Chebyshev Polynomial (Not in Base Package)

Calculates the Chebyshev polynomial of order n at the point x. Details

x is any real number.
n is the nonnegative order (integer) of the Chebyshev polynomial.
T(n,x) is the value of the nth Chebyshev polynomial at the point x.

Chebyshev Polynomial Details

The Chebyshev polynomial, Tn(x), is defined by

Tn(x) = cos(n arccos(x))

for n=0, 1, 2, ... and real numbers x.

Note  The result of this definition does not look like a polynomial at first glance, but you can use trigonometric rules to show that is a polynomial of degree n in the variable x.

These functions form the base of the so called Chebyshev approximation. For

it is

All form an orthogonal system over the weight function

The following diagram shows the first four Chebyshev polynomials of degrees 0, 1, 2, and 3.