1

I don't have a lot of ideas or a bold direction, I tried to generate such function in Desmos and it kind of seems to exist but I don't know the formula or even if it is elementary or not. I do know that:

$f(0)=0, f(x)=f(x+\pi)$

I also know that if $f(a)=b$

It means that $f(f(a))=f(b)\rightarrow f(b)=\sin(a)$

And if we apply this formula to itself a couple of times we get cool stuff:

$f(f(b))=f(\sin(a))\rightarrow f(\sin(a))=\sin(b)$

Apply this a general N times and get that $f$ of sine composed over itself repeatedly N times of (a) equals to sine composed over itself repeatedly N times of (b). A friend suggested I use the Weierstrass factorization theorem to prove no such function exists or no elementary function as this exists. But im not sure how it'd help, and he didn't solve the question either just brought up the idea. If anyone has the formula or the non-elementarity proof please answer

ELEMENTARY AND DISCRETE FUNCTIONS ONLY.

MJD
  • 65,394
  • 39
  • 298
  • 580
עמית חי לרמן
  • 83
  • 4
  • Please clarify your specific problem or provide additional details to highlight exactly what you need. As it's currently written, it's hard to tell exactly what you're asking. – CrSb0001 May 18 '23 at 16:59
  • 3
    See this thread: https://math.stackexchange.com/questions/1916184/find-fx-for-ffx-sin-x In the future, please consider searching for your question before posting a new one. – Charlie May 18 '23 at 17:02
  • 2
    Does this answer your question? Find $f(x)$ for $f(f(x))=\sin x$ - this was specified in Charlie's comment. – John Omielan May 18 '23 at 17:05
  • I specifically asked it to be an elementary function, so the suggested function in @Charlie 's link doesn't hold. But it's still cool and definitely a solution if I hadn't required the elementarity. Anyways thank you bout that but if someone has an elementary one it'd be better. – עמית חי לרמן May 18 '23 at 20:27

0 Answers0