Residues of products at simple poles and a weighted residue formula

Suppose \(f\) has a simple pole at \(a\) and \(g\) is analytic near \(a\). Prove \[ \operatorname{Res}(a,fg)=g(a)\operatorname{Res}(a,f). \]

Let \(D\) be a domain, let \(f\) be analytic in \(D\) except for simple poles \(a_1,\dots,a_n\), and let \(g\) be analytic in \(D\). Prove that for every piecewise smooth closed path \(\gamma\) avoiding the poles and null-homologous in \(D\), \[ \frac1{2\pi i}\int_\gamma f(z)g(z)\,dz =\sum_{k=1}^n n(\gamma,a_k)g(a_k)\operatorname{Res}(a_k,f). \]