Chebyshev rational functions
In-game article clicks load inline without leaving the challenge.
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:
R n ( x ) = d e f T n ( x − 1 x + 1 ) {\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
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 ) for n ≥ 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
( 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 ) for n ≥ 2 {\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}
( 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 {\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
Defining:
ω ( x ) = d e f 1 ( x + 1 ) x {\displaystyle \omega (x)\ {\stackrel {\mathrm {def} }{=}}\ {\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)\,\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
For an arbitrary function f(x) ∈ L2 ω the orthogonality relationship can be used to expand 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)\,\mathrm {d} x.}
Particular values
R 0 ( x ) = 1 R 1 ( x ) = x − 1 x + 1 R 2 ( x ) = x 2 − 6 x + 1 ( x + 1 ) 2 R 3 ( x ) = x 3 − 15 x 2 + 15 x − 1 ( x + 1 ) 3 R 4 ( x ) = x 4 − 28 x 3 + 70 x 2 − 28 x + 1 ( x + 1 ) 4 R n ( x ) = ( x + 1 ) − n ∑ m = 0 n ( − 1 ) m ( 2 n 2 m ) x n − m {\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
R n ( x ) = ∑ m = 0 n ( m ! ) 2 ( 2 m ) ! ( n + m − 1 m ) ( n m ) ( − 4 ) m ( x + 1 ) m {\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}}}}