This is not as clear cut a question as the earlier ones, and if you do not know an answer then it will be difficult to figure one out just based on intuition. (But perhaps possible).
If you are intrigued by the question and would like to explore what an answer could be, I would be interested to know how you tried to find an answer. Asked a colleague? Looked at a book (which?)? Looked online (where?)?
Here is the question:
A differentiable complex function automatically has derivatives of every order. (In contrast to differentiable real functions that need not have even second derivatives at any point.)
Can you describe this “miracle” as part of a more general phenomenon?
Combinatorics and more 
(spoiler alert)
My guess is that the answer you are looking for is – elliptic regularity. Anyway, that’s how I have always thought about it. As for how I think about elliptic regularity – I guess it is a reasonable intuition that convolving with a bump function will make anything smooth, and harmonic functions (more generally, solutions of elliptic PDE) are invariant under convolving with heat kernels (and suitable generalizations). But is this a test of your readers’ intuition, or a test of whether they’ve followed up the blog discussions linked to in your recent “Gina” posts?
Dear Danny
wow. do you mind I put your comment on hold to let other readers a chance? (just for 1 day)
In my complex analysis class we used analytic and holomorphic interchangeably to describe the kind of function you mention. It appears, on reading Wikipedia, that analytic is more general. According to my searches it means a function with a local Taylor series at each point. Such functions are always infinitely differentiable, whether they be complex analytic (that is, holomorphic), real analytic, or analytic in some other space. Complex analytic functions on a region do, however, have many special properties, such as being defined completely by the Taylor series at a single point, and obeying Cauchie’s integral formula. Is this generalized definition of analyticity close to what you were looking for?
I haven’t done functionanalysis in so long time – but my instant intuitive feeling is that complex space has phase and perhaps this phase is preserved upon derivation it is infinitely recursive – and real space simply is too simple to allow such a phenomena. A hunch and probably oh so wrong.
Dear Gil – please be my guest! (I did add “spoiler alert” and some space, but wordpress ate the space . . .) 🙂
The standard proof is by Cauchy’s integral formula, correct? Morally, this is an example of Fourier analysis on the circle. In terms of a general problem-solving technique, I’d call it “bootstrapping.” Then again, complex analysis isn’t my specialty.
We can (somewhat vaguely at first) conceive a strategy of associating a geometric object to every smooth complex function, and then mapping this set of geometric objects onto itself, such that each successive geometric map corresponds to taking a derivative of the prior function.
Hmmmm …. what geometric objects can we construct from complex-differentiable functions, that we cannot (in general) construct from ordinary differentiable functions?
One answer is that a complex-differentiable function specifies *two* (orthogonal) integral curves that pass through every point; these are the conformal meshes that are widely used in engineering).
We will take this two-dimensional orthogonal “mesh” (as contrasted with a one-dimensional “flow”) to be what geometrically distinguishes complex functions from real function.
This reduces the problem to purely geometric terms: “given the conformal mesh of a holomorphic function, show that there are no geometric obstructions to constructing the conformal mesh of the derivative of that function”.
For me it is most natural to conceive this construction as a paper-and-pencil (Pythagorean) construction … but I will keep Gil’s question in mind as I learn more about differential forms. Thanks!
I didn’t find a good answer. But I wonder if I am really wrong or just didn’t try hard enough. Therefore I write down some intuitions, in a quite fuzzy way.
1. (model theory) At first my idea was that maybe the useful difference is that C is a closed field, unlike R so maybe some model theory would have said that extending the language with a differentiation would be possible. But then, faced with the problem of connecting this model with “reality” I was stuck
2. (analysis) If one thinks of a holomorphic function as a (higher dimensional) real one, the condition of being holomorphic becomes “the differenital is conformal”, which is an ellipticity condition… but I guess this is answer too, is not good enough (more than explaining the miracle it reformulates it)
3. (geometry) Any primitive of the closed form dz/z on C*=C\{0}, gives an universal covering of C* by C. The exact covering diagram saying this, namely
0–>Z–>C–>C*–>0,
can be also identified with the one describing the exponential
exp:(C,+)–>(C*,x),
thus the integral \int_a f dz/z over nice closed curves a containing zero represents both the action of f on the homology near zero and the power n such that f(z)=z^n near zero.
Thus there is a topological control over the infinitesimal behavior of f, allowin bootstrapping (and infinite differentiability)
[maybe this can be adapted to some good gauge theory?]
What I had in mind was indeed solutions of elliptic PDE’s. As Danny said, I did not know about it and learned about it from John Baez over n-category cafe. Usually my best strategy for figuring out the answer to such a question is asking a colleauge.
A shared theme in this topic is that complex-differentiable functions are geometric objects in a sense that real-differentiable functions are not. This interplay between algebra, analysis, and geometry is why (as Shing-Tung Yao has said) “The most interesting geometric structure is the Kälher structure.”
Nowadays, the engineering community is slowly coming to the same appreciation. The role of geometry in classical engineering state-spaces is comprehensively set forth in V.I. Arnol’d’s Mathematical Methods of Classical Mechanics … it is becoming clear that a great challenge for engineers in the twenty-first century is to evolve an Arnol’d-style understanding of the quantum (Kälherian) state-spaces that are the operating basis for many modern technologies.
The mathematical arena for this understanding is at the intersection of geometry, algebra, information theory, quantum theory, and combinatorics … which is a fun place to work and learn.
In seeking to understand why Arnol’d’s Classical Mechanics is not widely taught (in engineering, anyway) … despite being viewed by many people as being among the finest texts on mechanics ever written … I found this account by Sergei Novikov of why Arnol’d’s geometric way of thinking did not become more popular:
——
Kolmogorov also asked [Arnol’d] to learn mechanics. Thus Arnol’d read a lot of books, starting from Appell and some Russian books
written by people in classical mechanics; however, as Arnol’d said, he could not understand what mechanics really was. Then he found the book of Landau and Lifshits (which was not yet famous at that time among the mathematical community). He told me that, after reading this book, he finally understood what mechanics was, and, after that, he understood how bad the book was.
Arnol’d himself wrote a brilliant book on mathematical understanding of classical mechanics. I would honestly say that I do not like that book, because he completely reconstructed the ideology. The book of Landau and his school was just a starting point to develop a great science; it contained many initial points allowing further progress. In Arnol’d’s reconstruction, the mathematics is, of course, much better –it is a very good book for pure mathematicians—but starting points for future research areas are missing. People who read Arnol’d’s book arrive at an endpoint.
For me, that is what many of the best mathematical writings are all about (like Gil’s question that began this post) … they suggest starting points.
Here is a related discussion over math overflow: http://mathoverflow.net/questions/3819/why-do-functions-in-complex-analysis-behave-so-well-as-opposed-to-functions-in