词条 | Chebyshev rational functions |
释义 |
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 {{math|n}} is defined as: where {{math|Tn(x)}} is a Chebyshev polynomial of the first kind. PropertiesMany properties can be derived from the properties of the Chebyshev polynomials of the first kind. Other properties are unique to the functions themselves. RecursionDifferential equationsOrthogonalityDefining: The orthogonality of the Chebyshev rational functions may be written: where {{math|cn {{=}} 2}} for {{math|n {{=}} 0}} and {{math|cn {{=}} 1}} for {{math|n ≥ 1}}; {{math|δnm}} is the Kronecker delta function. Expansion of an arbitrary functionFor an arbitrary function {{math|f(x) ∈ L{{su|b=ω|p=2|lh=0.8em}}}} the orthogonality relationship can be used to expand {{math|f(x)}}: where Particular valuesPartial fraction expansionReferences
| first= Ben-Yu | last= Guo | authorlink = |first2=Jie |last2=Shen |first3=Zhong-Qing |last3=Wang | year = 2002 | title = Chebyshev rational spectral and pseudospectral methods on a semi-infinite interval | journal = Int. J. Numer. Meth. Engng | volume = 53 | issue = | pages = 65–84 | doi = 10.1002/nme.392 | id = | url = http://www.math.purdue.edu/~shen/pub/GSW_IJNME02.pdf | accessdate = 2006-07-25 | citeseerx= 10.1.1.121.6069 1 : Rational functions |
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