Associated Legendre function/Catalogs
From Citizendium
<
Associated Legendre function
Revision as of 07:56, 10 September 2009 by
imported>Paul Wormer
(
diff
)
← Older revision
| Latest revision (diff) | Newer revision → (diff)
Jump to navigation
Jump to search
Main Article
Discussion
Related Articles
[?]
Bibliography
[?]
External Links
[?]
Citable Version
[?]
Catalogs
[?]
Proofs
[?]
An informational catalog, or several catalogs, about
Associated Legendre function
.
The associated Legendre functions through
l
= 6 are:
P
0
0
(
x
)
=
1
P
1
0
(
x
)
=
x
P
1
1
(
x
)
=
(
1
−
x
2
)
1
/
2
P
2
0
(
x
)
=
1
2
(
3
x
2
−
1
)
P
2
1
(
x
)
=
3
(
1
−
x
2
)
1
/
2
x
P
2
2
(
x
)
=
3
(
1
−
x
2
)
P
3
0
(
x
)
=
1
2
(
5
x
3
−
3
x
)
P
3
1
(
x
)
=
1
2
(
1
−
x
2
)
1
/
2
(
15
x
2
−
3
)
P
3
2
(
x
)
=
15
(
1
−
x
2
)
x
P
3
3
(
x
)
=
15
(
1
−
x
2
)
3
/
2
P
4
0
(
x
)
=
1
8
(
35
x
4
−
30
x
2
+
3
)
P
4
1
(
x
)
=
1
2
(
1
−
x
2
)
1
/
2
(
35
x
3
−
15
x
)
P
4
2
(
x
)
=
1
2
(
1
−
x
2
)
(
105
x
2
−
15
)
P
4
3
(
x
)
=
105
(
1
−
x
2
)
3
/
2
x
P
4
4
(
x
)
=
105
(
1
−
x
2
)
2
P
5
0
(
x
)
=
1
8
(
63
x
5
−
70
x
3
+
15
x
)
P
5
1
(
x
)
=
1
8
(
1
−
x
2
)
1
/
2
(
315
x
4
−
210
x
2
+
15
)
P
5
2
(
x
)
=
1
2
(
1
−
x
2
)
(
315
x
3
−
105
x
)
P
5
3
(
x
)
=
1
2
(
1
−
x
2
)
3
/
2
(
945
x
2
−
105
)
P
5
4
(
x
)
=
945
(
1
−
x
2
)
2
x
P
5
5
(
x
)
=
945
(
1
−
x
2
)
5
/
2
P
6
0
(
x
)
=
1
16
(
231
x
6
−
315
x
4
+
105
x
2
−
5
)
P
6
1
(
x
)
=
1
8
(
1
−
x
2
)
1
/
2
(
693
x
5
−
630
x
3
+
105
x
)
P
6
2
(
x
)
=
1
8
(
1
−
x
2
)
(
3465
x
4
−
1890
x
2
+
105
)
P
6
3
(
x
)
=
1
2
(
1
−
x
2
)
3
/
2
(
3465
x
3
−
945
x
)
P
6
4
(
x
)
=
1
2
(
1
−
x
2
)
2
(
10395
x
2
−
945
)
P
6
5
(
x
)
=
10395
(
1
−
x
2
)
5
/
2
x
P
6
6
(
x
)
=
10395
(
1
−
x
2
)
3
{\displaystyle {\begin{aligned}P_{0}^{0}(x)&=1\\\\P_{1}^{0}(x)&=x\\P_{1}^{1}(x)&=(1-x^{2})^{1/2}\\\\P_{2}^{0}(x)&={\tfrac {1}{2}}(3x^{2}-1)\\P_{2}^{1}(x)&=3(1-x^{2})^{1/2}x\\P_{2}^{2}(x)&=3(1-x^{2})\\\\P_{3}^{0}(x)&={\tfrac {1}{2}}(5x^{3}-3x)\\P_{3}^{1}(x)&={\tfrac {1}{2}}(1-x^{2})^{1/2}(15x^{2}-3)\\P_{3}^{2}(x)&=15(1-x^{2})x\\P_{3}^{3}(x)&=15(1-x^{2})^{3/2}\\\\P_{4}^{0}(x)&={\tfrac {1}{8}}(35x^{4}-30x^{2}+3)\\P_{4}^{1}(x)&={\tfrac {1}{2}}(1-x^{2})^{1/2}(35x^{3}-15x)\\P_{4}^{2}(x)&={\tfrac {1}{2}}(1-x^{2})(105x^{2}-15)\\P_{4}^{3}(x)&=105(1-x^{2})^{3/2}x\\P_{4}^{4}(x)&=105(1-x^{2})^{2}\\\\P_{5}^{0}(x)&={\tfrac {1}{8}}(63x^{5}-70x^{3}+15x)\\P_{5}^{1}(x)&={\tfrac {1}{8}}(1-x^{2})^{1/2}(315x^{4}-210x^{2}+15)\\P_{5}^{2}(x)&={\tfrac {1}{2}}(1-x^{2})(315x^{3}-105x)\\P_{5}^{3}(x)&={\tfrac {1}{2}}(1-x^{2})^{3/2}(945x^{2}-105)\\P_{5}^{4}(x)&=945(1-x^{2})^{2}x\\P_{5}^{5}(x)&=945(1-x^{2})^{5/2}\\\\P_{6}^{0}(x)&={\tfrac {1}{16}}(231x^{6}-315x^{4}+105x^{2}-5)\\P_{6}^{1}(x)&={\tfrac {1}{8}}(1-x^{2})^{1/2}(693x^{5}-630x^{3}+105x)\\P_{6}^{2}(x)&={\tfrac {1}{8}}(1-x^{2})(3465x^{4}-1890x^{2}+105)\\P_{6}^{3}(x)&={\tfrac {1}{2}}(1-x^{2})^{3/2}(3465x^{3}-945x)\\P_{6}^{4}(x)&={\tfrac {1}{2}}(1-x^{2})^{2}(10395x^{2}-945)\\P_{6}^{5}(x)&=10395(1-x^{2})^{5/2}x\\P_{6}^{6}(x)&=10395(1-x^{2})^{3}\\\end{aligned}}}
Categories
:
Subpages
Mathematics Extra Subpages
Physics Extra Subpages
Mathematics Catalogs
Physics Catalogs
All Content
Mathematics Content
Physics Content
Hidden categories:
Mathematics tag
Physics tag
Navigation menu
Personal tools
Log in
Namespaces
Page
Discussion
English
Views
Read
View source
View history
ZWI Export
More
Search
Read
Welcome to Citizendium
Citable Articles
All Articles
Random Article
All Recent Changes
About Us
Introduction
FAQ
Policies
Governance
Contact Us
Enquiries
Apply to Join
Personnel
How To
Forum
Start Article
Article Mechanics
How to Edit
Help
More Help
Quick Start
How to collaborate
How to format pages
Finance
Financial Report
Donate
Tools
What links here
Related changes
Special pages
Printable version
Permanent link
Page information
Cite this page