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Difference between revisions of "Chebyshev equation"

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The linear homogeneous second-order ordinary differential equation
 
The linear homogeneous second-order ordinary differential equation
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021860/c0218601.png" /></td> </tr></table>
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$$(1-x^2)\frac{d^2y}{dx^2}-x\frac{dy}{dx}+ay=0$$
  
 
or, in self-adjoint form,
 
or, in self-adjoint form,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021860/c0218602.png" /></td> </tr></table>
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$$\sqrt{1-x^2}\frac d{dx}\left(\sqrt{1-x^2}\frac{dy}{dx}\right)+ay=0,$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021860/c0218603.png" /> is a constant. Chebyshev's equation is a special case of the [[Hypergeometric equation|hypergeometric equation]].
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where $a$ is a constant. Chebyshev's equation is a special case of the [[Hypergeometric equation|hypergeometric equation]].
  
The points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021860/c0218604.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021860/c0218605.png" /> are regular singular points (cf. [[Regular singular point|Regular singular point]]) of the Chebyshev equation. Substituting the independent variable
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The points $x=-1$ and $x=1$ are regular singular points (cf. [[Regular singular point|Regular singular point]]) of the Chebyshev equation. Substituting the independent variable
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021860/c0218606.png" /></td> </tr></table>
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$$t=\arccos x\quad\text{for }|x|<1,$$
  
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$$t=\operatorname{Arcosh}|x|\quad\text{for }|x|>1$$
  
 
reduces this equation to a corresponding linear equation with constant coefficients:
 
reduces this equation to a corresponding linear equation with constant coefficients:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021860/c0218608.png" /></td> </tr></table>
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$$\frac{d^2y}{dt^2}+ay=0\quad\text{or}\quad\frac{d^2y}{dt^2}-ay=0,$$
  
so that Chebyshev's equation can be integrated in closed form. A fundamental systems of solutions to Chebyshev's equation on the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021860/c0218609.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021860/c02186010.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021860/c02186011.png" /> is a natural number, consists of the [[Chebyshev polynomials|Chebyshev polynomials]] (of the first kind) of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021860/c02186012.png" />,
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so that Chebyshev's equation can be integrated in closed form. A fundamental system of solutions to Chebyshev's equation on the interval $-1<x<1$ with $a=n^2$, where $n$ is a natural number, consists of the [[Chebyshev polynomials|Chebyshev polynomials]] (of the first kind) of degree $n$,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021860/c02186013.png" /></td> </tr></table>
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$$T_n(x)=\cos(n\arccos x),$$
  
and the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021860/c02186014.png" />, which are related to Chebyshev polynomials of the second kind. The polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021860/c02186015.png" /> is a real solution to Chebyshev's equation on the entire real line, with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021860/c02186016.png" />. Chebyshev's equation is also studied in complex domains.
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and the functions $U_n(x)=\sin(n\arccos x)$, which are related to Chebyshev polynomials of the second kind. The polynomial $T_n(x)$ is a real solution to Chebyshev's equation on the entire real line, with $a=n^2$. Chebyshev's equation is also studied in complex domains.

Latest revision as of 19:57, 21 November 2018

The linear homogeneous second-order ordinary differential equation

$$(1-x^2)\frac{d^2y}{dx^2}-x\frac{dy}{dx}+ay=0$$

or, in self-adjoint form,

$$\sqrt{1-x^2}\frac d{dx}\left(\sqrt{1-x^2}\frac{dy}{dx}\right)+ay=0,$$

where $a$ is a constant. Chebyshev's equation is a special case of the hypergeometric equation.

The points $x=-1$ and $x=1$ are regular singular points (cf. Regular singular point) of the Chebyshev equation. Substituting the independent variable

$$t=\arccos x\quad\text{for }|x|<1,$$

$$t=\operatorname{Arcosh}|x|\quad\text{for }|x|>1$$

reduces this equation to a corresponding linear equation with constant coefficients:

$$\frac{d^2y}{dt^2}+ay=0\quad\text{or}\quad\frac{d^2y}{dt^2}-ay=0,$$

so that Chebyshev's equation can be integrated in closed form. A fundamental system of solutions to Chebyshev's equation on the interval $-1<x<1$ with $a=n^2$, where $n$ is a natural number, consists of the Chebyshev polynomials (of the first kind) of degree $n$,

$$T_n(x)=\cos(n\arccos x),$$

and the functions $U_n(x)=\sin(n\arccos x)$, which are related to Chebyshev polynomials of the second kind. The polynomial $T_n(x)$ is a real solution to Chebyshev's equation on the entire real line, with $a=n^2$. Chebyshev's equation is also studied in complex domains.

How to Cite This Entry:
Chebyshev equation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Chebyshev_equation&oldid=15891
This article was adapted from an original article by N.Kh. Rozov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article