Built-in Mathematica Symbol

ChebyshevU

ChebyshevU[n, x]
gives the Chebyshev polynomial of the second kind

.

For certain special arguments, ChebyshevU automatically evaluates to exact values.

ChebyshevU[n, z] has a branch cut discontinuity in the complex z plane running from - to for noninteger n.

Compute the

ChebyshevU polynomial:

Compute the

ChebyshevU polynomial:

Evaluate for complex arguments and orders:

Evaluate for large orders:

Evaluate to high precision:

ChebyshevU threads element-wise over the list:

Simple cases give exact symbolic results even for arbitrary order:

ChebyshevU can be applied to power series:

ChebyshevU can be applied to Interval:

Approximate a function on the interval

:

Build a curve that passes through given points:

Light amplitude transmission through

layers of glass:

Get the list of coefficients in a ChebyshevU polynomial:

Use FunctionExpand to expand through trigonometric functions:

Derivative of ChebyshevU with respect to

:

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

Evaluate the function directly: