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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)

Quadratica
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    $e^{\sin x}$?.... – Randall May 29 '19 at 18:16
  • As a smart-aleck answer... The function $f(x) = e^5$ is periodic as well since constant functions are periodic too, loosely speaking "with period zero." That being said, if one or more of the exponents is $\omega(1)$, for example if one of the exponents is linear or super-linear, then consider the term whose exponent is of the largest order. That particular term will continually grow and will eventually dwarf all of the other terms combined in size and the function will be strictly increasing from then on implying it cannot be periodic. – JMoravitz May 29 '19 at 18:52
  • yes, a smart-alec answer but NOT the one required! – Quadratica May 29 '19 at 19:14

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

Ishan Deo
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  • of course using trigs raises the possibility of the whole function being periodic. I wonder if the given function can be periodic without using periodic exponents – Quadratica May 29 '19 at 19:16
  • @Quadratica that is a much better phrasing of your question. I would edit it this way. – Randall May 29 '19 at 23:16
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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.

Randall
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  • Thanks Randall, I appreciate your response and it makes perfect sense......I wish I had thought of it. – Quadratica May 30 '19 at 20:26
  • I will try this approach to investigate the periodicity, or otherwise, of the sum of two exponentials with non-periodic exponents. – Quadratica May 30 '19 at 20:39