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| [[Image:LegendreRational1.png|thumb|300px|Plot of the Legendre rational functions for n=0,1,2 and 3 for ''x'' between 0.01 and 100.]]
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| In [[mathematics]] the '''Legendre rational functions''' are a sequence of functions which are both [[rational functions|rational]] and [[orthogonal functions|orthogonal]]. A rational Legendre function of degree ''n'' is defined as:
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| :<math>R_n(x) = \frac{\sqrt{2}}{x+1}\,L_n\left(\frac{x-1}{x+1}\right)</math>
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| where <math>L_n(x)</math> is a [[Legendre polynomial]]. These functions are [[eigenfunction]]s of the singular [[Sturm-Liouville problem]]:
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| :<math>(x+1)\partial_x(x\partial_x((x+1)v(x)))+\lambda v(x)=0</math>
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| with eigenvalues
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| :<math>\lambda_n=n(n+1)\,</math>
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| == Properties==
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| Many properties can be derived from the properties of the Legendre polynomials of the first kind. Other properties are unique to the functions themselves.
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| === Recursion ===
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| :<math>R_{n+1}(x)=\frac{2n+1}{n+1}\,\frac{x-1}{x+1}\,R_n(x)-\frac{n}{n+1}\,R_{n-1}(x)\quad\mathrm{for\,n\ge 1}</math>
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| :<math>2(2n+1)R_n(x)=(x+1)^2(\partial_x R_{n+1}(x)-\partial_x R_{n-1}(x))+(x+1)(R_{n+1}(x)-R_{n-1}(x))</math>
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| === Limiting behavior ===
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| [[Image:LegendreRational2.png|thumb|300px|Plot of the seventh order (''n=7'') Legendre rational function multiplied by ''1+x'' for ''x'' between 0.01 and 100. Note that there are ''n'' zeroes arranged symmetrically about ''x=1'' and if ''x''<sub>0</sub> is a zero, then ''1/x''<sub>0</sub> is a zero as well. These properties hold for all orders.]]
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| It can be shown that
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| :<math>\lim_{x\rightarrow \infty}(x+1)R_n(x)=\sqrt{2}</math>
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| and
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| :<math>\lim_{x\rightarrow \infty}x\partial_x((x+1)R_n(x))=0</math>
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| === Orthogonality ===
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| :<math>\int_{0}^\infty R_m(x)\,R_n(x)\,dx=\frac{2}{2n+1}\delta_{nm}</math> | |
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| where <math>\delta_{nm}</math> is the [[Kronecker delta]] function.
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| == Particular values ==
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| :<math>R_0(x)=1\,</math>
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| :<math>R_1(x)=\frac{x-1}{x+1}\,</math>
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| :<math>R_2(x)=\frac{x^2-4x+1}{(x+1)^2}\,</math>
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| :<math>R_3(x)=\frac{x^3-9x^2+9x-1}{(x+1)^3}\,</math> | |
| :<math>R_4(x)=\frac{x^4-16x^3+36x^2-16x+1}{(x+1)^4}\,</math>
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| == References ==
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| {{cite journal
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| | last = Zhong-Qing
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| | first = Wang
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| | authorlink =
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| | coauthors = Ben-Yu, Guo
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| | year = 2005
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| | month =
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| | title = A mixed spectral method for incompressible viscous fluid flow in an infinite strip
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| | journal = Mat. apl. comput.
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| | volume = 24
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| | issue = 3
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| | pages =
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| | doi = 10.1590/S0101-82052005000300002
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| | id =
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| | url = http://www.scielo.br/scielo.php?script=sci_arttext&pid=S0101-82052005000300002&lng=en&nrm=iso
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| | format = PDF
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| | accessdate = 2006-08-08
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| }}
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| [[Category:Rational functions]]
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