# Difference between revisions of "Associated Legendre functions"

(New page: The '''associated Legendre functions''' <math>P^m_n(x)</math> are most conveniently defined in terms of derivatives of the Legendre polynomials: <math> P^m_n(x)= (1-x^2)^{m/2} \frac{d...) |
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The '''associated Legendre functions''' <math>P^m_n(x)</math> are | The '''associated Legendre functions''' <math>P^m_n(x)</math> are | ||

− | most conveniently defined in terms of derivatives of the | + | polynomials which are most conveniently defined in terms of derivatives of the |

[[Legendre polynomials]]: | [[Legendre polynomials]]: | ||

<math> P^m_n(x)= (1-x^2)^{m/2} \frac{d^m}{dx^m} P_n(x) </math> | <math> P^m_n(x)= (1-x^2)^{m/2} \frac{d^m}{dx^m} P_n(x) </math> | ||

+ | |||

+ | The first associated Legendre polynomials are: | ||

+ | |||

+ | |||

+ | |||

+ | :<math>P_0^0 (x) =1</math> | ||

+ | |||

+ | |||

+ | :<math>P_1^0 (x) =x</math> | ||

+ | |||

+ | |||

+ | :<math>P_1^1 (x) =-(1-x^2)^{1/2}</math> | ||

+ | |||

+ | |||

+ | :<math>P_2^0 (x) =\frac{1}{2}(3x^2-1)</math> | ||

+ | |||

+ | |||

+ | :<math>P_2^1 (x) =-3x(1-x^2)^{1/2}</math> | ||

+ | |||

+ | |||

+ | :<math>P_2^2 (x) =3(1-x^2)</math> | ||

+ | |||

+ | ''etc''. | ||

[[category: mathematics]] | [[category: mathematics]] |

## Revision as of 13:01, 20 June 2008

The **associated Legendre functions** are
polynomials which are most conveniently defined in terms of derivatives of the
Legendre polynomials:

The first associated Legendre polynomials are:

*etc*.