I want to calculate the Fourier series for the periodic function $f(x)=\sqrt{1-x^2}$ for $0<x<1$.
The DC term is easy to find, it’s just $a_n$ and $b_n$ that I can’t quite figure out.
Basically, my question is how is this integral solved?:
$$\int_{0}^{1} \sqrt{1-x^2}\cos(2n\pi x)\,dx $$
I’ve tried the substitution $x=\sin(\theta)$ but I didn’t really get anywhere.
Also, does this lead to Bessel Functions in any way? I saw some similarities when I looked them up, but I couldn’t reach an answer.
I would appreciate it if anyone could help me with this.
Thanks.