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Complex analysis is the good twin and real analysis the evil one: beautiful formulas and elegant theorems seem to blossom spontaneously in the complex domain, while toil and pathology rule the reals. ~ Charles Pugh

People often like to talk about elegant "miracles" in Complex Analysis. However, what's are "pathological" objects/properties in Complex Analysis?

EDIT (09/13/18): Also posted as https://math.stackexchange.com/questions/2912320/most-pathological-object-in-complex-analysis

EDIT: Changed the wording of the question.

Qi Zhu
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    First, this does not seem to be on-topic here, since it is not of research level; better ask it on MSE, where you do have an account. Second, who is Charles Pugh, given that Google only finds 8 results about this name, 4 of which from the 18th and 19th centuries? – Alex M. Sep 12 '18 at 16:42
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    See this for the original. – Robert Israel Sep 12 '18 at 16:46
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    Charles Chapman Pugh – Robert Israel Sep 12 '18 at 16:47
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    @Alex M. I apologize, if MathOverflow is the wrong forum for this question. I've posted it on MSE before where it was suggested to me to post it on MathOverflow. I thought my question did seem similar to many of those "top voted topics". Charles Pugh is an author who has written a book about Real Analysis as Robert Israel has commented. Again, I'm very sorry, if this is the wrong forum for such a question. – Qi Zhu Sep 12 '18 at 17:03
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    @AlexM. To be honest, I have read far more well-received big-list questions on MO than on MSE. Questions like these can be found in great number in the all time highest votes lists. I do not qualify as someone who can judge whether this is on-topic, but I wonder whether we have double standards, or whether the acceptability of questions has changed over time. – M. Winter Sep 12 '18 at 17:07
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    This question is perfectly fine for MO, and why does it matter who Charles Pugh is? If his quote is a good way to frame the question, then why not use it? – arsmath Sep 12 '18 at 17:14
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    The obvious example IMO are multivalued functions and branch cuts. –  Sep 12 '18 at 17:55
  • Could you be a bit more specific: what precisely do you mean by "pathological"? – j.c. Sep 12 '18 at 18:07
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    @j.c. I think the point of the question is for OP to not specify that and leave it open to interpretation. –  Sep 12 '18 at 18:28
  • @j.c. Exactly what Zachary Selk said! I believe it's best to leave it open to interpretation. – Qi Zhu Sep 12 '18 at 21:08
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    Since you're asking specifically for "the most pathological object / property", I rather think clarification would be helpful for answerers. Or, you might change your question to ask for a list of examples. – j.c. Sep 12 '18 at 23:30
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    For my money, a function with essential singularities seems "pathological". Functions that are meromorphic on the full extended complex plane are basically algebraic objects with all sorts of wonderful rigidity. On the other hand, the big Picard theorem says that if a function has an essential singularity at $z$, the range of $f$ on any neighborhood of $z$ misses at most one point from $\mathbb{C}$! These beasts also lie behind holomorphic functions on e.g. the disk which can't be extended to bigger domains. – dorebell Sep 13 '18 at 02:09
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    I always see this subject exactly the other way around: complex analysis is the dystopian autocracy of the Laurent series, where nothing unexpected, interesting can ever hope to happen. – leftaroundabout Sep 14 '18 at 16:56

7 Answers7

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I would say that Mandelbrot set (like similar fractal objects coming from complex dynamics) can be seen as a "pathological" object, at least from the point of view of regularity (the boundary is nowhere differentiable, for instance).

Francesco Polizzi
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    However, compared to similar objects in real dynamics (or indeed in higher-dimensional complex parameter spaces), the Mandelbrot set is wonderfully well-behaved and well-organised. We even have a (conjectural) complete description of its topology, which would bring with it a complete classification of the different dynamics in the complex quadratic family. – Lasse Rempe Sep 17 '18 at 22:29
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I don't know how you want to define "pathological", but some of the corollaries of Runge's theorem give you functions with interesting properties. One of mine: there is a rational function $f$ such that for every holomorphic function $g$ on the open unit disk $\mathbb D$, $g$ or $g-f$ has a zero in $\mathbb D$. This is American Mathematical Monthly problem 6520, solution at www.jstor.org/stable/2323638

Robert Israel
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A natural boundary is probably rather pathological, in the same spirit as that continuous-everywhere-but-differentiable-nowhere and smooth-everywhere-but-analytic-nowhere functions are in real analysis. In particular, it consists of a function which has a property (analytic) that one might intuitively expect would lead to another (analytic continuation, to at least some extent), but doesn't.

A simple example is the series function:

$$f(z) = \sum_{n=0}^{\infty} z^{2^n}$$

This function is defined on $|z| < 1$, but the circle $|z| = 1$ is singular, and the series thus both converges on the maximal domain and forbids any extension beyond it. The latter curve is thus a natural boundary - an enclosing wall singularity that prevents any further extension of the function's domain to an area of nontrivial measure.

The_Sympathizer
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    In this case, I had to revise my intuition several times. At first analytic continuation was counter-intuitive because I expected analytic functions to behave like continuous ones (having partitions of unity). After that, natural boundaries were counter-intuitive because I expected analytic continuation to always be possible. Then when I found out about domains of holomorphy in several complex variables, that was counter-intuitive because you do get some analytic continuation for free as long as the complex dimension $> 1$. – Robert Furber Sep 13 '18 at 11:50
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In an old MO question of mine, I had wondered the following (I'm quoting my question):

Let D ⊂ ℂ be the closed unit disc in the complex plane, and let C be a continuously embedded path in D between the points -1 and 1. The curve C splits D into two halves $D_1$ and $D_2$.

Let f : D→ℂ be a continuous function that is holomorphic on the interiors of $D_1$ and $D_2$.
Is f then necessarily holomorphic?

The answer turns out to be no.

Community
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André Henriques
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  • For me this is more about the ability of Jordan curves to be surprisingly pathological, than about complex analysis per se. – Tim Seguine Sep 14 '18 at 09:13
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I'd say that multivariable complex analysis is more complicated relative to single variable complex analysis than multivariable real analysis is to single variable real analysis. There are new phenomena that could charitably be called 'rich' and uncharitably be called 'pathological.'

For instance, it's well known that there are no non-constant holomorphic functions on 1D compact complex manifolds, but there are always non-constant meromorphic functions. In higher dimensions there are compact complex manifolds without even any non-constant meromorphic functions.

Another thing is related to The_Sympathizer's answer: Any open set in $\mathbb{C}$ can be the 'domain of holomorphicity' of a holomorphic function, i.e. a domain which beyond which the function cannot be analytically extended. In higher dimensions this is no longer true and characterizing the open sets which are domains of holomorphicity becomes somewhat complicated.

James Hanson
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The rigidity of complex domains in higher dimension For example the unit ball in $\mathbb{C}^2$ is not holomorphic equivalent to the unit cube.

Ali Taghavi
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    How is this a "pathology"? Surely it makes more sense to view the one-variable theory as being an exception to what happens in general – Yemon Choi Sep 14 '18 at 23:55
  • @YemonChoi you are right this is a pathology in higher dimension in comparison to one dimension. On the other hand I called this situation a pathology since this situation (at the same dimension) can not occured in the real case. – Ali Taghavi Sep 15 '18 at 06:17
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    "Pathology" means something bad, or wild, or contradicting one's intuition (e.g. Peano's space-filling curve). I don't see why anyone would strongly expect two homeomorphic domains in ${\mathbb C}^n$ to be biholomorphic just because it works for $n=1$, and hence I still don't see this as pathological – Yemon Choi Sep 15 '18 at 12:46
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Another complex dynamics example:

Suppose $0 < \lambda < \frac{1}{e}$. The Julia set of $\lambda e^z$ can be divided into a set $E$ of "endpoints" and a collection of "hairs" connecting these endpoints to $\infty$. Mayer proved in 1990 that $E$ is totally separated, but $E \cup \{\infty\}$ is connected.

Micah
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    In fact, the Julia set in question is homeomorphic to the Lelek fan, a well-known topological object that has this property. See https://link.springer.com/article/10.1007/s40315-016-0169-8 and https://londmathsoc.onlinelibrary.wiley.com/doi/abs/10.1112/blms.12176 for some recent further results in these directions. – Lasse Rempe Sep 17 '18 at 22:31
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    Moreover - and even more surprisingly - the set of endpoints has full Hausdorff dimension (2), while the union of hairs only has Hausdorff dimension 1 (i.e., the same as the topological dimension). This seemingly "paradoxical" result was proved by Karpinska; see https://www.sciencedirect.com/science/article/pii/S0764444299803218 . It also generalises to much larger classes of entire functions. – Lasse Rempe Sep 17 '18 at 22:33