Chebyshev rational functions
In mathematics, the Chebyshev rational functions are a sequence of functions which are both rational and orthogonal. They are named after Pafnuty Chebyshev. A rational Chebyshev function of degree n is defined as:
where Tn(x) is a Chebyshev polynomial of the first kind.
Properties
Many properties can be derived from the properties of the Chebyshev polynomials of the first kind. Other properties are unique to the functions themselves.
Recursion
Differential equations
Orthogonality
Defining:
The orthogonality of the Chebyshev rational functions may be written:
where cn = 2 for n = 0 and cn = 1 for n ≥ 1; δnm is the Kronecker delta function.
Expansion of an arbitrary function
For an arbitrary function f(x) ∈ L2
ω the orthogonality relationship can be used to expand f(x):
where
Particular values
Partial fraction expansion
gollark: GTech™ manufactures more teleporters every 1251982 years for our teleporter network.
gollark: If only you had a replicator to replicate your replicator with, and also a pattern for a replicator on a memory disk or something.
gollark: Well, yes.
gollark: If only you had a replicator to let you replicate your replicator.
gollark: GTech™ retroactively altered the progression to hinder our competitors.
References
- Guo, Ben-Yu; Shen, Jie; Wang, Zhong-Qing (2002). "Chebyshev rational spectral and pseudospectral methods on a semi-infinite interval" (PDF). Int. J. Numer. Meth. Engng. 53: 65–84. CiteSeerX 10.1.1.121.6069. doi:10.1002/nme.392. Retrieved 2006-07-25.
This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.