# Pathology in Complex Analysis

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.

• 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
• Charles Chapman Pugh – Robert Israel Sep 12 '18 at 16:47
• @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
• 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
• @j.c. I think the point of the question is for OP to not specify that and leave it open to interpretation. – user69208 Sep 12 '18 at 18:28

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).

• 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-Gillen Sep 17 '18 at 22:29

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

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.

• 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

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.

• 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 9:13

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.

• Complex tori that aren't abelian varieties are examples of the pathology in your second paragraph, right? – Robert Furber Sep 13 '18 at 11:53
• Yeah, those are the example I know of. – James Hanson Sep 13 '18 at 14:02
• As I said in response to Ali Taghavi's answer: surely it makes more sense to view the one-variable theory as being an exception to what happens in general – Yemon Choi Sep 15 '18 at 12:45

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.

• 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 6:17
• "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

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.