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:
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
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. Methods Eng. 53 (1): 65–84. Bibcode:2002IJNME..53...65G. CiteSeerX10.1.1.121.6069. doi:10.1002/nme.392. S2CID 9208244. Retrieved 2006-07-25.