Chebyshev polynomials - Revision history

Carl McBride: Added orthogonality condition

2008-07-07T10:29:20Z

Added orthogonality condition

← Older revision		Revision as of 12:29, 7 July 2008
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	:<math>\left. T_6 (x)\right. =32x^6 - 48x^4 + 18x^2 -1</math>		:<math>\left. T_6 (x)\right. =32x^6 - 48x^4 + 18x^2 -1</math>
			==Orthogonality==
			The Chebyshev polynomials are orthogonal polynomials with respect to the weighting function
			<math>(1-x^2)^{-1/2}</math> such that

			:<math>\int_{-1}^{1} \frac{T_m (x)T_n (x) }{ \sqrt{1-x^2}} \mathrm{d} x= \left\{ \begin{array}{lll}
			\frac{1}{2}\pi \delta_{(mn)} & ; & m \neq 0, n\neq 0 \\
			\pi & ; & m=n=0 \end{array} \right.</math>

			where <math>\delta_{(mn)}</math> is the [[Kronecker delta]].
	==Applications in statistical mechanics==		==Applications in statistical mechanics==
	*[[Computational implementation of integral equations]]		*[[Computational implementation of integral equations]]
	==See also==		==See also==
	*[http://mathworld.wolfram.com/ChebyshevPolynomialoftheFirstKind.html Chebyshev Polynomial of the First Kind -- from Wolfram MathWorld]]		*[http://mathworld.wolfram.com/ChebyshevPolynomialoftheFirstKind.html Chebyshev Polynomial of the First Kind -- from Wolfram MathWorld]
	[[category: mathematics]]		[[category: mathematics]]

Carl McBride: Added applications section.

2008-07-07T10:04:47Z

Added applications section.

← Older revision		Revision as of 12:04, 7 July 2008
Line 29:		Line 29:

	:<math>\left. T_6 (x)\right. =32x^6 - 48x^4 + 18x^2 -1</math>		:<math>\left. T_6 (x)\right. =32x^6 - 48x^4 + 18x^2 -1</math>
			==Applications in statistical mechanics==
			*[[Computational implementation of integral equations]]
	==See also==		==See also==
	*[http://mathworld.wolfram.com/ChebyshevPolynomialoftheFirstKind.html Chebyshev Polynomial of the First Kind -- from Wolfram MathWorld]]		*[http://mathworld.wolfram.com/ChebyshevPolynomialoftheFirstKind.html Chebyshev Polynomial of the First Kind -- from Wolfram MathWorld]]
	[[category: mathematics]]		[[category: mathematics]]

Carl McBride: New page: '''Chebyshev polynomials''' of the first kind are a set of orthogonal polynomials defined as the solutions to the Chebyshev differential equation and denoted $T_n(x)$ . They are ...

2007-05-31T09:57:31Z

New page: '''Chebyshev polynomials''' of the first kind are a set of orthogonal polynomials defined as the solutions to the Chebyshev differential equation and denoted <math>T_n(x)</math>. They are ...

New page

'''Chebyshev polynomials''' of the first kind are a set of orthogonal polynomials defined as the solutions to the Chebyshev differential equation and denoted
<math>T_n(x)</math>.
They are used as an approximation to a least squares fit,
and are a special case of the ultra-spherical polynomial (Gegenbauer polynomial)
with <math>\alpha=0</math>.
Chebyshev polynomial of the first kind, <math>T_n (z)</math> can be defined by the contour integral

:<math>T_n (z) = \frac{1}{4 \pi i} \oint \frac{(1-t^2)t^{-n-1}}{(1-2tz+t^2)} {\rm d}t</math>

The first seven Chebyshev polynomials of the first kind are:

:<math>\left. T_0 (x) \right. =1</math>

:<math>\left. T_1 (x) \right. =x</math>

:<math>\left. T_2 (x) \right. =2x^2 -1</math>

:<math>\left. T_3 (x) \right. =4x^3 - 3x</math>

:<math>\left. T_4 (x) \right. =8x^4 - 8x^2 +1</math>

:<math>\left. T_5 (x) \right. =16x^5 - 20x^3 +5x</math>

:<math>\left. T_6 (x)\right. =32x^6 - 48x^4 + 18x^2 -1</math>

==See also==
*[http://mathworld.wolfram.com/ChebyshevPolynomialoftheFirstKind.html Chebyshev Polynomial of the First Kind -- from Wolfram MathWorld]]
[[category: mathematics]]

Chebyshev polynomials - Revision history

Carl McBride: Added orthogonality condition

Carl McBride: Added applications section.

Carl McBride: New page: '''Chebyshev polynomials''' of the first kind are a set of orthogonal polynomials defined as the solutions to the Chebyshev differential equation and denoted T_n(x). They are ...

Carl McBride: New page: '''Chebyshev polynomials''' of the first kind are a set of orthogonal polynomials defined as the solutions to the Chebyshev differential equation and denoted $T_n(x)$ . They are ...