Simple pole and a product of functions
by Hidenori
Proposition
Suppose $f$ has a simple pole at $z_0$ and $g$ is holomorphic at $z_0$. Prove that
\[\begin{align*} \Res_{z = z_0}(f(z)g(z)) = g(z_0)\Res_{z = z_0}(f(z)). \end{align*}\]Solution
By Corollary 9.6[A first course in complex analysis], $f(z) = \tilde{f}(z)/(z - z_0)$ for some $\tilde{f}$ that is holomorphic in an open disk around $z_0$.
Then $fg = g\tilde{f}/(z - z_0)$ where $g\tilde{f}$ is holomorphic at $z_0$. Thus $fg$ has a simple pole at $z_0$. By Proposition 9.11(b)[A first course in complex analysis],
\[\begin{align*} \Res_{z = z_0}(fg) &= \lim_{z \rightarrow z_0} (z - z_0)fg \\ &= \lim_{z \rightarrow z_0} g\tilde{f} \\ &= g(z_0)\tilde{f}(z_0) \\ &= g(z_0)\lim_{z \rightarrow z_0}(z - z_0)f(z) \\ &= g(z_0)\Res_{z = z_0}(f(z)). \end{align*}\]Subscribe via RSS