A Bohl function is a linear combination of terms of the form $t^ke^{\lambda t}$ where $k$ is a non negative integer and $\lambda \in \Bbb C$. We denote the set of exponents of a Bohl function $p$ as $\sigma(p)$. For instance if $p(t) = te^t + 3t^2e^{3t}$ then $\sigma(p) = \{ 1,3\} $.
Now it says that $\sigma(pq) \subseteq \sigma(p) + \sigma(q) $. Now I was wondering how we can proof this. It seems wrong to me because if $p=q=e^x$ then $\sigma(pq)=\{2\} \neq \{1\} + \{1\}$. Maybe I don't really understand how to work out $\sigma(p)+\sigma(q)$. If anyone could help me to explain where I am going wrong I would be very thankful. Also, I can't find a single piece of information on Bohl functions anywhere else. Is there any area of mathematics where these are commonly encountered or encountered at all? Thanks in advance!
As a little background. This came up in a course on system control. Here we use the fact that the entries of $e^{At}$ are Bohl functions.
