Can a combination of exponentials be periodic?

Question

I know that $e^x$ is not periodic but what about a combination of exp functions, such as $e^{3x^2+2/x}+5e^{x^2}-e^{\sqrt(x)}$, can they be periodic (all exponents REAL and period >0)

Answer 1

One way to do that is if the powers of $\exp$ were periodic - like $e^{\sin x}$.

Answer 2

Given the function $g: \mathbb{R} \to \mathbb{R}$ define $f: \mathbb{R} \to \mathbb{R}$ by $f(x)=e^{g(x)}$. If $f(x)$ is periodic with period $p$, then so is $g$. In this version of your question (can $f$ be periodic if $g$ isn't?), the answer to your question is then "no." (You can modify the domains if you need to. I'm just keeping it simple.)

The proof is simple. By assumption $f(x+p)=f(x)$. This says $e^{g(x+p)}=e^{g(x)}$. Take the log of both sides to get $g(x+p)=g(x)$. So, $g$ is periodic.