Legendre polynomials - Revision history

Carl McBride: Added applications section.

2008-07-07T10:06:25Z

Added applications section.

← Older revision		Revision as of 12:06, 7 July 2008
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	:<math>x^6= \frac{1}{231}[33P_0 (x) + 110P_2(x)+ 72P_4(x)+ 16P_6(x)]</math>		:<math>x^6= \frac{1}{231}[33P_0 (x) + 110P_2(x)+ 72P_4(x)+ 16P_6(x)]</math>
			==Applications in statistical mechanics==
			*[[Computational implementation of integral equations]]
			*[[Order parameters]]
			*[[Lebwohl-Lasher model]]
			*[[Rotational relaxation]]
	==See also==		==See also==
	*[[Associated Legendre function]]		*[[Associated Legendre function]]

2008-06-23T10:36:16Z

Slight style change.

← Older revision		Revision as of 12:36, 23 June 2008
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	:<math>P_n (z) = \frac{1}{2 \pi i} \oint ( 1-2tz + t^2)^{1/2}~t^{-n-1} {\rm d}t</math>		:<math>P_n (z) = \frac{1}{2 \pi i} \oint ( 1-2tz + t^2)^{1/2}~t^{-n-1} {\rm d}t</math>

	~~'''~~Legendre polynomials~~'''~~ can also be defined (Ref 1) using ~~'''~~Rodrigues formula~~'''~~ as:		Legendre polynomials can also be defined (Ref 1) using Rodrigues formula, used for producing a series of orthogonal polynomials, as:

	:<math> P_n(x) = \frac{1}{2^n n!} \frac{d^n}{dx^n} (x^2 - 1)^n </math>		:<math> P_n(x) = \frac{1}{2^n n!} \frac{d^n}{dx^n} (x^2 - 1)^n </math>

2008-06-23T08:16:01Z

References: Slight tidy of reference.

← Older revision		Revision as of 10:16, 23 June 2008
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	==References==		==References==

	# B. P. Demidotwitsch, I.A. Maron, and E.S. Schuwalowa, "Métodos numéricos de		# B. P. Demidotwitsch, I. A. Maron, and E. S. Schuwalowa, "Métodos numéricos de Análisis", Ed. Paraninfo, Madrid (1980) (translated from Russian text)
	Análisis", (Ed. Paraninfo, Madrid, 1980) (~~original in~~ Russian)

2008-06-20T17:02:13Z

2008-06-20T16:58:30Z

Normalisation

← Older revision		Revision as of 18:58, 20 June 2008
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	:<math>P_n (z) = \frac{1}{2 \pi i} \oint ( 1-2tz + t^2)^{1/2}~t^{-n-1} {\rm d}t</math>		:<math>P_n (z) = \frac{1}{2 \pi i} \oint ( 1-2tz + t^2)^{1/2}~t^{-n-1} {\rm d}t</math>

	'''Legendre polynomials''' can also be defined using '''Rodrigues formula''' as:		'''Legendre polynomials''' can also be defined (Ref 1) using '''Rodrigues formula''' as:

	:<math> P_n(x) = \frac{1}{2^n n!} \frac{d^n}{dx^n} (x^2 - 1)^n </math>		:<math> P_n(x) = \frac{1}{2^n n!} \frac{d^n}{dx^n} (x^2 - 1)^n </math>
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	:<math> \int_{-1}^{1} P_n(x) P_m(x) d x = 0, </math> for <math> m \ne n </math>		:<math> \int_{-1}^{1} P_n(x) P_m(x) d x = 0, </math> for <math> m \ne n </math>

			whereas

			:<math>\int_{-1}^{1} P_n(x) P_n(x) d x = \frac{2}{2n+1} </math>

	The first seven Legendre polynomials are:		The first seven Legendre polynomials are:

2008-06-20T16:54:11Z

range definition for the so-called shifted polynomials

← Older revision		Revision as of 18:54, 20 June 2008
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	:<math>P_6 (x) =\frac{1}{16}(231x^6 -315x^4 + 105x^2 -5)</math>		:<math>P_6 (x) =\frac{1}{16}(231x^6 -315x^4 + 105x^2 -5)</math>

	"shifted" Legendre polynomials (which obey the orthogonality relationship):		"shifted" Legendre polynomials (which obey the orthogonality relationship
			in the range [0:1]):

	:<math>\overline{P}_0 (x) =1</math>		:<math>\overline{P}_0 (x) =1</math>

2008-06-20T16:52:08Z

orthogonality definitions

← Older revision		Revision as of 18:52, 20 June 2008
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	:<math> P_n(x) = \frac{1}{2^n n!} \frac{d^n}{dx^n} (x^2 - 1)^n </math>		:<math> P_n(x) = \frac{1}{2^n n!} \frac{d^n}{dx^n} (x^2 - 1)^n </math>

			Legendre polynomials form an orthogonal system in the range [-1:1], i.e.:

			:<math> \int_{-1}^{1} P_n(x) P_m(x) d x = 0, </math> for <math> m \ne n </math>

	The first seven Legendre polynomials are:		The first seven Legendre polynomials are:

2008-06-20T16:47:53Z

Rodrigues

← Older revision		Revision as of 18:47, 20 June 2008
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	:<math>P_n (z) = \frac{1}{2 \pi i} \oint ( 1-2tz + t^2)^{1/2}~t^{-n-1} {\rm d}t</math>		:<math>P_n (z) = \frac{1}{2 \pi i} \oint ( 1-2tz + t^2)^{1/2}~t^{-n-1} {\rm d}t</math>

			'''Legendre polynomials''' can also be defined using '''Rodrigues formula''' as:

			:<math> P_n(x) = \frac{1}{2^n n!} \frac{d^n}{dx^n} (x^2 - 1)^n </math>

	The first seven Legendre polynomials are:		The first seven Legendre polynomials are:

2008-06-20T12:28:39Z

Style: expanded acronym

← Older revision		Revision as of 14:28, 20 June 2008
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	'''Legendre polynomials''' (~~aka.~~ Legendre functions of the first kind, Legendre coefficients, or zonal harmonics)		'''Legendre polynomials''' (also known as Legendre functions of the first kind, Legendre coefficients, or zonal harmonics)
	are solutions of the [[Legendre differential equation]].		are solutions of the [[Legendre differential equation]].
	The Legendre polynomial, <math>P_n (z)</math> can be defined by the contour integral		The Legendre polynomial, <math>P_n (z)</math> can be defined by the contour integral

2008-06-20T11:00:47Z

Moved first assoc.'s to their new page

@@ Line 59: / Line 59: @@
 :<math>x^6= \frac{1}{231}[33P_0 (x) + 110P_2(x)+ 72P_4(x)+ 16P_6(x)]</math>
 ==See also==
 *[[Associated Legendre function]]
 *[http://mathworld.wolfram.com/LegendrePolynomial.html Legendre Polynomial -- from Wolfram MathWorld]
 [[category: mathematics]]

← Older revision		Revision as of 19:02, 20 June 2008
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	*[http://mathworld.wolfram.com/LegendrePolynomial.html Legendre Polynomial -- from Wolfram MathWorld]		*[http://mathworld.wolfram.com/LegendrePolynomial.html Legendre Polynomial -- from Wolfram MathWorld]
	[[category: mathematics]]		[[category: mathematics]]

			==References==

			# B. P. Demidotwitsch, I.A. Maron, and E.S. Schuwalowa, "Métodos numéricos de
			Análisis", (Ed. Paraninfo, Madrid, 1980) (original in Russian)