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 $T_{n}(x)$. 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, $T_{n}(z)$ can be defined by the contour integral

$T_{n}(z)={\frac {1}{4\pi i}}\oint {\frac {(1-t^{2})t^{-n-1}}{(1-2tz+t^{2})}}{\rm {d}}t$

The first seven Chebyshev polynomials of the first kind are:

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

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \left.T_{1}(x)\right.=x}

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

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\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 (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_6 (x)\right. =32x^6 - 48x^4 + 18x^2 -1}

Orthogonality

The Chebyshev polynomials are orthogonal polynomials with respect to the weighting function 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 (1-x^2)^{-1/2}} such that

$\int _{-1}^{1}{\frac {T_{m}(x)T_{n}(x)}{\sqrt {1-x^{2}}}}\mathrm {d} x=\left\{{\begin{array}{lll}{\frac {1}{2}}\pi \delta _{(mn)}&;&m\neq 0,n\neq 0\\\pi &;&m=n=0\end{array}}\right.$

where 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 \delta_{(mn)}} is the Kronecker delta.

Applications in statistical mechanics

• Computational implementation of integral equations

See also

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