Let \(\gamma\) be a smooth curve joining two distinct points \(a, b\in \CC\).
Prove that the function \[ f(z) \definedas \int_\gamma {g(w) \over w-z} \,dw \] is analytic in \(\CC\setminus\gamma\).
Solution
Toward applying Morera, let \(T \subseteq \CC\sm \gamma\) be a triangle, so that \(z\in T\) and \(w\in \gamma\) implies \(z-w\neq 0\). Then \[ \oint_T f(z) \dz &= \oint_T \int_\gamma {g(w)\over w-z}\dw\dz \\ &= \int_\gamma \oint_T {g(w)\over w-z}\dz\dw \\ &= \int_\gamma g(w) \qty{ \oint_T {1 \over w-z}\dz} \dw \\ &= \int_\gamma g(w) \cdot 0 \dw \\ &= 0 ,\] where the exchange of integrals is justified by compactness of \(\gamma, T\), and the inner integral vanishes because for a fixed \(w\in \gamma\), the function \(z\mapsto {1\over w-z}\) has a simple pole at \(w\), and so is holomorphic in \(\gamma^c\) and vanishes by Goursat.