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File:Chebyshev Polynomials of the 2nd Kind (n=0-5, x=(-1,1)).svg. Add languages. Page contents not supported in other languages. ... Printable version; Page information;
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 ().
The Chebyshev nodes of the second kind, also called the Chebyshev extrema, are the extrema of the Chebyshev polynomials of the first kind, which are also the zeros of the Chebyshev polynomials of the second kind. Both of these sets of numbers are commonly referred to as Chebyshev nodes in literature. [1]
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:
The zeroes of the elliptic rational function will be the zeroes of the polynomial in the numerator of the function. The following derivation of the zeroes of the elliptic rational function is analogous to that of determining the zeroes of the Chebyshev polynomials (Lutovac, Tošić & Evans 2001, § 12.6). Using the fact that for any z
Like the Chebyshev polynomials, it is named after Pafnuty Chebyshev. The two most common types of discrete Chebyshev transforms use the grid of Chebyshev zeros, the zeros of the Chebyshev polynomials of the first kind () and the grid of Chebyshev extrema, the extrema of the Chebyshev polynomials of the first kind, which are also the zeros of ...
Somewhat more general Laguerre polynomial sequences are orthogonal with respect to gamma distributions. The Chebyshev polynomials of the first kind are orthogonal with respect to the measure . The Chebyshev polynomials of the second kind are orthogonal with respect to the Wigner semicircle distribution.
It is a solution of a second-order linear ordinary differential equation (ODE). Every second-order linear ODE with three regular singular points can be transformed into this equation. For systematic lists of some of the many thousands of published identities involving the hypergeometric function, see the reference works by Erdélyi et al. (1953 ...
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