// Workers AI · traducción al español
Transformación Jacobi
En matemáticas, la transformación Jacobi es una transformación integral nombrada en honor del matemático Carl Gustav Jacob Jacobi, que utiliza polinomios Jacobi como núcleos de la transformación.
Traducción generada por IA. El artículo original en inglés continúa abajo.
// Workers AI · dad joke mode What did the Jacobi transform say? "I'm transforming into a new form.
From Wikipedia, the free encyclopedia
In mathematics, the Jacobi transform is an integral transform named after the mathematician Carl Gustav Jacob Jacobi , which uses Jacobi polynomials
P
n
α
,
β
(
x
)
{\displaystyle P_{n}^{\alpha ,\beta }(x)}
[ 1] [ 2] [ 3] [ 4]
The Jacobi transform of a function
F
(
x
)
{\displaystyle F(x)}
[ 5]
J
{
F
(
x
)
}
=
f
α
,
β
(
n
)
=
∫
−
1
1
(
1
−
x
)
α
(
1
+
x
)
β
P
n
α
,
β
(
x
)
F
(
x
)
d
x
{\displaystyle J\{F(x)\}=f^{\alpha ,\beta }(n)=\int _{-1}^{1}(1-x)^{\alpha }\ (1+x)^{\beta }\ P_{n}^{\alpha ,\beta }(x)\ F(x)\ dx}
The inverse Jacobi transform is given by
J
−
1
{
f
α
,
β
(
n
)
}
=
F
(
x
)
=
∑
n
=
0
∞
1
δ
n
f
α
,
β
(
n
)
P
n
α
,
β
(
x
)
,
where
δ
n
=
2
α
+
β
+
1
Γ
(
n
+
α
+
1
)
Γ
(
n
+
β
+
1
)
n
!
(
α
+
β
+
2
n
+
1
)
Γ
(
n
+
α
+
β
+
1
)
{\displaystyle J^{-1}\{f^{\alpha ,\beta }(n)\}=F(x)=\sum _{n=0}^{\infty }{\frac {1}{\delta _{n}}}f^{\alpha ,\beta }(n)P_{n}^{\alpha ,\beta }(x),\quad {\text{where}}\quad \delta _{n}={\frac {2^{\alpha +\beta +1}\Gamma (n+\alpha +1)\Gamma (n+\beta +1)}{n!(\alpha +\beta +2n+1)\Gamma (n+\alpha +\beta +1)}}}
Some Jacobi transform pairs
F
(
x
)
{\displaystyle F(x)\,}
f
α
,
β
(
n
)
{\displaystyle f^{\alpha ,\beta }(n)\,}
x
m
,
m
<
n
{\displaystyle x^{m},\ m<n\,}
0
{\displaystyle 0}
x
n
{\displaystyle x^{n}\,}
n
!
(
α
+
β
+
2
n
+
1
)
δ
n
{\displaystyle n!(\alpha +\beta +2n+1)\delta _{n}}
P
m
α
,
β
(
x
)
{\displaystyle P_{m}^{\alpha ,\beta }(x)\,}
δ
n
δ
m
,
n
{\displaystyle \delta _{n}\delta _{m,n}}
(
1
+
x
)
a
−
β
{\displaystyle (1+x)^{a-\beta }\,}
(
n
+
α
n
)
2
α
+
a
+
1
Γ
(
a
+
1
)
Γ
(
α
+
1
)
Γ
(
a
−
β
+
1
)
Γ
(
α
+
a
+
n
+
2
)
Γ
(
a
−
β
+
n
+
1
)
{\displaystyle {\binom {n+\alpha }{n}}2^{\alpha +a+1}{\frac {\Gamma (a+1)\Gamma (\alpha +1)\Gamma (a-\beta +1)}{\Gamma (\alpha +a+n+2)\Gamma (a-\beta +n+1)}}}
(
1
−
x
)
σ
−
α
,
ℜ
σ
>
−
1
{\displaystyle (1-x)^{\sigma -\alpha },\ \Re \sigma >-1\,}
2
σ
+
β
+
1
n
!
Γ
(
α
−
σ
)
Γ
(
σ
+
1
)
Γ
(
n
+
β
+
1
)
Γ
(
α
−
σ
+
n
)
Γ
(
β
+
σ
+
n
+
2
)
{\displaystyle {\frac {2^{\sigma +\beta +1}}{n!\Gamma (\alpha -\sigma )}}{\frac {\Gamma (\sigma +1)\Gamma (n+\beta +1)\Gamma (\alpha -\sigma +n)}{\Gamma (\beta +\sigma +n+2)}}}
(
1
−
x
)
σ
−
β
P
m
α
,
σ
(
x
)
,
ℜ
σ
>
−
1
{\displaystyle (1-x)^{\sigma -\beta }P_{m}^{\alpha ,\sigma }(x),\ \Re \sigma >-1\,}
2
α
+
σ
+
1
m
!
(
n
−
m
)
!
Γ
(
n
+
α
+
1
)
Γ
(
α
+
β
+
m
+
n
+
1
)
Γ
(
σ
+
m
+
1
)
Γ
(
α
−
β
+
1
)
Γ
(
α
+
β
+
n
+
1
)
Γ
(
α
+
σ
+
m
+
n
+
2
)
Γ
(
α
−
β
+
m
+
1
)
{\displaystyle {\frac {2^{\alpha +\sigma +1}}{m!(n-m)!}}{\frac {\Gamma (n+\alpha +1)\Gamma (\alpha +\beta +m+n+1)\Gamma (\sigma +m+1)\Gamma (\alpha -\beta +1)}{\Gamma (\alpha +\beta +n+1)\Gamma (\alpha +\sigma +m+n+2)\Gamma (\alpha -\beta +m+1)}}}
2
α
+
β
Q
−
1
(
1
−
z
+
Q
)
−
α
(
1
+
z
+
Q
)
−
β
,
Q
=
(
1
−
2
x
z
+
z
2
)
1
/
2
,
|
z
|
<
1
{\displaystyle 2^{\alpha +\beta }Q^{-1}(1-z+Q)^{-\alpha }(1+z+Q)^{-\beta },\ Q=(1-2xz+z^{2})^{1/2},\ |z|<1\,}
∑
n
=
0
∞
δ
n
z
n
{\displaystyle \sum _{n=0}^{\infty }\delta _{n}z^{n}}
(
1
−
x
)
−
α
(
1
+
x
)
−
β
d
d
x
[
(
1
−
x
)
α
+
1
(
1
+
x
)
β
+
1
d
d
x
]
F
(
x
)
{\displaystyle (1-x)^{-\alpha }(1+x)^{-\beta }{\frac {d}{dx}}\left[(1-x)^{\alpha +1}(1+x)^{\beta +1}{\frac {d}{dx}}\right]F(x)\,}
−
n
(
n
+
α
+
β
+
1
)
f
α
,
β
(
n
)
{\displaystyle -n(n+\alpha +\beta +1)f^{\alpha ,\beta }(n)}
{
(
1
−
x
)
−
α
(
1
+
x
)
−
β
d
d
x
[
(
1
−
x
)
α
+
1
(
1
+
x
)
β
+
1
d
d
x
]
}
k
F
(
x
)
{\displaystyle \left\{(1-x)^{-\alpha }(1+x)^{-\beta }{\frac {d}{dx}}\left[(1-x)^{\alpha +1}(1+x)^{\beta +1}{\frac {d}{dx}}\right]\right\}^{k}F(x)\,}
(
−
1
)
k
n
k
(
n
+
α
+
β
+
1
)
k
f
α
,
β
(
n
)
{\displaystyle (-1)^{k}n^{k}(n+\alpha +\beta +1)^{k}f^{\alpha ,\beta }(n)}
↑ Debnath, L. "On Jacobi Transform." Bull. Cal. Math. Soc 55.3 (1963): 113-120. ↑ Debnath, L. "SOLUTION OF PARTIAL DIFFERENTIAL EQUATIONS BY JACOBI TRANSFORM." BULLETIN OF THE CALCUTTA MATHEMATICAL SOCIETY 59.3-4 (1967): 155. ↑ Scott, E. J. "Jacobi transforms." (1953). ↑ Shen, Jie; Wang, Yingwei; Xia, Jianlin (2019). "Fast structured Jacobi-Jacobi transforms" . Math. Comp . 88 (318): 1743– 1772. doi :10.1090/mcom/3377 ↑ Debnath, Lokenath, and Dambaru Bhatta. Integral transforms and their applications. CRC press, 2014.
![{\displaystyle (1-x)^{-\alpha }(1+x)^{-\beta }{\frac {d}{dx}}\left[(1-x)^{\alpha +1}(1+x)^{\beta +1}{\frac {d}{dx}}\right]F(x)\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6cf3f023f7d443653e8a4664e7c2e71e76cb4e07)
![{\displaystyle \left\{(1-x)^{-\alpha }(1+x)^{-\beta }{\frac {d}{dx}}\left[(1-x)^{\alpha +1}(1+x)^{\beta +1}{\frac {d}{dx}}\right]\right\}^{k}F(x)\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f37708dcabf8079699227184c4cbfbcac88a868d)
