# Questions tagged [analysis-and-odes]

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5 questions
1answer
130 views

### A first order ODE problem

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 $F'(x)<0$ ...
0answers
88 views

### Is $\lim_{z\to0} \exp_{\sqrt{2}}^{\circ z}(\xi)$ continuous?

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 ...
2answers
614 views

### The continuous Taylor series; are they just Taylor series?

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 ...
0answers
93 views

### Solving the difference equation in exotic scenarios

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 ...
1answer
113 views

### Entire composite square roots of functions of finite order

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 ...