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In contrast to other topics in analysis such as functional analysis with its vast amount of counterexamples to intuitively correct looking statements (see here for an example), everything in complex analysis seems to be very well-behaved (for example holomorphic functions are always analytic). But is this maxim always right? Do you know any holomorphic functions which behave in a way one wouldn't expect at first sight?

EDIT: As you can see in the answers, I came up with something myself. But I would be glad if you knew more examples where strange stuff happens.

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I'm trying to decide whether one would expect the zeta-function to have zeros off the half-line, or whether one would expect this not to happen. – Gerry Myerson Dec 9 '12 at 11:50
@GerryMyerson: Actually, I thought more of something like the famous $e^(-x^{-2})$ of real analysis. Are there any caveats regarding the analytic continuation of holomorphic functions or issues with the radius of convergence, which can also arise in complex analysis? – Dominik Dec 9 '12 at 11:59
I find the result (by Carleman I think) that any continuous function on $\mathbb{R}$ can be uniformly approximated by entire functions rather counter intuitive, given all the restrictive theorems about entire functions. – WimC Dec 9 '12 at 12:03
There are no counterexamples in complex analysis, the functions are too nice! – user51427 Dec 9 '12 at 12:20
Does anyone expect at first sight that near an essential singularity, an analytic function will take on all complex values (except perhaps 1) infinitely many times? – Katie Dobbs Dec 9 '12 at 13:41
up vote 7 down vote accepted

Voronin's Universality Theorem about the Riemann Zeta Function really surprised me the first time I heard about it.

This theorem essentially says that the Zeta Function in some sense encodes all the possible behaviors of holomorphic functions and also all the possible behaviors of curves.

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What the hell! This is absolutely impressive. – YoTengoUnLCD Feb 28 at 3:38

Now I recall this function which struck me as particularly weird.

Consider $$f(z)=\sum_{k=0}^\infty z^{n!}.$$

This power series converges for $|z|<1$, but for every $z=e^{i2\pi q}$ with $q \in \Bbb Q$ we see that there is a $k$ with $z^n=1$ for each $k \ge n$, so that $$\lim\limits_{r \rightarrow 1}f(re^{i2\pi q})=\lim\limits_{r \rightarrow 1}(const+\sum_{k=0}^\infty r^{n!})=+\infty$$

Thus, the set of singularities is dense on the unit circle and $f$ cannot be extend beyond the unit circle.

This came to my mind when I saw the prime zeta function, which cannot be extended beyond $Re(z)=0$ because it has pole at each $\rho/n$, where $\rho$ is a nontrivial Zeta function zero and $n$ is a positive integer.

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Other examples of 'Lacunary functions'. – Raymond Manzoni Dec 9 '12 at 19:09

From another math.SE question, a function $f(z) = \sum a_n z^n$ which is convergent on the interior of the unit disc and extends continuously to the entire closed disc, but doesn't converge at all the points on the boundary.

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I used to think that the functions in complex analysis were in general too "nice" in the sense of rigidly controlled to leave much room for counter-intuitive situations, until I started reading Pommerenke's book on Boundary Behaviour of Conformal Maps. As he states in the preface:

"Then the conformal map has many unexpected properties, for instance almost all the boundary is mapped onto almost nothing and vice versa."

Admittedly, he is talking about situations where the domain is not necessarily bounded by a piecewise smooth curve, and considers meromorphic and not only holomorphic functions, but the extend to which the boundary behaviour of such "nice" functions can be badly behaved came as a bit of a shock to me at the time.

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Interesting. Could you also provide an explicit example which illustrates this bad behaviour? – Dominik Dec 9 '12 at 13:02
Not off-hand. I suspect that he must give an example somewhere, but a quick scan of the book doesn't reveal one easily. – Old John Dec 9 '12 at 13:05

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