^{inf ty} _{n =0} (2 n +1)/(2) \tilde f(n) P _{n(x)}

Associated Legendre transform

Associated Legendre transform is defined as

l{J} _{n,m} \{f(x)\} = \tilde f(n,m) = \int ^{1} _{-1} ( 1- x ^{2)} ^{-m/2} P ^{m(x)} _{n} f(x) dx

The inverse Legendre transform is given by

l{J} ^{-1} _{n,m} \{\tilde f(n,m)\} = f(x) = \sum ^{inf ty} _{n =0} (2 n +1)/(2) ((n - m)!)/((n + m)!) \tilde f(n,m)( 1- x ^{2)} ^{m/2} P ^{m(x)} _{n}

Some Legendre transform pairs

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

Notes and References

1. Book: Debnath . Lokenath . Dambaru Bhatta. Integral transforms and their applications. . 2007 . Chapman & Hall/CRC . Boca Raton . 9781482223576 . 2nd.
2. Churchill . R. V. . The Operational Calculus of Legendre Transforms . Journal of Mathematics and Physics . 1954 . 33 . 1-4 . 165–178 . 10.1002/sapm1954331165. 2027.42/113680 . free .
3. 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.