legendre transform integral transform
Legendre transform (integral transform) GudangMovies21 Rebahinxxi LK21
In mathematics, Legendre transform" target="_blank">transform is an integral transform" target="_blank">transform named after the mathematician Adrien-Marie Legendre, which uses Legendre polynomials
P
n
(
x
)
{\displaystyle P_{n}(x)}
as kernels of the transform" target="_blank">transform. Legendre transform" target="_blank">transform is a special case of Jacobi transform" target="_blank">transform.
The Legendre transform" target="_blank">transform of a function
f
(
x
)
{\displaystyle f(x)}
is
J
n
{
f
(
x
)
}
=
f
~
(
n
)
=
∫
−
1
1
P
n
(
x
)
f
(
x
)
d
x
{\displaystyle {\mathcal {J}}_{n}\{f(x)\}={\tilde {f}}(n)=\int _{-1}^{1}P_{n}(x)\ f(x)\ dx}
The inverse Legendre transform" target="_blank">transform is given by
J
n
−
1
{
f
~
(
n
)
}
=
f
(
x
)
=
∑
n
=
0
∞
2
n
+
1
2
f
~
(
n
)
P
n
(
x
)
{\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" target="_blank">transform
Associated Legendre transform" target="_blank">transform is defined as
J
n
,
m
{
f
(
x
)
}
=
f
~
(
n
,
m
)
=
∫
−
1
1
(
1
−
x
2
)
−
m
/
2
P
n
m
(
x
)
f
(
x
)
d
x
{\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" target="_blank">transform is given by
J
n
,
m
−
1
{
f
~
(
n
,
m
)
}
=
f
(
x
)
=
∑
n
=
0
∞
2
n
+
1
2
(
n
−
m
)
!
(
n
+
m
)
!
f
~
(
n
,
m
)
(
1
−
x
2
)
m
/
2
P
n
m
(
x
)
{\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)}