Associated Legendre functions: Difference between revisions
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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> | |||
:<math>P_2^2 (x) =3(1-x^2)</math> | |||
''etc''. | |||
[[category: mathematics]] | [[category: mathematics]] | ||
Revision as of 12:01, 20 June 2008
The associated Legendre functions Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P^m_n(x)} are polynomials which are most conveniently defined in terms of derivatives of the Legendre polynomials:
The first associated Legendre polynomials are:
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P_0^0 (x) =1}
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P_1^0 (x) =x}
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P_1^1 (x) =-(1-x^2)^{1/2}}
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P_2^0 (x) =\frac{1}{2}(3x^2-1)}
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P_2^2 (x) =3(1-x^2)}
etc.