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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 ().
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;
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:
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 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
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 ...
Depending on the bilinear form, the vector space may contain null vectors, non-zero self-orthogonal vectors, in which case perpendicularity is replaced with hyperbolic orthogonality. In the case of function spaces , families of functions are used to form an orthogonal basis , such as in the contexts of orthogonal polynomials , orthogonal ...