Periodic Function and change of scale

Question

I am confused by a discussion with a colleague. The discussion is about the period of a periodic function.

For example, the periodic function $$f(x)=\sin(x), \quad x\in (0,\infty)$$ has period $2\pi$. If I change the scale and build the function, $$g(x)=\sin(\ln x),\quad x\in (0,\infty)$$ is this new function, g, periodic? If it is, what is the period?

EDIT

I will clarify my point. If I change the scale of the function $g$, let's say, $\ln x =u$ then I will have function $$h(u)=\sin u, \quad u\in \mathbb R$$ and now $h$ is periodic on $u\in \mathbb R $.

So, my point is can I say that $g$ is not periodic in $x$-domain but it is in $\log$-domain?

I'm pretty sure periodic functions must be defined over the whole real line. Since g is not defined for negative x then g cannot be periodic — Nick Castillo, Mar 02 '20 at 12:31
@guy3141 That may be true, but I think it's beside the point that OP is trying to make. — Bladewood, Mar 02 '20 at 20:45

score 3 · Accepted Answer · answered Mar 02 '20 at 12:27

3

$g$ is not periodic as the difference between two consecutive roots is unbounded as we consider the roots going to $\infty$.

answered Mar 02 '20 at 12:27

mathcounterexamples.net

70,018

Could I say that $g$ is not periodic at $x$ but it is in $\ln$-scale? – Arnaldo Mar 02 '20 at 12:31
I don’t know what means to be periodic in $\ln$-scale... – mathcounterexamples.net Mar 02 '20 at 12:32
I mean, if I call $\ln x=u$ then $\sin(\ln x)=\sin(u), u\in \mathbb R$, then $\sin(u)$ is periodic. – Arnaldo Mar 02 '20 at 12:34
That is true. But then you’re not using the standard definition for being periodic. – mathcounterexamples.net Mar 02 '20 at 12:36
I think I am using the standard defition but in different scale of the domain. Isn't it? – Arnaldo Mar 02 '20 at 12:37
Which is not the standard definition... – mathcounterexamples.net Mar 02 '20 at 12:38

score 2 · Answer 2 · answered Mar 02 '20 at 12:31

2

Functions such as $\sin(\ln x)$ are called log-periodic rather than periodic.

answered Mar 02 '20 at 12:31

J.G.

115,835

It was just a example, I could write $g(x)=\sin (e^x)$ and the ask the same question. What do you think? – Arnaldo Mar 02 '20 at 12:36
@Arnaldo While I've encountered no discussion of such functions, it would make sense to call them something like exp-periodic. Again, though, it's not periodic, because no constant $p$ satisfies $\sin e^x=\sin e^{x+p}$ for all $x$. – J.G. Mar 02 '20 at 12:42

Periodic Function and change of scale

2 Answers2