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:
Plot of the Chebyshev rational functions for n = 0, 1, 2, 3, 4 for 0.01 ≤ x ≤ 100 , log scale.
R
n
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d
e
f
T
n
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1
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{\displaystyle R_{n}(x)\ {\stackrel {\mathrm {def} }{=}}\ T_{n}\left({\frac {x-1}{x+1}}\right)}
where Tn (x ) is a Chebyshev polynomial of the first kind.
Properties
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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
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R
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1
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R
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R
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1
{\displaystyle R_{n+1}(x)=2\left({\frac {x-1}{x+1}}\right)R_{n}(x)-R_{n-1}(x)\quad {\text{for}}\,n\geq 1}
Differential equations
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R
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R
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{\displaystyle (x+1)^{2}R_{n}(x)={\frac {1}{n+1}}{\frac {\mathrm {d} }{\mathrm {d} x}}R_{n+1}(x)-{\frac {1}{n-1}}{\frac {\mathrm {d} }{\mathrm {d} x}}R_{n-1}(x)\quad {\text{for }}n\geq 2}
(
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R
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n
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{\displaystyle (x+1)^{2}x{\frac {\mathrm {d} ^{2}}{\mathrm {d} x^{2}}}R_{n}(x)+{\frac {(3x+1)(x+1)}{2}}{\frac {\mathrm {d} }{\mathrm {d} x}}R_{n}(x)+n^{2}R_{n}(x)=0}
Orthogonality
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Plot of the absolute value of the seventh-order (n = 7 ) Chebyshev rational function for 0.01 ≤ x ≤ 100 . Note that there are n zeroes arranged symmetrically about x = 1 and if x 0 is a zero, then 1 / x 0 is a zero as well. The maximum value between the zeros is unity. These properties hold for all orders.
Defining:
ω
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1
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{\displaystyle \omega (x)\ {\stackrel {\mathrm {def} }{=}}\ {\frac {1}{(x+1){\sqrt {x}}}}}
The orthogonality of the Chebyshev rational functions may be written:
∫
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∞
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n
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{\displaystyle \int _{0}^{\infty }R_{m}(x)\,R_{n}(x)\,\omega (x)\,\mathrm {d} x={\frac {\pi c_{n}}{2}}\delta _{nm}}
where cn = 2 for n = 0 and cn = 1 for n ≥ 1 ; δnm is the Kronecker delta function.
Expansion of an arbitrary function
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For an arbitrary function f (x ) ∈ L 2 ω the orthogonality relationship can be used to expand f (x ) :
f
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∑
n
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∞
F
n
R
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{\displaystyle f(x)=\sum _{n=0}^{\infty }F_{n}R_{n}(x)}
where
F
n
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2
c
n
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∫
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f
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R
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ω
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{\displaystyle F_{n}={\frac {2}{c_{n}\pi }}\int _{0}^{\infty }f(x)R_{n}(x)\omega (x)\,\mathrm {d} x.}
Particular values
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R
0
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x
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=
1
R
1
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=
x
−
1
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1
R
2
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2
−
6
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1
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2
R
3
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x
3
−
15
x
2
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15
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1
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3
R
4
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x
4
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28
x
3
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70
x
2
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28
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1
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4
R
n
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{\displaystyle {\begin{aligned}R_{0}(x)&=1\\R_{1}(x)&={\frac {x-1}{x+1}}\\R_{2}(x)&={\frac {x^{2}-6x+1}{(x+1)^{2}}}\\R_{3}(x)&={\frac {x^{3}-15x^{2}+15x-1}{(x+1)^{3}}}\\R_{4}(x)&={\frac {x^{4}-28x^{3}+70x^{2}-28x+1}{(x+1)^{4}}}\\R_{n}(x)&=(x+1)^{-n}\sum _{m=0}^{n}(-1)^{m}{\binom {2n}{2m}}x^{n-m}\end{aligned}}}
Partial fraction expansion
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R
n
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∑
m
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n
(
m
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2
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2
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!
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{\displaystyle R_{n}(x)=\sum _{m=0}^{n}{\frac {(m!)^{2}}{(2m)!}}{\binom {n+m-1}{m}}{\binom {n}{m}}{\frac {(-4)^{m}}{(x+1)^{m}}}}
References
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