[[Image:ChebychevRational1.png|thumb|300px|Plot of the Chebyshev rational functions for for 0.01 ≤ x ≤ 100 , log scale.]] 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

[[Image:ChebychevRational2.png|thumb|300px|Plot of the absolute value of the seventh-order ( ) Chebyshev rational function for 0.01 ≤ x ≤ 100 . Note that there are n zeroes arranged symmetrically about and if x0 is a zero, then 1⁄x0 is a zero as well. The maximum value between the zeros is unity. These properties hold for all orders.]] Defining: The orthogonality of the Chebyshev rational functions may be written: where for and 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