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The Chebyshev polynomials are two sequences of polynomials related to the cosine and sine functions, notated as and . They can be defined several ways that have the same end result; in this article the polynomials are defined by starting with trigonometric functions: The Chebyshev polynomials of the first kind are given by Similarly, define the Chebyshev polynomials of the second kind as

The Chebyshev polynomials form an orthogonal basis which (among other things) implies that the coefficients can be determined easily through the application of ...

The Chebyshev polynomials are orthogonal polynomials used in many disparate areas of pure and applied mathematics

A two-stage numerical procedure using Chebyshev polynomials and ... symbol software mathematica to connect monomials with Chebyshev polynomials and employed ...

Axiom and Mathematica calculate a particular Tn(x) if issued, and use no remember tables. For numerical computations, both exact and approximate, they use.

Is there functionality in Mathematica to expand a function into a series with Chebyshev polynomials? The Series function only approximates with Taylor ...

Nov 01, 2021 · The Chebyshev polynomials of the second kind. U n ( x) are solutions of the differential equation. ( 1 − x 2) y ″ − 3 x y ′ + n ( n + 2) y = 0, and they are orthogonal on the interval [-1, 1] with weight function. ρ ( x) = ( 1 − x 2) 1 / 2: ∫ 1 − 1 U m ( x) U n ( x) √ 1 − x 2 d x = { π / 2, if n = m, 0, if n ≠ m.

Explicit polynomials are given for integer n. For certain special arguments, ChebyshevU automatically evaluates to exact values. ChebyshevU can be evaluated to arbitrary numerical precision.

ChebyshevT[n, x] gives the Chebyshev polynomial of the first kind Tn (x).

Explicit polynomials are given for integer n. For certain special arguments, ChebyshevT automatically evaluates to exact values. ChebyshevT can be evaluated to arbitrary numerical precision.

In particular, what are Chebyshev polynomials, what properties do they hold, ... Software- Mathematica, Chebyshev Polynomials, graphing utility.

Chebyshev polynomials are usually used for either approximation of continuous functions or function expansion. For the case of functions that are solutions of linear ordinary differential equations with polynomial coefficients (a typical case for special functions), the problem of computing Chebyshev series is efficiently solved by means of Clenshaw’s method.

28.12.2020 · Mathematica Stack Exchange is a question and answer site for users of Wolfram Mathematica. It only takes a minute to sign up. ... The way to use Nest[] for computing the Chebyshev polynomial of the first kind is to recognize that a three-term recurrence is equivalent to repeated multiplication by a certain $2\times 2$ matrix.

MATHEMATICA? Box 1 shows a simple program for these. Chebyshev polynomials from n = 0 to 6, written in MAPLE V,. Release 2. (Our column in the next issue is ...

17.12.2021 · Chebyshev Polynomial of the Second Kind. A modified set of Chebyshev polynomials defined by a slightly different generating function.They arise in the development of four-dimensional spherical harmonics in angular momentum theory. They are a special case of the Gegenbauer polynomial with .They are also intimately connected with trigonometric …

I want to convert a polynomial in "standard form" to Chebyshev form. Here's one way to do it: (* My polynomial *) pa = Sum [a [i]*t^i, {i, 0, 5}] (* "standard form" coefficients of Chebyshev polynomial of same degree *) pb = CoefficientList [Sum [b [i]*ChebyshevT [i, t], {i, 0, 5}], t] (* helper variable *) bs = Table [b [i], {i, 0, 5}] Solve [Table [a [i - 1] == pb [ [i]], {i, 1, 6}], bs]

ChebyshevT[n, x] gives the Chebyshev polynomial of the first kind Tn (x).