mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, the Legendre rational functions are a sequence of

orthogonal functions In mathematics, orthogonal functions belong to a function space that is a vector space equipped with a bilinear form. When the function space has an interval (mathematics), interval as the domain of a function, domain, the bilinear form may be the ...

on . They are obtained by composing the

Cayley transform In mathematics, the Cayley transform, named after Arthur Cayley, is any of a cluster of related things. As originally described by , the Cayley transform is a mapping between skew-symmetric matrices and special orthogonal matrices. The transform ...

with

Legendre polynomials In mathematics, Legendre polynomials, named after Adrien-Marie Legendre (1782), are a system of complete and orthogonal polynomials with a wide number of mathematical properties and numerous applications. They can be defined in many ways, and t ...

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

R_n(x) = \frac\,P_n\left(\frac\right)

where

P_n(x)

is a Legendre polynomial. These functions are

eigenfunction In mathematics, an eigenfunction of a linear operator ''D'' defined on some function space is any non-zero function f in that space that, when acted upon by ''D'', is only multiplied by some scaling factor called an eigenvalue. As an equation, th ...

s of the singular Sturm–Liouville problem:

right) + \lambda v(x) = 0

with eigenvalues

\lambda_n=n(n+1)\,

Properties

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

Recursion

R_(x)=\frac\,\frac\,R_n(x)-\frac\,R_(x)\quad\mathrm

and

2 (2n+1) R_n(x) = \left(x+1\right)^2 \left(\frac R_(x) - \frac R_(x)\right) + (x+1) \left(R_(x) - R_(x)\right)

Limiting behavior

It can be shown that

\lim_(x+1)R_n(x)=\sqrt

and

\lim_x\partial_x((x+1)R_n(x))=0

Orthogonality

\int_^\infty R_m(x)\,R_n(x)\,dx=\frac\delta_

where

\delta_

is the

Kronecker delta In mathematics, the Kronecker delta (named after Leopold Kronecker) is a function of two variables, usually just non-negative integers. The function is 1 if the variables are equal, and 0 otherwise: \delta_ = \begin 0 &\text i \neq j, \\ 1 &\ ...

function.

Particular values

\begin
R_0(x) &= \frac\,1 \\
R_1(x) &= \frac\,\frac \\
R_2(x) &= \frac\,\frac \\
R_3(x) &= \frac\,\frac \\
R_4(x) &= \frac\,\frac
\end

References

* {{cite journal , last = Zhong-Qing , first = Wang , author2 = Ben-Yu, Guo , year = 2005 , title = A mixed spectral method for incompressible viscous fluid flow in an infinite strip , journal = Computational & Applied Mathematics , publisher = Sociedade Brasileira de Matemática Aplicada e Computacional , volume = 24 , issue = 3 , doi = 10.1590/S0101-82052005000300002 , doi-access = free Rational functions