In an old mathematics book (Ritt, 1948, p.5) I have come across the notion of "monogenic analytic" and "fluent" functions. These are complex valued functions. Has anyone heard of these terms before? I'm thinking it's related to analytic continuation.
According to the material, a monogenic analytic function (m.a.f) is one which has been analytically continued as far as possible, and an "element" of a m.a.f is a power series of the function about some point $X_0$ representing the function in the appropriate domain about $X_0$.
The definition of a fluent function goes like this:
Let $f(X)$ be a monogenic analytic function and let $P(X-X_0)$ be an element of $f$. Then $f$ is said to be fluent if, for all $\epsilon>0$, every curve $\gamma(t)$ with $\gamma(0)=X_0$, there exists a curve $\gamma_1(t)$ with $\gamma_1(0)=X_0$ such that $|\gamma(t) - \gamma_1(t)|<\epsilon$, for all $0\leq t\leq 1$, and such that $P(X-X_0)$ can be continued along the entire curve $\gamma_1(t)$.
This all sounds very much like the book is decsribing analytic continuation, but I am worried the terminology used in the book is out-dated, and there is a more modern approach. I'm trying to get to grips with the material.