Linked Questions

107 votes
9 answers
36k views

solving $f(f(x))=g(x)$

This question is of course inspired by the question How to solve f(f(x))=cosx and Joel David Hamkins' answer, which somehow gives a formal trick for solving equations of the form $f(f(x))=g(x)$ on a ...
Kevin Buzzard's user avatar
74 votes
15 answers
18k views

$f(f(x))=\exp(x)-1$ and other functions "just in the middle" between linear and exponential

The question is about the function $f(x)$ so that $f(f(x))=\exp (x)-1$. The question is open ended and it was discussed quite recently in the comment thread in Aaronson's blog here http://...
Gil Kalai's user avatar
  • 24.7k
57 votes
8 answers
10k views

Does the formal power series solution to $f(f(x))= \sin( x) $ converge?

I have spent some time using gp-pari. There is, of course, a formal power series solution to $ f(f(x)) = \sin x.$ It is displayed below, identified by the symbol $g$ because I am not entirely sure ...
Will Jagy's user avatar
  • 25.7k
26 votes
5 answers
3k views

Do complex iterates of functions have any meaning?

Using a method explained in this answer to How to solve $f(f(x)) = \cos(x)$?, it is possible to calculate not only integer and real iterates of functions but also complex ones, for example, the $i$-th ...
Anixx's user avatar
  • 10.1k
13 votes
6 answers
4k views

Finding f such that f(f(x))=g(x) given g

Suppose $g(x)$ is a smooth increasing function defined for $x \ge 0$ such that $g(x) \ge x$ for all $x$. Does there exist a function $f$ with similar properties such that $f(f(x))=g(x)$ for all $x \ge ...
David Corwin's user avatar
  • 15.4k
32 votes
0 answers
2k views

$f\circ f=g$ revisited

This may be related to solving $f(f(x))=g(x)$. Let $C(\mathbb{R})$ be the linear space of all continuous functions from $\mathbb{R}$ to $\mathbb{R}$, and let $\mathcal{S}:=\{g\in C(\mathbb{R}) ; \...
Ady's user avatar
  • 4,060
3 votes
1 answer
496 views

Prove that these two definitions of "natural" integration constant coincide when both converge

These are two possible definitions of antiderivative (integral) incorporating a supposedly natural choice of an integration constant (see this question for further details). The first one is based on ...
Anixx's user avatar
  • 10.1k
2 votes
1 answer
763 views

What about the other $f$ such that $f(f(x)) = \sin(x)$?

This question is inspired by the big MO question here; and also inspired by the big MO question here. The premise of this question requires a backdrop on fractional iteration; so I'll start slow. ...
user avatar
0 votes
1 answer
228 views

Is it possible to numericaly solve functional equation

Given a functional equation of form $f(f(x))=T(x)$ is there any good ways to solve it numerically? If not then at least approximate in some small region $x\in(-a;a)$. E.g. with the equation $f(f(x))=x+...
Warmist's user avatar
0 votes
0 answers
190 views

Does the functional square root of the cosine admit a vector-based interpretation?

In linear algebra, the cosine of the angle between two vectors $a$ and $b$ is defined as $$\cos(a,b) = \frac{\langle a, b \rangle}{||a||\cdot||b||} .$$ The functional square root of the cosine has at ...
Max Lonysa Muller's user avatar