In last week’s post on polynomial approximations for sine, I showed that the polynomial based on Chebyshev series was much better than a couple alternatives. I calculated a few terms of the Chebyshev series for sin(π*x*) but didn’t include the calculations in that blog post. I calculated the series coefficients numerically, but this post will show how to calculate the coefficients analytically.

## Generalities

The Chebyshev series for a function *f*(*x*) on [−1, 1] is given by

where *T*_{n}(*x*) is the *n*th Chebyshev polynomial of the first kind. The coefficients are given by

One way of defining the polynomials *T*_{n}(*x*) is

and so the change of variables *x* = cos θ lets us conclude

## Series for sin(π*x*)

Now for our particular function, *f*(*x*) = sin(π*x*), we know by symmetry that the coefficients with even subscripts will be zero. This is because sine is an odd function, and *T*_{n} is an even function when *n* is even,

Using equation 10.9.2 here we can prove that if *n* = 2*k *+ 1 then

where *J*_{n} is the *n*th Bessel function of the first kind.

(The preceding sentence was the conclusion to a fair amount of fumbling around on my part. As is often the case in mathematics, the length of the write-up is unrelated to the length of the discovery process.)