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In mathematics the Chebyshev rational functions are a sequence of functions which are both rational and orthogonal . A rational Chebyshev function of degree n is defined as:
R
n
(
x
)
≡
T
n
(
x
−
1
x
+
1
)
{\displaystyle R_{n}(x)\equiv T_{n}\left({\frac {x-1}{x+1}}\right)}
where
T
n
(
x
)
{\displaystyle T_{n}(x)}
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
R
n
+
1
(
x
)
=
2
x
−
1
x
+
1
R
n
(
x
)
−
R
n
−
1
(
x
)
f
o
r
n
≥
1
{\displaystyle R_{n+1}(x)=2\,{\frac {x-1}{x+1}}R_{n}(x)-R_{n-1}(x)\quad \mathrm {for\,n\geq 1} }
Differential equations
(
x
+
1
)
2
R
n
(
x
)
=
1
n
+
1
d
d
x
R
n
+
1
(
x
)
−
1
n
−
1
d
d
x
R
n
−
1
(
x
)
f
o
r
n
≥
2
{\displaystyle (x+1)^{2}R_{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\geq 2} }
(
x
+
1
)
2
x
d
2
d
x
2
R
n
(
x
)
+
(
3
x
+
1
)
(
x
+
1
)
2
d
d
x
R
n
(
x
)
+
n
2
R
n
(
x
)
=
0
{\displaystyle (x+1)^{2}x{\frac {d^{2}}{dx^{2}}}\,R_{n}(x)+{\frac {(3x+1)(x+1)}{2}}{\frac {d}{dx}}\,R_{n}(x)+n^{2}R_{n}(x)=0}
Orthogonality
Defining:
ω
(
x
)
≡
1
(
x
+
1
)
x
{\displaystyle \omega (x)\equiv {\frac {1}{(x+1){\sqrt {x}}}}}
The orthogonality of the Chebyshev rational functions may be written:
∫
0
∞
R
m
(
x
)
R
n
(
x
)
ω
(
x
)
d
x
=
π
c
n
2
δ
n
m
{\displaystyle \int _{0}^{\infty }R_{m}(x)\,R_{n}(x)\,\omega (x)\,dx={\frac {\pi c_{n}}{2}}\delta _{nm}}
where
c
n
{\displaystyle c_{n}}
n=0 and
c
n
{\displaystyle c_{n}}
n
≥
1
{\displaystyle n\geq 1}
δ
n
m
{\displaystyle \delta _{nm}}
Kronecker delta function.
Expansion of an arbitrary function
For an arbitrary function
f
(
x
)
∈
L
ω
2
{\displaystyle f(x)\in L_{\omega }^{2}}
f
(
x
)
{\displaystyle f(x)}
f
(
x
)
=
∑
n
=
0
∞
F
n
R
n
(
x
)
{\displaystyle f(x)=\sum _{n=0}^{\infty }F_{n}R_{n}(x)}
where
F
n
=
2
c
n
π
∫
0
∞
f
(
x
)
R
n
(
x
)
ω
(
x
)
d
x
{\displaystyle F_{n}={\frac {2}{c_{n}\pi }}\int _{0}^{\infty }f(x)R_{n}(x)\omega (x)\,dx}
Particular values
R
0
(
x
)
=
1
{\displaystyle R_{0}(x)=1\,}
R
1
(
x
)
=
x
−
1
x
+
1
{\displaystyle R_{1}(x)={\frac {x-1}{x+1}}\,}
R
2
(
x
)
=
x
2
−
6
x
+
1
(
x
+
1
)
2
{\displaystyle R_{2}(x)={\frac {x^{2}-6x+1}{(x+1)^{2}}}\,}
R
3
(
x
)
=
x
3
−
15
x
2
+
15
x
−
1
(
x
+
1
)
3
{\displaystyle R_{3}(x)={\frac {x^{3}-15x^{2}+15x-1}{(x+1)^{3}}}\,}
R
4
(
x
)
=
x
4
−
28
x
3
+
70
x
2
−
28
x
+
1
(
x
+
1
)
4
{\displaystyle R_{4}(x)={\frac {x^{4}-28x^{3}+70x^{2}-28x+1}{(x+1)^{4}}}\,}
References
Ben-Yu, Guo (2002). "Chebyshev rational spectral and pseudospectral methods on a semi-infinite interval" (PDF) . Int. J. Numer. Meth. Engng . 53 : 65– 84. doi :10.1002/nme.392 . Retrieved 2006-07-25 .
