Suppose that \(f: \RR\to\RR\) is a continuous function that vanishes outside of some finite interval. For each \(z\in \CC\), define \[ g(z) = \int_{-\infty}^\infty f(t) e^{-izt} \,dt .\]
Show that \(g\) is entire.
Solution
By Fubini: \[ \oint_T g(z)\dz &\da \oint_T \int_\RR f(t)e^{-izt} \dt\dz \\ &\da \int_\RR \oint_T f(t)e^{-izt} \dz\dt \\ &\da \int_\RR f(t) \qty{ \oint_T e^{-izt} \dz } \dt \\ &\da \int_\RR f(t) \cdot 0 \dt \\ &= 0 ,\] where the inner integral vanishes because \(z\mapsto e^{-izt}\) is entire by Goursat’s theorem. So \(g\) is entire by Morera.