Definition talk:Analytic Function

This is another one of those pages, like Abel's Theorem, which would benefit enormously from a complex perspective. It's something we're still working up to, I know, but there's no time like now to figure out exactly how we'd like to do this.

What I mean by this is that I have two equivalent definitions in two separate textbooks, one of which is rather formal, and comes from Hille's Analytic Function Theory: Volume 1. I produce it here:

Let $f(z)$ be a single-valued continuous function in a domain $D$. We say that $f(z)$ is complex-differentiable at a point $z_0 \in D$ if

$ \lim_{h \to 0} h^{-1}(f(z_o+h) - f(z_0))$

exists as a finite number and is independent of how the complex increment $h$ tends to $0$. The limit, when it exists, will be denoted $f'(z_0)$ and called the derivative of $f$ at $z_0$.

The function $f(z) \ $ is differentiable in $D$ if it is differentiable at every point in $D$. Such a function is called analytic.

I'm going to add this to the page Definition:Differentiable in just a moment. However, I earlier mentioned I had another, more intuitive understanding of an analytic function, which comes from T. Needham's Visual Complex Analysis, and requires the definition of a term he himself coined, amplitwist. I wouldn't mention it, except that this text won awards for being so clear and easy to understand. Basically, his definition, which is equivalent to that above, is:

Analytic mappings are precisely those whose local effect is an amplitwist: all the infinitesimal complex numbers emanating from a single point are amplified and twisted the same amount. This can be made precise with a few terms he coined himself, such as amplitwist, and give an excellent and much more useful, I've found, understanding of what an analytic function is.

My question is, do we bother with this extra, rather atypical of the general literature, definition, or maintain only the kind of definition given above?

Until this is resolved, I'm including ONLY the material from the Hille text, copied directly from Definition:Differentiable Zelmerszoetrop 02:03, 27 January 2009 (UTC)

I tend to think that we are better off focusing on those definitions we need for proofs. So if there is a proof that involves the other definition or is clarified by the other definition, I would include it. If not, I would skip it. --Cynic (talk) 04:45, 27 January 2009 (UTC)

When I've got up to complex analysis, the plan is to add the Cauchy-Riemann (I think that's the guys) equations and stuff, and connect the defn for analytic in to those. Thing is, there are so many directions to come at this subject that there is no "right" one. As long as we join them all together with equivalence proofs we're covered. --Matt Westwood 06:25, 27 January 2009 (UTC)

Rather than duplicating material, I've separated out the work that specifies the definition of "differentiable" from that defining "Derivative" and "Analytic". I have a plan! --Matt Westwood 07:55, 27 January 2009 (UTC)

I've been doing some remembering of my own indoctrination in this subject some few years ago. I remember a construction that was exactly your "amplitwist", and it was shown (IIRC) to be a direct consequence of analyticness and vice versa. I'll have to dig it out, but I'm having to take these things in small steps at the moment, I'm tired. --Matt Westwood 22:04, 13 February 2009 (UTC)

Analytic vs. holomorphic

It is fairly standard terminology these days to say that an analytic function is one that can be expressed by a power series, while a holomorphic function is one that is complex differentiable (on an open set). The two definitions are equivalent, but this is not trivial - it requires the development of Cauchy's integral theorem.

I would suggest defining analytic and holomorphic separately, with a note of course that these two definitions coincide, with a reference to Analytic Functions are Holomorphic, an article that is yet to be created.

As an aside, I believe Bers used to refer to the equivalence of these definitions, and some others, as the Fundamental Theorem of Complex Analysis, as in the recent book 'Complex Analysis in the spirit of Lipman Bers', by Kra et al. lasserempe 09:13, 16 February 2009 (UTC)

All sounds sensible to me. Go for it. --Matt Westwood 19:05, 16 February 2009 (UTC)