Legendre rational functions: Difference between revisions

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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 [ {{closed-open|0, ∞)}}. They are obtained by composing the [[Cayley transform]] with [[Legendre polynomials]].

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

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

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

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

<math display="block">(x+1) \frac{d}{dx}\left(x \frac{d}{dx} \left[\left(x+1\right) v(x)\right]\right) + \lambda v(x) = 0</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>

with eigenvalues

:<math display="block">\lambda_n=n(n+1)\,</math>

:<math>\lambda_n=n(n+1)\,</math>

== Properties==

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=== Recursion ===

:<math display="block">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>

:<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>

and

:<math display="block">2 (2n+1) R_n(x) = \left(x+1\right)^2 \left(\partial_xfrac{d}{dx} R_{n+1}(x) - \partial_xfrac{d}{dx} R_{n-1}(x)\right) + (x+1) \left(R_{n+1}(x) - R_{n-1}(x)\right)</math>

:<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>

=== 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

:<math display="block">\lim_{x\rightarrow to\infty}(x+1)R_n(x)=\sqrt{2}</math>

:<math>\lim_{x\rightarrow \infty}(x+1)R_n(x)=\sqrt{2}</math>

and

:<math display="block">(\lim_{x+1)\partial_x(to\infty}x\partial_x((x+1)vR_n(x)))+\lambda v(x)=0</math>

:<math>\lim_{x\rightarrow \infty}x\partial_x((x+1)R_n(x))=0</math>

=== Orthogonality ===

:<math display="block">\int_{0}^\infty R_m(x)\,R_n(x)\,dx=\frac{2}{2n+1}\delta_{nm}</math>

where <math>\delta_{nm}</math> is the [[Kronecker delta]] function.

== Particular values ==

<math display="block">\begin{align}

:<math>R_0(x)=1\,</math>

:<math>R_1R_0(x) &= \frac{x-1\sqrt{2}}{x+1}\,</math>1 \\

:<math>R_2R_1(x) &= \frac{x^\sqrt{2-4x}}{x+1}\,\frac{x-1}{(x+1)^2} \\,</math>

:<math>R_3R_2(x) &= \frac{\sqrt{2}}{x+1}\,\frac{x^3-9x^2+9x-4x+1}{(x+1)^32} \\,</math>

:<math>R_4R_3(x) &= \frac{\sqrt{2}}{x+1}\,\frac{x^43-16x^3+36x9x^2-16x+9x-1}{(x+1)^43} \\,</math>

R_4(x) &= \frac{\sqrt{2}}{x+1}\,\frac{x^4-16x^3+36x^2-16x+1}{(x+1)^4}

\end{align}</math>

== References ==

* {{cite journal

| last = Zhong-Qing

| first = Wang

| authorlinkauthor2 = Ben-Yu, Guo

| year = 2005

|author2=Ben-Yu, Guo

| year = 2005

| title = A mixed spectral method for incompressible viscous fluid flow in an infinite strip

| journal = Mat.Computational apl.& comput.Applied Mathematics

| publisher = Sociedade Brasileira de Matemática Aplicada e Computacional

| volume = 24

| issue = 3

| pages =

| doi = 10.1590/S0101-82052005000300002

| iddoi-access = free

| url = http://www.scielo.br/scielo.php?script=sci_arttext&pid=S0101-82052005000300002&lng=en&nrm=iso

| format = PDF

| accessdate = 2006-08-08

}}