Suppose that \(f\) is holomorphic on a punctured open set \(\Omega\setminus\theset{w_0}\) and let \(T\subset \Omega\) be a triangle containing \(w_0\). Prove that if \(f\) is bounded near \(w_0\), then \(\int_T f(z) ~dz = 0\).
Solution
Without loss of generality assume \(w_0 = 0\). If \(\abs{f(z)} \leq M\) for \(\abs{z} < \eps\), pick \(T\) contained in \(\DD_\eps\), then \[ \abs{\oint_T f(z)\dz } \leq \oint_T \abs{f(z)}\dz \leq \oint_T M \dz = M \mu(T) ,\] and by homotopy invariance, this integral doesn’t change as \(T\) is perturbed. Shrinking \(T\), we can make \(\mu(T)\to 0\).