Difference between revisions of "Chebyshev U"

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(Created page with " =Properties= <div class="toccolours mw-collapsible mw-collapsed" style="width:800px"> <strong>Theorem (Orthogonality):</strong> The following formulas hold: $$\int_{-1}^1 \df...")
 
 
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The Chebyshev polynomials of the second kind are orthogonal polynomials defined by
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$$U_n(x) = \sin(n \mathrm{arcsin}(x)),$$
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where $\sin$ denotes [[sine]] and $\mathrm{arcsin}$ denotes [[arcsin]].
  
 
=Properties=
 
=Properties=
<div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
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[[Orthogonality of Chebyshev U on (-1,1)]]<br />
<strong>Theorem (Orthogonality):</strong> The following formulas hold:
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[[Relationship between Chebyshev U and hypergeometric 2F1]]<br />
$$\int_{-1}^1 \dfrac{T_m(x)T_n(x)}{\sqrt{1-x^2}} dx = \left\{ \begin{array}{ll}
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[[Relationship between Chebyshev U and Gegenbauer C]]<br />
0 &; m \neq n \\
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[[U n(x)=(-i/2)(x+i sqrt(1-x^2))^n+(-i/2)(x-i sqrt(1-x^2))^n]]<br />
\dfrac{\pi}{2} &; m=n\neq 0 \\
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[[U n(x)=Sum (-1)^k n!/((2k+1)!(n-2k-1)!)(1-x^2)^(k+1/2)x^(n-2k-1)]]<br />
\pi &; m=n=0
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\end{array} \right.$$
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=References=
and
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* {{BookReference|Special Functions for Scientists and Engineers|1968|W.W. Bell|prev=Chebyshev T|next=T n(x)=(1/2)(x+i sqrt(1-x^2))^n+(1/2)(x-i sqrt(1-x^2))^n}}: $(7.2)$
$$\int_{-1}^1 \dfrac{U_m(x)U_n(x)}{\sqrt{1-x^2}} dx = \left\{ \begin{array}{ll}
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0 &; m \neq n \\
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{{:Orthogonal polynomials footer}}
\dfrac{\pi}{2} &; m=n\neq 0\\
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0 &; m=n=0.
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[[Category:SpecialFunction]]
\end{array} \right.$$
 
<div class="mw-collapsible-content">
 
<strong>Proof:</strong> █
 
</div>
 
</div>
 

Latest revision as of 19:33, 15 March 2018

The Chebyshev polynomials of the second kind are orthogonal polynomials defined by $$U_n(x) = \sin(n \mathrm{arcsin}(x)),$$ where $\sin$ denotes sine and $\mathrm{arcsin}$ denotes arcsin.

Properties

Orthogonality of Chebyshev U on (-1,1)
Relationship between Chebyshev U and hypergeometric 2F1
Relationship between Chebyshev U and Gegenbauer C
U n(x)=(-i/2)(x+i sqrt(1-x^2))^n+(-i/2)(x-i sqrt(1-x^2))^n
U n(x)=Sum (-1)^k n!/((2k+1)!(n-2k-1)!)(1-x^2)^(k+1/2)x^(n-2k-1)

References

Orthogonal polynomials