Essential Singularities and
The Casorati-Weierstrauss Theorem
It
seems as though I’ve been neglecting my primary academic focuses, physics and
math, on this website. This is not due
to a dulling interest in the subject, but rather the difficulty of putting
theoretical physics and math online, combined with the lack of interest in
these subjects. However, to remedy this
situation, I’ve decided to post information on some topics in complex analysis
that I’ve found particularly fascinating recently. I’ll try to remove most technical language from
here and the nuts-and-bolts mathematical work, in order to make it more
accessible to the random people who somehow or other find this site, but if
you’re more interested there’s plenty of other fascinating information out
there.
Background information on
singularities-
Let’s
take a function f(z) in the complex plane (the set of
numbers of the form a+bi). Now, imagine
taking the value of this function at z0, which means taking the limit as f(z) approaches z0- because you’re working in a plane, the
limit at a point has to be the same regardless of the direction you approach
from to exist. If this doesn’t happen,
you’ve got a singularity. I’m only
looking at points here (branch cuts may or may not be discussed on this blog
later), so you’ve got an isolated singularity if this fails to happen.
There
are three types of singularities- removable singularities, poles, and essential
singularities. Removable ones are the
ones that aren’t really singularities; in real analysis they typically appear
when something in the denominator cancels with something in the numerator of a
fraction. For these ones, you can simply
define a value for f(z0) equal to the limit at z0, and
then f is holomorphic (complex differentiable) at z0. Also, there is a formula, called Riemann’s
theorem on removable singularities, which basically says that if the absolute
value of a function is bounded on a disk around a point (excluded that specific
point), then the point is a removable
singularity. In mathematical
language: Suppose that f is holomorphic
in an open set Ω except possibly at a point z0 in Ω. If f is bounded on Ω-{z0}, then z0 is a
removable singularity.
For
this discussion, poles aren’t really important; all you really need to
understand about them is a singularity is a pole if the |f(z)| →∞ as z→z0 from any
direction. So now, what are essential
singularities? Really, really bad
singularities- but as you’re going to see (hopefully) they’re also really,
really cool. The short definition is that
any singularity that isn’t removable or a pole is an essential
singularity. A slightly better
definition is f(z0) has different values, some of which can be ∞,
depending on how z→z0. For example, think about
e1/z at z0=0- if you approach this on the real axis, it goes to infinity, but
if you approach it on the imaginary axis it oscillates rapidly.
The Casorati-Weierstrauss Theorem-
In
mathematical language- Suppose f is holomorphic in the punctured disc
Dr(z0)-{z0} and has an essential singularity at z0. Then, the image of Dr(z0)-{z0} under f is
dense in the complex plane
If
that didn’t make much sense to you, it’s ok.
This is a rather technical formulation- a slightly easier version to understand
is that if f(z) has an essential singularity at z0, and you draw a circle of
ANY radius, and pick any complex number α, it’s possible to find a series of points z1,z2,z3… that are
inside your circle such that the limit of f(z1),f(z2),f(z3)…is α, (f(zn)→ α as n→∞), including α =∞. Wow- for every
complex number, there’s a series whose function converges to it, in absolutely
any circle around an essential singularity- no matter how small! Now that’s incredible. You’re probably thinking that can’t possible
true- what if you draw a really, really tiny circle- surely there can’t be
series that converge to every complex number in there- but there are. But if you’re taking this on faith, you’re
not a very good mathematician- and so here’s a proof of this incredibly
remarkable result.
Proof-
Note: This is a very informal proof and is intended to only give a
general idea of the formal approach one would take. If I wrote up the entire formal proof this
would be way too long and boring for most people, and the people who would be
interested have probably already seen it.
If you’d like a more formal proof, I’d recommend looking in any complex
analysis book or bothering a local math professor.
Suppose
the Casorati-Weierstrauss Theorem is not true.
Then there exists an α in the
complex plane and δ and ε in the real numbers greater than
zero such that there does not exist a point z such that |z-z0|< δ and |f(z)- α |< ε.
Now
define a function g(z)=1/(f(z)- α) and consider it in the disk
|z-z0|< δ . We know that in this disk, |f(z)-
α |≥ ε. This implies that g(z)
is holomorphic in 0<|z-z0|< δ (because the denominator is never zero). Also, we know that |g(z)|≤1/ ε.
Therefore, g(z) is bounded and holomorphic, and
by Riemann’s theorem on removable singularities, f(z0) is a removable
singularity. Therefore we can define a
value of g(z0).
Now
if we solve for f(z), we find f(z)= α +1/g(z). At z=z0, f(z0)
either does not have a singularity (if g(z0)≠0) or a pole (if
g(z0)=0). But this contradicts the
assumption that z0 is an essential singularity.
Thus the Casorati-Weierstrauss Theorem has been proven by contradiction.
So again, this just reinforces how
incredibly nonintuitive math can be occasionally- and the fascinating results
of complex analysis!