200.104.240.11: /* Definition */ people understand what "not" means without extra formatting to help them. also rm unnecessary brackets

2014-11-22T13:22:32Z

Definition: people understand what "not" means without extra formatting to help them. also rm unnecessary brackets

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195.37.61.178: /* Contour integral representation */

2014-02-21T12:17:24Z

Contour integral representation

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en>Cuzkatzimhut: Reverted 1 edit by 2001:630:63:4:2c81:db1b:208d:eb83 (talk): What was the point in selective name switch? (TW)

2014-01-15T11:52:50Z

Reverted 1 edit by 2001:630:63:4:2c81:db1b:208d:eb83 (talk): What was the point in selective name switch? (TW)

← Older revision		Revision as of 13:52, 15 January 2014
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	~~Jerrie is what you would call me but My family~~ and ~~i don~~'~~t like when consumers use my full identify~~. ~~The job I've been occupying~~ for ~~years is an actual people manager~~. ~~Guam is~~ where ~~I've always been living~~. ~~What I seriously like doing~~ is ~~fish are suffering from acne~~ and ~~I'll~~ be ~~initiating something else along~~ can. ~~Go to~~ the ~~best website to find from more~~: ~~http~~://~~prometeu~~.~~net~~<br><br>~~my blog~~ ... [http://~~prometeu~~.~~net Clash Of Clans Cheat~~]		{{For\|the most common case of integer degree\|Legendre polynomials\|associated Legendre polynomials}}
			{{no footnotes\|date=January 2013}}

			In mathematics, the '''Legendre functions''' ''P''<sub>λ</sub>, ''Q''<sub>λ</sub> and '''associated Legendre functions''' ''P''{{su\|p=μ\|b=λ}}, ''Q''{{su\|p=μ\|b=λ}} are generalizations of [[Legendre polynomials]] to non-integer degree.
			[[File:Associated Legendre Poly.svg\|thumb\|500px\|Associated Legendre polynomial curves for l=5.]]

			==Differential equation==
			Associated Legendre functions are solutions of the '''general Legendre equation'''

			:<math>(1-x^2)\,y'' -2xy' + \left[\lambda(\lambda+1) - \frac{\mu^2}{1-x^2}\right]\,y = 0,\,</math>

			where the complex numbers λ and μ are called the degree and order of the associated Legendre functions, respectively. The Legendre polynomials are the associated Legendre functions of order μ=0.

			This is a second order linear equation with three regular singular points (at 1, −1, and ∞). Like all such equations, it can be converted into the [[hypergeometric differential equation]] by a change of variable, and its solutions can be expressed using [[hypergeometric function]]s.

			==Definition==
			These functions may actually be defined for general complex parameters and argument:

			:<math>P_{\lambda}^{\mu}(z) = \frac{1}{\Gamma(1-\mu)} \left[\frac{1+z}{1-z}\right]^{\mu/2} \,_2F_1 (-\lambda, \lambda+1; 1-\mu; \frac{1-z}{2}),\qquad \text{for } \ \|1-z\|<2</math>

			where <math>\Gamma</math> is the [[gamma function]] and <math> _2F_1</math> is the [[hypergeometric function]].

			The second order differential equation has a second solution,
			<math>Q_\lambda^{\mu}(z)</math>, defined as:

			:<math>Q_{\lambda}^{\mu}(z) = \frac{\sqrt{\pi}\ \Gamma(\lambda+\mu+1)}{2^{\lambda+1}\Gamma(\lambda+3/2)}\frac{e^{i\mu\pi}(z^2-1)^{\mu/2}}{z^{\lambda+\mu+1}} \,_2F_1 \left(\frac{\lambda+\mu+1}{2}, \frac{\lambda+\mu+2}{2}; \lambda+\frac{3}{2}; \frac{1}{z^2}\right),\qquad \text{for}\ \ \|z\|>1.</math>

			==Integral representations==
			The Legendre functions can be written as contour integrals. For example
			:<math>P_\lambda(z) =\frac{1}{2\pi i}
			\int_{1,z} \frac{(t^2-1)^\lambda}{2^\lambda(t-z)^{\lambda+1}}dt</math>
			where the contour circles around the points 1 and ''z'' in the positive direction and does not circle around −1.
			For real x, we have
			:<math>P_s(x) = \frac{1}{2\pi}\int_{-\pi}^{\pi}\left(x+\sqrt{x^2-1}\cos\theta\right)^s d\theta = \frac{1}{\pi}\int_0^1\left(x+\sqrt{x^2-1}(2t-1)\right)^s\frac{dt}{\sqrt{t(1-t)}},\qquad s\in\mathbb{C}</math>

			==Legendre function as characters==
			The real integral representation of <math>P_s</math> are very useful in the study of harmonic analysis on <math>L^1(G//K)</math> where <math>G//K</math> is the [[Homogeneous space\|double coset space]] of <math>SL(2,\mathbb{R})</math> (see [[Zonal spherical function]]). Actually the Fourier transform on <math>L^1(G//K)</math> is given by
			:<math>L^1(G//K)\ni f\mapsto \hat{f}</math>
			where
			:<math>\hat{f}(s)=\int_1^\infty f(x)P_s(x)dx,\qquad -1\leq\Re(s)\leq 0 </math>

			==References==

			* {{AS ref\|8\|332}}
			* {{citation\|first1=Richard\|last1=Courant\|authorlink1=Richard Courant\|first2=David\|last2=Hilbert\|authorlink2=David Hilbert\|year=1953\|title=Methods of Mathematical Physics, Volume 1\|publisher=Interscience Publisher, Inc\|publication-place=New York}}.
			*{{dlmf\|first=T. M. \|last=Dunster\|id=14\|title=Legendre and Related Functions}}
			*{{eom\|id=L/l058030\|first=A.B.\|last= Ivanov}}
			*{{Citation \| last1=Snow \| first1=Chester \| title=Hypergeometric and Legendre functions with applications to integral equations of potential theory \| url=http://babel.hathitrust.org/cgi/pt?id=mdp.39015023896346 \| publisher=U. S. Government Printing Office \| location=Washington, D.C. \| series=National Bureau of Standards Applied Mathematics Series, No. 19 \| mr=0048145 \| year=1952\|origyear=1942}}
			*{{Citation \| last1=Whittaker \| first1=E. T. \|authorlink1=E. T. Whittaker\| last2=Watson \| first2=G. N. \| authorlink2=G. N. Watson\|title=A Course in Modern Analysis \| publisher=[[Cambridge University Press]] \| isbn=978-0-521-58807-2 \| year=1963 }}

			==External links==
			*[http://functions.wolfram.com/HypergeometricFunctions/LegendrePGeneral/ Legendre function P] on the Wolfram functions site.
			*[http://functions.wolfram.com/HypergeometricFunctions/LegendreQGeneral/ Legendre function Q] on the Wolfram functions site.
			*[http://functions.wolfram.com/HypergeometricFunctions/LegendreP2General/ Associated Legendre function P] on the Wolfram functions site.
			*[http://functions.wolfram.com/HypergeometricFunctions/LegendreQ2General/ Associated Legendre function Q] on the Wolfram functions site.

			[[Category:Hypergeometric functions]]

158.227.111.4: /* Hermite functions */

2012-08-29T10:42:12Z

Hermite functions

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Jerrie is what you would call me but My family and i don't like when consumers use my full identify. The job I've been occupying for years is an actual people manager. Guam is where I've always been living. What I seriously like doing is fish are suffering from acne and I'll be initiating something else along can. Go to the best website to find from more: http://prometeu.net<br><br>my blog ... [http://prometeu.net Clash Of Clans Cheat]

Hermite polynomials - Revision history

200.104.240.11: /* Definition */ people understand what "not" means without extra formatting to help them. also rm unnecessary brackets

195.37.61.178: /* Contour integral representation */

en>Cuzkatzimhut: Reverted 1 edit by 2001:630:63:4:2c81:db1b:208d:eb83 (talk): What was the point in selective name switch? (TW)

158.227.111.4: /* Hermite functions */