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Plot of the Chebyshev polynomial of the first kind () with = in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D. The Chebyshev polynomials are two sequences of polynomials related to the cosine and sine functions, notated as () and ().
Here we plot the Chebyshev nodes of the first kind and the second kind, both for n = 8. For both kinds of nodes, we first plot the points equi-distant on the upper half unit circle in blue. Then the blue points are projected down to the x-axis. The projected points, in red, are the Chebyshev nodes.
File:Chebyshev Polynomials of the 2nd Kind (n=0-5, x=(-1,1)).svg. Add languages. Page contents not supported in other languages. File; Talk;
The most commonly used orthogonal polynomials are orthogonal for a measure with support in a real interval. This includes: The classical orthogonal polynomials (Jacobi polynomials, Laguerre polynomials, Hermite polynomials, and their special cases Gegenbauer polynomials, Chebyshev polynomials and Legendre polynomials).
Chebyshev's equation is the second order linear differential equation + = where p is a real (or complex) constant. The equation is named after Russian mathematician Pafnuty Chebyshev. The solutions can be obtained by power series:
In applied mathematics, a discrete Chebyshev transform (DCT) is an analog of the discrete Fourier transform for a function of a real interval, converting in either direction between function values at a set of Chebyshev nodes and coefficients of a function in Chebyshev polynomial basis. Like the Chebyshev polynomials, it is named after Pafnuty ...
The Dickson polynomials with parameter α = 1 are related to Chebyshev polynomials T n (x) = cos (n arccos x) of the first kind by [1] (,) = (). Since the Dickson polynomial D n (x,α) can be defined over rings with additional idempotents, D n (x,α) is often not related to a Chebyshev polynomial.
In numerical analysis Chebyshev–Gauss quadrature is an extension of Gaussian quadrature method for approximating the value of integrals of the following kind: ∫ − 1 + 1 f ( x ) 1 − x 2 d x {\displaystyle \int _{-1}^{+1}{\frac {f(x)}{\sqrt {1-x^{2}}}}\,dx}