Legendre rational functions: Difference between revisions

Revision as of 11:59, 8 August 2006

Revision as of 16:26, 8 August 2006

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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 [[rational functions|rational]] and [[orthogonal functions|orthogonal]]. A rational Legendre function of degree ''n'' is defined as:

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=== Limiting behavior ===

[[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''₀ is a zero, then ''1/x''₀ is a zero as well. These properties hold for all orders.]]

It can be shown that