Jump to content

Legendre rational functions

From Wikipedia, the free encyclopedia
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

Properties

[edit]

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

[edit]

and

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 and

Orthogonality

[edit]

where is the Kronecker delta function.

Particular values

[edit]

References

[edit]
  • Zhong-Qing, Wang; Ben-Yu, Guo (2005). "A mixed spectral method for incompressible viscous fluid flow in an infinite strip". Computational & Applied Mathematics. 24 (3). Sociedade Brasileira de Matemática Aplicada e Computacional. doi:10.1590/S0101-82052005000300002.