ChebyshevT

ChebyshevT[n,x]

gives the Chebyshev polynomial of the first kind TemplateBox[{n, x}, ChebyshevT].

Details

  • Mathematical function, suitable for both symbolic and numerical manipulation.
  • Explicit polynomials are given for integer n.
  • TemplateBox[{n, {cos,  , theta}}, ChebyshevT]=cos(ntheta).
  • For certain special arguments, ChebyshevT automatically evaluates to exact values.
  • ChebyshevT can be evaluated to arbitrary numerical precision.
  • ChebyshevT automatically threads over lists.
  • ChebyshevT[n,z] has a branch cut discontinuity in the complex z plane running from to if n is not an integer.
  • ChebyshevT can be used with Interval and CenteredInterval objects. »

Examples

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Basic Examples  (7)

Evaluate numerically:

Compute the 10^(th) Chebyshev polynomial:

Plot over a subset of the reals:

Plot over a subset of the complexes:

Series expansion at the origin:

Asymptotic expansion at Infinity:

Asymptotic expansion at a singular point:

Scope  (44)

Numerical Evaluation  (6)

Evaluate numerically:

Evaluate to high precision:

The precision of the output tracks the precision of the input:

Complex number input:

Evaluate efficiently at high precision:

Compute worst-case guaranteed intervals using Interval and CenteredInterval objects:

Or compute average-case statistical intervals using Around:

Compute the elementwise values of an array:

Or compute the matrix ChebyshevT function using MatrixFunction:

Specific Values  (7)

Values of ChebyshevT at fixed points:

ChebyshevT for symbolic n:

Values at zero:

Values at infinity:

Find the first positive maximum of ChebyshevT[5,x]:

Compute the associated ChebyshevT[7,x] polynomial:

Compute the associated ChebyshevT[1/2,x] polynomial for half-integer n:

Visualization  (3)

Plot the ChebyshevT function for various orders:

Plot the real part of TemplateBox[{3, z}, ChebyshevT]:

Plot the imaginary part of TemplateBox[{3, z}, ChebyshevT]:

Plot the Chebyshev polynomial as a function of two variables:

Function Properties  (14)

ChebyshevT is defined for all real values from the interval [-1,]:

ChebyshevT is defined for all complex values:

TemplateBox[{1, x}, ChebyshevT] achieves all real and complex values:

Real range of TemplateBox[{2, x}, ChebyshevT]:

It achieves all complex values:

Chebyshev polynomial of an odd order is odd:

Chebyshev polynomial of an even order is even:

ChebyshevT threads elementwise over lists:

Chebyshev polynomials are analytic:

In general, ChebyshevT is neither analytic nor meromorphic:

TemplateBox[{2, x}, ChebyshevT] is neither non-decreasing nor non-increasing:

TemplateBox[{2, x}, ChebyshevT] is not injective:

TemplateBox[{1, x}, ChebyshevT] is:

TemplateBox[{2, x}, ChebyshevT] is not surjective:

TemplateBox[{1, x}, ChebyshevT] is:

TemplateBox[{2, x}, ChebyshevT] is neither non-negative nor non-positive:

TemplateBox[{n, x}, ChebyshevT] has singularities and discontinuities for when is not an integer:

TemplateBox[{2, x}, ChebyshevT] is convex:

TraditionalForm formatting:

Differentiation  (3)

First derivative with respect to x:

Higher derivatives with respect to x:

Plot the higher derivatives with respect to x when n=5:

Formula for the ^(th) derivative with respect to x:

Integration  (4)

Compute the indefinite integral using Integrate:

Verify the anti-derivative:

Definite integral:

Definite integral of ChebyshevT over a period for odd integers is 0:

More integrals:

Series Expansions  (3)

Find the Taylor expansion using Series:

Plots of the first three approximations around :

General term in the series expansion using SeriesCoefficient:

Taylor expansion at a generic point:

Function Identities and Simplifications  (4)

ChebyshevT is defined through the identity:

The ordinary generating function of ChebyshevT:

The exponential generating function of ChebyshevT:

Recurrence relations:

Generalizations & Extensions  (2)

ChebyshevT can be applied to power series:

ChebyshevT can be applied to Interval:

Applications  (4)

Plot the first 10 Chebyshev polynomials:

Find a minimax approximation to the function Clip[4 x]:

Get an expansion for a function in the Chebyshev polynomials:

The values of the function at the Chebyshev nodes:

Find the Chebyshev coefficients:

Show the error:

Solve a differential equation with the ChebyshevT function as the inhomogeneous part:

Properties & Relations  (7)

Use FullSimplify with ChebyshevT:

Derivative of ChebyshevT is expressed in terms of ChebyshevU:

ChebyshevT can be represented as a DifferenceRoot:

General term in the series expansion of ChebyshevT:

The generating function for ChebyshevT:

The exponential generating function for ChebyshevT:

Possible Issues  (1)

Cancellations in the polynomial form may lead to inaccurate numerical results:

Evaluate the function directly:

Neat Examples  (1)

Plot the first few BanchoffChmutov surfaces:

Wolfram Research (1988), ChebyshevT, Wolfram Language function, https://reference.wolfram.com/language/ref/ChebyshevT.html (updated 2022).

Text

Wolfram Research (1988), ChebyshevT, Wolfram Language function, https://reference.wolfram.com/language/ref/ChebyshevT.html (updated 2022).

CMS

Wolfram Language. 1988. "ChebyshevT." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2022. https://reference.wolfram.com/language/ref/ChebyshevT.html.

APA

Wolfram Language. (1988). ChebyshevT. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/ChebyshevT.html

BibTeX

@misc{reference.wolfram_2024_chebyshevt, author="Wolfram Research", title="{ChebyshevT}", year="2022", howpublished="\url{https://reference.wolfram.com/language/ref/ChebyshevT.html}", note=[Accessed: 22-December-2024 ]}

BibLaTeX

@online{reference.wolfram_2024_chebyshevt, organization={Wolfram Research}, title={ChebyshevT}, year={2022}, url={https://reference.wolfram.com/language/ref/ChebyshevT.html}, note=[Accessed: 22-December-2024 ]}