= Solution
The boundary data are odd except at isolated endpoint choices, which do not change <integrals>. Thus every cosine <Fourier coefficient>, including $a_0$, vanishes. For the sine <Fourier coefficients>,
$$
b_n=\frac2\pi\int_0^\pi\sin(nt)\,dt=\frac{2[1-(-1)^n]}{\pi n}
=\begin{cases}4/(\pi n),&n\text{ odd},\\0,&n\text{ even}.\end{cases}
$$
Consequently the extension is
$$
\boxed{y_{f_0}(r,\theta)=\frac4\pi\sum_{\substack{n\ge1\\n\text{ odd}}}\frac{r^n\sin(n\theta)}n.}
$$
It converges absolutely for every $r<1$; its limiting boundary values at jumps are the midpoint of the two one-sided limits.
Back to article page
