# “fluent” functions

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.

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I also retagged based on guessing that you are referring to Joseph Ritt's 1948 opus Integration in Finite Terms. –  Willie Wong May 31 '12 at 10:49

A fluent is the terminology Newton used for the "thing you take a derivative of". Assuming the fundamental theorem of calculus holds, you have the following relations

fluent::fluxion = function::derivative = antiderivative::function

Ah, with the context it seems that the above isn't exactly correct. From my understanding the concept of a fluent function (unfortunately the definition of in Chapter 1 of Ritt's book isn't very precise about what the domains of various functions are) is essentially an analytic function defined on some open domain $\Omega\subseteq \mathbb{C}$ such that by analytic continuation it can be extended to a possibly multivalued function on almost all of $\mathbb{C}$. Meromorphic functions and functions with branch cuts are obviously in the class of fluent functions.

Perhaps illustrative would be a class of functions which is not fluent: the class of lacunary functions is one such.

It is a terminology that has not been used much since: a search on MathReviews for the combination of "fluent" and "function" returned zero results. And skimming through the book it appears that the main use of the idea of "fluent" functions is to put Liouville's work on firmer and more concrete ground. For the purpose of actually developing the general framework of differential algebra it does not seem to be necessary.

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and what is '::' ? –  Yrogirg May 31 '12 at 11:45
some undefined symbol signifying a relation. –  Willie Wong May 31 '12 at 12:05
What about "monogenic analytic" functions? Is the term still in use? I have heard of Monodromy and the Monodromy theorem, which may be related, but not sure. –  pbs May 31 '12 at 13:50
@pbs: sort of, but the meaning has evolved and is now applied to also other situations since the time of Weierstrass. For the purpose of the book a "monogenic analytic function" is a maximally extended (via analytic continuation) analytic function. Monogenic still carries the meaning of "complex differentiable", while "analytic" here refers to the analytic continuation. Best I can tell, it is still used in the fields of analytic function theory, as well as the study of hypercomplex functions. –  Willie Wong May 31 '12 at 14:01