(related to Corollary: Reciprocity of Complex Exponential Function, Non-Zero Property)
By virtue of the functional equation of the complex exponential function, we have \[\exp(x)\cdot \exp(-x)=\exp(x-x)=\exp(0).\quad\quad( * )\] From the result that \[\exp(0)=1\] it follows \[\exp(-x)=\exp(x)^{-1}=\frac{1}{\exp(x)}.\] In particular, \(\exp(x)\neq 0\) for all \(x\in\mathbb C\), otherwise we would have \(0\cdot\exp(-x)=0=1\), which cannot be fulfilled for any complex number \(\exp(-x)\).
