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 ChebyshevU function using MatrixFunction:

Values of ChebyshevU at fixed points:

ChebyshevU for symbolic n:

Values at zero:

Values at infinity:

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

Compute the associated ChebyshevU[7,x] polynomial:

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

Plot the ChebyshevU function for various orders:

Plot the real part of :

Plot the imaginary part of :

Plot the Chebyshev polynomial as a function of two variables:

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

ChebyshevU is defined for all complex values besides :

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:

ChebyshevU threads elementwise over lists:

Chebyshev polynomials are analytic:

In general, ChebyshevU 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 ChebyshevU 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:

ChebyshevU is defined through the following trigonometric identity:

The ordinary generating function of ChebyshevU:

The exponential generating function of ChebyshevU:

Recurrence relations: