ChebyshevT
ChebyshevT[n,x]
gives the Chebyshev polynomial of the first kind .
Details

- Mathematical function, suitable for both symbolic and numerical manipulation.
- 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.
- ChebyshevT automatically threads over lists.
- ChebyshevT[n,z] has a branch cut discontinuity in the complex z plane running from
to
.
Examples
open allclose allBasic Examples (7)
Compute the 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:
Scope (36)
Numerical Evaluation (4)
Specific Values (7)
Values of ChebyshevT at fixed points:
ChebyshevT for symbolic n:
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 (4)
Plot the ChebyshevT function for various orders:
Plot as real parts of two parameters vary:
Types 2 and 3 of ChebyshevT function have different branch cut structures:
Function Properties (7)
ChebyshevT is defined for all real values from the interval [-1,∞]:
ChebyshevT is defined for all complex values:
Chebyshev polynomial of an odd order is odd:
Chebyshev polynomial of an even order is even:
ChebyshevT threads elementwise over lists:
TraditionalForm formatting:
Differentiation (3)
Integration (4)
Compute the indefinite integral using Integrate:
Definite integral of ChebyshevT over a period for odd integers is 0:
Series Expansions (3)
Find the Taylor expansion using Series:
Plots of the first three approximations around :
General term in the series expansion using SeriesCoefficient:
Function Identities and Simplifications (4)
ChebyshevT is defined through the identity:
The ordinary generating function of ChebyshevT:
The exponential generating function of ChebyshevT:
Generalizations & Extensions (2)
Applications (2)
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:
Text
Wolfram Research (1988), ChebyshevT, Wolfram Language function, https://reference.wolfram.com/language/ref/ChebyshevT.html.
BibTeX
BibLaTeX
CMS
Wolfram Language. 1988. "ChebyshevT." Wolfram Language & System Documentation Center. Wolfram Research. 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