A composite square root of a function $g$ is a function $f$ such that $f(f(z)) = g(z)$. Not surprisingly, for arbitrary $g$ a function like this is hard to find. Specifically I am looking at functions $g$ such that $g$ has finite nonzero order, so that
$$0 < \limsup_{r\to\infty} \dfrac{\log\log ...
The difference equation, as referenced in the title, is a very specific object I'm referring to. If you have a holomorphic function $\phi$ on a domain $G$, then a solution $F$ to the difference equation of $\phi$ is a function holomorphic on $G'$ (for some domain $G'$) such that
$$F(z+1) - F(z) ...
I first posed this question when I was a first year student. I came up with some ad hoc arguments as to why the result is true (a bit of numerical experimentation), but never had a proof. I forgot about it until the other day, and thought it was still an interesting question. First, some motivati...
This question has arisen in a bunch of my research, to the side of my research actually, I keep on getting curious about how it should be answered. I'll frame it in an anachronistic sense, but the question is more general. I'm a firm believer that most general problems are solved by solving speci...
Fix $a>0$ and $b>0$. Does the following ODE
\begin{equation}
G(x)^2+2axG(x)G'(x)+2aG'(x)(x-b)=0 \tag{*}
\end{equation}
have a solution, say, $F(x)$, that satisfies $F(x)>0$ and $F'(x)<0$ on $(b,\infty)$?
I tried to solve it by Mathematica, and it gives $G$ as solution to
\begin{equation} x=e^{-...
In connection with this - should there be a separate tag for ODEs. (To be able to distinguish them from other topics in ca.classical-analysis-and-odes.)
Fix $a>0$ and $b>0$. Does the following ODE
\begin{equation}
G(x)^2+2axG(x)G'(x)+2aG'(x)(x-b)=0 \tag{*}
\end{equation}
have a solution, say, $F(x)$, that satisfies $F(x)>0$ and $F'(x)<0$ on $(b,\infty)$?
I tried to solve it by Mathematica, and it gives $G$ as solution to
\begin{equation} x=e^{-...
