Legendre rational functions
From Wikipedia, the free encyclopedia
In mathematics the Legendre rational functions are a sequence of functions which are both rational and orthogonal. A rational Legendre function of degree n is defined as:
where Ln(x) is a Legendre polynomial. These functions are eigenfunctions of the singular Sturm-Liouville problem:
with eigenvalues
.
Contents |
[edit] 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.
[edit] Recursion
and
[edit] Limiting behavior
It can be shown that
and
[edit] Orthogonality
where δnm is the Kronecker delta function.
[edit] Particular values
[edit] References
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 on 2006-08-08.
