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Often a specific technical term is coined in mathematics because a concept is so often repeated. The usual modern presentation method of Definition- Theorem - Proof often starts with the discovery method of Object - Property - Pattern, just finding and naming things is the first step.

In another thread, I came across the concept of 'holomorphic'. An answer to How to express in closed form? led to my asking the question Complex conjugate of z without knowing z=x+iy.

I had never heard the term 'holomorphic' before, but wikipedia helped. Looking at complex analysis texts online, I couldn't find anything in their indexes. Lots of papers use it, and there are lots of questions here on math.SE, so it must be part of everyday (higher) math. The definition includes 'analytic' which (I think ) means you can take an infinite sequence of derivatives (which means you can create a Taylor series for it)). But then I don't see what's special about that (I'm sure that's obvious but I just don't know; so you can compute values quickly?) and that means something even -more- for complex functions.

So the question here is...why is 'holomorphic' such a big deal? And is it in fact? If a complex function is holomorphic you can do ..what with it? if it is not holomorphic, is there anything else you can do with it? What specific areas of mathematics deal with it? is it simply basic complex analysis or part of some particular branch? (I'm actually not even sure that it is particular to complex analysis)

(My background is not analysis at all (cs, combinatorics, logic) so I know basic real analysis but not complex at all.)

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Holomorphic and (complex) analytic mean the same thing. This is a big question and I think to answer it you should probably start by reading a good complex analysis textbook, e.g. Needham's Visual Complex Analysis: usf.usfca.edu/vca/ –  Qiaochu Yuan May 2 '11 at 20:10
    
@Qioachu: yes, that text is in my 'shopping cart'. So I should consider those synonymous in the complex context? Is that why I can't find it in the index of that book? –  Mitch May 2 '11 at 20:15
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Essentially you can replace any instance of "holomorphic" by "complex differentiable." A function is holomorphic at a point if it's complex differentiable in an open set about that point. Usually, we're dealing with functions differentiable in an open set (exceptions to this are rather uninteresting, or at least not that useful), so we interchange the two freely. "Analytic" at a point means that the function admits a power series representation there (not just infinitely differentiable: $e^{-1/x^2}$ at $x=0$). It turns out that being analytic is literally equivalent to being holomorphic. –  Riley E May 2 '11 at 20:16
    
(continued from above) Actually, the example I gave above doesn't quite work in this case. On the real line, it's infinitely differentiable at the origin, but it's not differentiable there in the complex plane. –  Riley E May 2 '11 at 20:20
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@Qiaochu: I wouldn't say they mean the same thing. In my view they are a priori different concepts: representable as power series on the one hand, and the derivative in the sense of a limit of difference quotients on the other. It is then a theorem that (in complex analysis) they turn out to be equivalent. (edit: I now see that Riley E has made similar remarks, I should have refreshed this page.) –  wildildildlife May 2 '11 at 20:25

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up vote 15 down vote accepted

Briefly, holomorphic functions are immensely useful because, on the one hand, they are surprisingly common (since any power series, for example, whose coefficients grow reasonably slowly defines a holomorphic function), and on the other hand one can prove very strong theorems about them. There is a web of results including Cauchy's integral formula and the identity theorem which assert that holomorphic functions are astonishingly rigid: given information about a holomorphic function in a very small part of its domain, one can extract information about the function's behavior in other a priori unrelated parts of their domain (and this is what allows things like contour integration to work).

For that reason, holomorphic functions are a powerful tool to apply to a problem when they do apply. For example, analytic number theorists frequently construct holomorphic or meromorphic functions which carry number-theoretic information, such as the Riemann zeta function, to prove theorems like the prime number theorem. Since you say you have a background in combinatorics, you might enjoy reading Flajolet and Sedgewick's Analytic Combinatorics, a thorough exposition of (among other things) ways to use complex analysis to provide asymptotics for combinatorial sequences.

Here is a simple example. Let $f_n$ denote the number of ways that $n$ horses can win a race, with ties. It turns out that this sequence has generating function

$$F(z) = \sum_{n \ge 0} \frac{f_n}{n!} z^n = \frac{1}{2 - e^z}.$$

This function is meromorphic with poles at $z = \log 2 + 2 \pi i k, k \in \mathbb{Z}$, each of which has residue $-\frac{1}{2}$. In fact, it turns out that $F(z)$ admits an infinite partial fraction decomposition

$$F(z) = \sum_{k \in \mathbb{Z}} \frac{1}{2(\log 2 + 2 \pi i k - z)}.$$

And by expanding the terms on the RHS in a geometric series, this gives an asymptotic expansion for $\frac{f_n}{n!}$ with leading term $\frac{1}{2 (\log 2)^{n+1}}$. In other words,

$$f_n \sim \frac{n!}{2 (\log 2)^{n+1}}.$$

The pole at $z = \log 2$ dominates the asymptotic expansion: the leading term in the error of the above expression is given by the other poles nearest the origin, which occur at $z = \log 2 \pm 2 \pi i$. Because these poles have nonzero imaginary part, if you plot the error in the above approximation you'll find that it oscillates. It is not so easy to explain why this should be the case without complex analysis.

A famous example is Hardy and Ramanujan's asymptotic formula for the partition function

$$p(n) \sim \frac{1}{4n \sqrt{3}} e^{ \pi \sqrt{ \frac{2n}{3} } }$$

which is proven using a much more sophisticated version of the above argument.

But really, there is too much to say about holomorphic functions, so again I suggest that you read a textbook. Besides Needham's book, I also personally enjoyed Stein and Shakarchi, which is very user-friendly and has good applications.

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+1: Cool example! –  wildildildlife May 2 '11 at 20:35
    
Nice combinatorial examples...but (and this is the nature of my ignorance) I don't see where the concept of holomorphic is used, especially since I don't even see that particular string in you answer. Is meromorphic a particular kind of holomorphic? I have a passing acquaintance with Flajolet/Sedgewick but the concept (or at least word) holomorphic never arose there. It all seemed to be about poles (and presumably a meromorphic function can have poles (well...by definition) without being holomophic). –  Mitch May 2 '11 at 21:42
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@Mitch: a holomorphic function is one which is complex-differentiable everywhere, whereas a meromorphic function is one which is complex-differentiable everywhere except at some isolated points. So "meromorphic" is a generalization of "holomorphic." The notion of complex-differentiability is still central here, but just on a different domain ($\mathbb{C}$ minus some collection of points). Again, you really should read a textbook if you want to learn this stuff properly. –  Qiaochu Yuan May 2 '11 at 21:48

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