Legendre rational functions

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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:

where is a Legendre polynomial. These functions are eigenfunctions of the singular Sturm-Liouville problem:

with eigenvalues


Many properties can be derived from the properties of the Legendre polynomials of the first kind. Other properties are unique to the functions themselves.



Limiting behavior[edit]

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 x0 is a zero, then 1/x0 is a zero as well. These properties hold for all orders.

It can be shown that



where is the Kronecker delta function.

Particular values[edit]


Zhong-Qing, Wang; Ben-Yu, Guo (2005). "A mixed spectral method for incompressible viscous fluid flow in an infinite strip" (PDF). Mat. apl. comput. 24 (3). doi:10.1590/S0101-82052005000300002. Retrieved 2006-08-08.