Chebyshev polynomials

Chebyshev polynomials of the first kind are a set of orthogonal polynomials defined as the solutions to the Chebyshev differential equation and denoted Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle T_{n}(x)} . They are used as an approximation to a least squares fit, and are a special case of the ultra-spherical polynomial (Gegenbauer polynomial) with $\alpha =0$. Chebyshev polynomial of the first kind, Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle T_{n}(z)} can be defined by the contour integral

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle T_{n}(z)={\frac {1}{4\pi i}}\oint {\frac {(1-t^{2})t^{-n-1}}{(1-2tz+t^{2})}}{\rm {d}}t}

The first seven Chebyshev polynomials of the first kind are:

$\left.T_{0}(x)\right.=1$

$\left.T_{1}(x)\right.=x$

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left. T_2 (x) \right. =2x^2 -1}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left. T_3 (x) \right. =4x^3 - 3x}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left. T_4 (x) \right. =8x^4 - 8x^2 +1}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left. T_5 (x) \right. =16x^5 - 20x^3 +5x}

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \left.T_{6}(x)\right.=32x^{6}-48x^{4}+18x^{2}-1}

Applications in statistical mechanics

• Computational implementation of integral equations

See also

• Chebyshev Polynomial of the First Kind -- from Wolfram MathWorld]