Legendre rational functions: Difference between revisions

Revision as of 14:00, 30 March 2021

Revision as of 23:59, 9 July 2022

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{{Short description|Sequence of orthogonal functions on [0, ∞)}}

[[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 [[orthogonal functions]] on  [0,  ∞). They are obtained by composing the [[Cayley transform]] with [[Legendre polynomials]].

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

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

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

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