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I am currently conducting independent research on iterated functions and have successfully solved equations of the form $f^a + f^b + ... + f^n = c$, but I am struggling with the specific equation $$f^2(x)=1+x^2$$, where $f^2(x)$ is equivalent to $f(f(x))$ I am interested in learning how to solve this equation using well-known methods, and would appreciate any guidance or support.

For the case $f^1 + f^2 + f^3 + c = 0$, I proposed a function $f = A+Bx$, where $A$ and $B$ are constants. Substituting this function into the equation, I arrived at:

$$(A+Bx) + (A+AB+B^2x)+(A+AB+AB^2+B^3)x + c=0$$ $$A(3+2B+B^2+(B+B^2+B^3)x)+C=0$$ Since $A$ is a constant and we want to solve for $A$, we can set the coefficient of $x$ equal to zero: $$(B+B^2+B^3)x=0$$ This gives us the solutions: $$B=0, \frac{-1\pm \sqrt{3} i}{2}$$ Then $$A = \frac{c}{2 \left(\frac{-1 \pm \sqrt 3 i }{2}\right) + \left( \frac{-1 \pm \sqrt 3 i }{2} \right)^2 }$$

And with $B=0$ then $f(x) = -\frac{c}{3}$.

I would also like to learn more about the properties and applications of iterated functions, and would be grateful for any resources or references on this topic.

Please note that this is not a school assignment and there is no urgency in receiving a response. I am conducting this research on my own and do not have access to textbooks or resources on this topic, hence my need to seek help on this platform.

Thank you in advance for your assistance.

Josué Adalid Juárez Botello
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  • Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer. – Community Mar 30 '23 at 04:34
  • Welcome to MSE. Here's how to ask a good question. Follow these guidelines to get help in this forum. It's particularly important that you share your own work and thoughts on the problem to show that you have made a serious effort by yourself before asking for help, and you're not just trying to get others to solve it for you. This is not a homework service. – jjagmath Mar 30 '23 at 04:35
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    I used this wonderful site and found a link to the same question, it have been already answered here in Math Stack Exchange. – jjagmath Mar 30 '23 at 05:25

1 Answers1

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I believe the best reference to study this kind of equations is the book

Iterative Functional Equations, by Marek Kuczma, Bogdan Choczewski and Roman Ger.

About the equation you claim to have solved, notice that solving for an equation usually means to find all solutions, but you only found some solutions, assuming $f$ have a very particular form, so it could be way more solutions than the ones you found.

For the specific case of $f(f(x))=x^2+1$ look at Hagen von Eitzen answer in this question of MSE:

if $\ f(f(x))= x^2 + 1$ , then $\ f(6)= $?

jjagmath
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