gives the Chebyshev polynomial of the first kind .
- 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 .
- ChebyshevT can be used with Interval and CenteredInterval objects. »
Examplesopen allclose all
Basic 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:
Numerical Evaluation (5)
The precision of the output tracks the precision of the input:
Evaluate efficiently at high precision:
ChebyshevT can be used with Interval and CenteredInterval objects:
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:
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 (14)
ChebyshevT is defined for all real values from the interval [-1,∞]:
ChebyshevT is defined for all complex values:
achieves all real and complex values:
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:
is neither non-decreasing nor non-increasing:
is neither non-negative nor non-positive:
has singularities and discontinuities for when is not an integer:
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)
ChebyshevT can be applied to power series:
ChebyshevT can be applied to Interval:
Plot the first 10 Chebyshev polynomials:
Find a minimax approximation to the function Clip[4 x]:
Solve a differential equation with the ChebyshevT function as 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:
Wolfram Research (1988), ChebyshevT, Wolfram Language function, https://reference.wolfram.com/language/ref/ChebyshevT.html (updated 2022).
Wolfram Language. 1988. "ChebyshevT." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2022. https://reference.wolfram.com/language/ref/ChebyshevT.html.
Wolfram Language. (1988). ChebyshevT. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/ChebyshevT.html