Legendre rational functions: Difference between revisions

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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.]]

In [[mathematics]] the '''Legendre rational functions''' are a sequence of functions which are both [[rationalorthogonal functions|rational]] andon [[orthogonal0, functions|orthogonal]]∞). AThe rationalare Legendreobtained functionby ofcomposing degreethe ''n''[[Cayley istransform]] definedwith as:[[Legendre polynomials]].

A rational Legendre function of degree ''n'' is defined as:

:<math>R_n(x) = \frac{\sqrt{2}}{x+1}\,L_n\left(\frac{x-1}{x+1}\right)</math>

where <math>L_n(x)</math> is a [[Legendre polynomial]]. These functions are [[eigenfunction]]s of the singular [[Sturm-Liouville problem]]:

:<math>(x+1)\partial_x(x\partial_x((x+1)v(x)))+\lambda v(x)=0</math>