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	~~:''This article~~ is ~~'''~~not~~''' about~~ the ~~Chebyshev rational functions used in the design of [[elliptic filter]]s~~. ~~For those functions, see [[Elliptic rational functions]].''~~		Hi there, I am Alyson Boon even though it is not the title on my beginning certification. Invoicing is my occupation. Doing ballet is some thing she would by no means give up. Some time in the past he selected to reside in North Carolina and he doesn't plan on altering it.<br><br>Look at my web blog; psychic phone readings ([http://www.khuplaza.com/dent/14869889 www.khuplaza.com])

	[[Image:ChebychevRational1.png\|thumb\|300px\|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 functions\|rational]] and [[orthogonal functions\|orthogonal]]. They are named after [[Pafnuty Chebyshev]]. ~~A rational Chebyshev function of degree ''n''~~ is ~~defined as:~~

	~~:<math>R_n(x)\ \stackrel{\mathrm{def}}{=}\ T_n\left(\frac{x-1}{x+1}\right)</math>~~

	~~where <math>T_n(x)</math> 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 ===~~

	~~:<math>R_{n+1}(x)=2\,\frac{x-1}{x+1}R_n(x)-R_{n-1}(x)\quad\mathrm{for\,n\ge 1}</math>~~

	~~=== Differential equations ===~~

	~~:<math>(x+1)^2R_n(x)=\frac{1}{n+1}\frac{d}{dx}\,R_{n+1}(x)-\frac{1}{n-1}\frac{d}{dx}\,R_{n-1}(x)~~
	~~\quad\mathrm{for\,n\ge 2}</math>~~

	~~:<math>(x+1)^2x\frac{d^2}{dx^2}\,R_n(x)+\frac{(3x+1)(x+1)}{2}\frac{d}{dx}\,R_n(x)+n^2R_{n}(x) = 0</math>~~

	~~=== Orthogonality ===~~
	~~[[Image:ChebychevRational2.png\|thumb\|300px\|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 ''x''<sub>0</sub> is a zero, then 1/''x''<sub>0</sub> is a zero as well. The maximum value between the zeros is unity. These properties hold for all orders.]]

	~~Defining:~~

	:<~~math~~>~~\omega(x) \ \stackrel{\mathrm{def}}{=}\ \frac{1}{(x+1)\sqrt{x}}~~<~~/math~~>

	~~The orthogonality of the Chebyshev rational functions may be written:~~

	~~:<math>\int_{0}^\infty R_m(x)\,R_n(x)\,\omega~~(~~x)\,dx=\frac{\pi c_n}{2}\delta_{nm}</math>~~

	~~where <math>c_n</math> equals 2 for ''n'' = 0 and <math>c_n</math> equals 1 for <math>n \ge 1</math> and <math>\delta_{nm}</math> is the~~ [~~[Kronecker delta]] function.~~

	~~=== Expansion of an arbitrary function ===~~
	~~For an arbitrary function <math>f(x)\in L_\omega^2</math> the orthogonality relationship can be used to expand <math>f(x)</math>:~~

	~~:<math>f(x)=\sum_{n=0}^\infty F_n R_n(x)</math>~~

	~~where~~

	~~:<math>F_n=\frac{2}{c_n\pi}\int_{0}^\infty f(x)R_n(x)\omega(x)\,dx.</math>~~

	~~== Particular values ==~~

	:~~<math>R_0(x)=1\,<~~/~~math>~~
	~~:<math>R_1(x)=\frac{x-1}{x+1}\,<~~/~~math>~~
	~~:<math>R_2(x)=\frac{x^2-6x+1}{(x+1)^2}\,</math>~~
	~~:<math>R_3(x)=\frac{x^3-15x^2+15x-1}{(x+1)^3}\,</math>~~
	~~:<math>R_4(x)=\frac{x^4-28x^3+70x^2-28x+1}{(x+1)^4}\,</math>~~
	~~:<math>R_n(x)=\frac{1}{(x+1)^n}\sum_{m=0}^{n} (-1)^m{2n \choose 2m}x^{n-m}\,</math>~~

	~~== Partial fraction expansion ==~~

	~~:<math>R_n(x)=\sum_{m=0}^{n} \frac{(m!)^2}{(2m)!}{n+m-1 \choose m}{n \choose m}\frac{(-4)^m}{(x+1)^m} </math>~~

	~~== References ==~~
	*{{cite journal
	~~\| last = Ben-Yu~~
	~~\| first = Guo~~
	~~\| authorlink =~~
	~~\| coauthors = Jie, Shen; Zhong-Quing, Wang~~
	~~\| year = 2002~~
	~~\| month =~~
	~~\| 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~~
	~~\| format = PDF~~
	~~\| accessdate = 2006-07-25~~
	}}

	~~[[Category:Rational functions]~~]

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en>AnomieBOT: Dating maintenance tags: {{Unreferenced}}

en>Cowgummi: Removed broken link to reference

85.75.77.117 at 20:59, 9 July 2012