 Solution

The identity theorem says analytic functions agreeing on a set with an interior accumulation point agree throughout a connected domain. Analytic continuation is an analytic function on a larger connected domain agreeing on a nonempty overlap. Factor the denominator as $(t+i)(t-z)$. For $\Im z<0$, contour closure below giveswith a removable value at $-i$; this entire expression is the continuation. For $\Im z>0$, the displayed real-axis integral encloses only $-i$ and equals $2\pi i{e}^{-5}/(z+i)$, so it is not that continuation. Finally $1/n\to 0$ only at a boundary point of $D$, so the zeros do not determine $G$; for example $G(z)=\sin (\pi /z)$ is nonzero and has them all.