1
Because I can't find a function $h:\mathbb R\mapsto\mathbb R$ with the property
$$h^{\circ 2}(x)=x^2+1$$
I'm looking for a function that almost has that property - that is, I would like to find a closed-form (and preferably elementary) function $h:\mathbb R\mapsto\mathbb R$ satisfying
$$\lim_{x\t...
1
Let $s:\mathbb R\to\mathbb R$ be the function defined as
$$s(x)=x+(-1)^{\lfloor x\rfloor}$$
Find a function $t:\mathbb R\to \mathbb R$ such that $t^{\circ 2}=s$, or prove that no such function exists.
I was fairly sure that I could construct such a function $t$ by taking multiple cases a...
5
I've been doing a lot of research about functional half-iteration, and I posed the following question to myself:
Consider the function $q:\mathbb R\mapsto\mathbb R$ defined as
$$q(x)=x^2+1$$
Does $q^{\circ 1/2}$ exist? Does a continuous $q^{\circ 1/2}$ exist? What about a differentiable $...
Find the general $f^n(x)$ where
$f^1(x)=e^x$
$f^{a}(f^{b}x)=f^{(a+b)}(x)$
where $n,x\in\mathbb R$
I'm fairly confident that no two $f^n(x)$ with different values of $n$ will intersect, since then we could use the argument that if $f^n(x)=x$ at some point $A$, then $f^m(x)=x$ a...
I was scrolling through MSE on the cell phone earlier today and these popped up:
and the list went on. Is this an artifact of the search engine, or is there some reason for the sudden "interest" in it?
It should be important to notice that most of those questions, as far as I recall, are...
