Difference between revisions of "Associated Legendre functions"

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(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>
 +
 +
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:<math>P_2^2 (x) =3(1-x^2)</math>
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''etc''.
  
 
[[category: mathematics]]
 
[[category: mathematics]]

Revision as of 12:01, 20 June 2008

The associated Legendre functions P^m_n(x) are polynomials which are most conveniently defined in terms of derivatives of the Legendre polynomials:

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


The first associated Legendre polynomials are:


P_0^0 (x) =1


P_1^0 (x) =x


P_1^1 (x) =-(1-x^2)^{1/2}


P_2^0 (x) =\frac{1}{2}(3x^2-1)


P_2^1 (x) =-3x(1-x^2)^{1/2}


P_2^2 (x) =3(1-x^2)

etc.