Legendrovi polinomi
Legendrovi polinómi [ležándrovi ~] so rešitve Legendrove diferencialne enačbe :
d
d
x
[
(
1
−
x
2
)
d
d
x
P
n
(
x
)
]
+
n
(
n
+
1
)
P
n
(
x
)
=
0
.
{\displaystyle {\mathrm {d} \over \mathrm {d} x}\left[(1-x^{2}){\mathrm {d} \over \mathrm {d} x}P_{n}(x)\right]+n(n+1)P_{n}(x)=0\!\,.}
Imenovani so po Adrien-Marieu Legendru . Ta navadna diferencialna enačba je pogosto rabljena v fiziki in na drugih tehničnih področjih. Pojavi se pri reševanju Laplaceove enačbe in sorodnih parcialnih diferencialnih enačbah v sfernih koordinatah.
P
n
(
x
)
=
1
2
n
n
!
d
n
d
x
n
[
(
x
2
−
1
)
n
]
.
{\displaystyle P_{n}(x)={1 \over 2^{n}n!}{\mathrm {d} ^{n} \over \mathrm {d} x^{n}\left[(x^{2}-1)^{n}\right]\!\,.}
Ortogonalnost
Pomembna značilnost Legendrovih polinomov je, da so ortogonalni v L2 na intervalu −1 ≤ x ≤ 1:
∫
−
1
1
P
m
(
x
)
P
n
(
x
)
d
x
=
2
2
n
+
1
δ
m
n
,
{\displaystyle \int _{-1}^{1}P_{m}(x)P_{n}(x)\,\mathrm {d} x={2 \over {2n+1}\delta _{mn}\!\,,}
(kjer je δmn oznaka za Kroneckerjevo delto , ki je 1, ko je m = n in 0 sicer).
Zgledi Legendrovih polinomov
Prvih nekaj Legendrovih polinomov:
n
P
n
(
x
)
{\displaystyle P_{n}(x)\,}
0
1
{\displaystyle 1\,}
1
x
{\displaystyle x\,}
2
1
2
(
3
x
2
−
1
)
{\displaystyle {\begin{matrix}{\frac {1}{2}\end{matrix}(3x^{2}-1)\,}
3
1
2
(
5
x
3
−
3
x
)
{\displaystyle {\begin{matrix}{\frac {1}{2}\end{matrix}(5x^{3}-3x)\,}
4
1
8
(
35
x
4
−
30
x
2
+
3
)
{\displaystyle {\begin{matrix}{\frac {1}{8}\end{matrix}(35x^{4}-30x^{2}+3)\,}
5
1
8
(
63
x
5
−
70
x
3
+
15
x
)
{\displaystyle {\begin{matrix}{\frac {1}{8}\end{matrix}(63x^{5}-70x^{3}+15x)\,}
6
1
16
(
231
x
6
−
315
x
4
+
105
x
2
−
5
)
{\displaystyle {\begin{matrix}{\frac {1}{16}\end{matrix}(231x^{6}-315x^{4}+105x^{2}-5)\,}
7
1
16
(
429
x
7
−
693
x
5
+
315
x
3
−
35
x
)
{\displaystyle {\begin{matrix}{\frac {1}{16}\end{matrix}(429x^{7}-693x^{5}+315x^{3}-35x)\,}
8
1
128
(
6435
x
8
−
12012
x
6
+
6930
x
4
−
1260
x
2
+
35
)
{\displaystyle {\begin{matrix}{\frac {1}{128}\end{matrix}(6435x^{8}-12012x^{6}+6930x^{4}-1260x^{2}+35)\,}
9
1
128
(
12155
x
9
−
25740
x
7
+
18018
x
5
−
4620
x
3
+
315
x
)
{\displaystyle {\begin{matrix}{\frac {1}{128}\end{matrix}(12155x^{9}-25740x^{7}+18018x^{5}-4620x^{3}+315x)\,}
10
1
256
(
46189
x
10
−
109395
x
8
+
90090
x
6
−
30030
x
4
+
3465
x
2
−
63
)
{\displaystyle {\begin{matrix}{\frac {1}{256}\end{matrix}(46189x^{10}-109395x^{8}+90090x^{6}-30030x^{4}+3465x^{2}-63)\,}
Glej tudi
The article is a derivative under the Creative Commons Attribution-ShareAlike License .
A link to the original article can be found here and attribution parties here
By using this site, you agree to the Terms of Use . Gpedia ® is a registered trademark of the Cyberajah Pty Ltd