Chebyshev rational functions

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This article is not about the Chebyshev rational functions used in the design of elliptic filters. For those functions, see Elliptic rational functions.
Plot of the Chebyshev rational functions for n = 0, 1, 2, 3 and 4 for x between 0.01 and 100.

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 is a Chebyshev polynomial of the first kind.

Properties[edit]

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[edit]

Differential equations[edit]

Orthogonality[edit]

Plot of the absolute value of the seventh order (n = 7) Chebyshev rational function 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. 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 equals 2 for n = 0 and equals 1 for and is the Kronecker delta function.

Expansion of an arbitrary function[edit]

For an arbitrary function the orthogonality relationship can be used to expand :

where

Particular values[edit]

Partial fraction expansion[edit]

References[edit]