Chebyshev polynomials

The Chebyshev polynomials named after Pafnuty Chebyshev, compose a polynomial sequence, and are defined by

<math>T_n(\cos(\theta))=\cos(n\theta)</math>

for n = 0, 1, 2, 3, .... . That cos(nx) is an nth-degree polynomial in cos(x) can be seen by observing that cos(nx) is the real part of one side of De Moivre's formula, and the real part of the other side is a polynomial in cos(x) and sin(x), in which all powers of sin(x) are even.

These polynomials are orthogonal with respect to the weight

<math>\frac{dx}{\sqrt{1-x^2}},</math>

on the interval [-1,1], i.e., we have

<math>\int_{-1}^1 T_n(x)T_m(x)\,\frac{dx}{\sqrt{1-x^2}}=0\quad\mbox{if}\ n\neq m.</math>