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:

ChebyshevT can be used with Interval and CenteredInterval objects:

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:

Plot the ChebyshevT function for various orders:

Plot the real part of :

Plot the imaginary part of :

Plot as real parts of two parameters vary:

Types 2 and 3 of ChebyshevT function have different branch cut structures:

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

ChebyshevT is defined for all complex values:

achieves all real and complex values:

Real range of :

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 not injective:

is:

is not surjective:

is:

is neither non-negative nor non-positive:

has singularities and discontinuities for when is not an integer:

is convex:

TraditionalForm formatting:

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 derivative with respect to x:

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:

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:

ChebyshevT is defined through the identity:

The ordinary generating function of ChebyshevT:

The exponential generating function of ChebyshevT:

Recurrence relations: