Residues of sums and products

Let \(U\subset\mathbb C\) be open and let \(f,g:U\to\mathbb C\) be analytic except for isolated singularities. Let \(z_0\in U\).

1. Is the residue of \(f+g\) at \(z_0\) equal to the sum of the residues of \(f\) and \(g\) at \(z_0\)?

2. Is the residue of \(fg\) at \(z_0\) equal to the product of the residues of \(f\) and \(g\) at \(z_0\)?