# Why do we categorize all other (iso.) singularities as “essential”?

This is something I've been wondering since my complex analysis course, but it never bothered me enough to ask; however, now would be better than never.

When dealing with isolated singularities, we classify each of these points as removable, pole (of order $k$), or essential. It easy to see that all isolated singularities must be of one of these three categories by construction:

We define any isolated singularity that isn't removable or pole as an essential singularity.

Why is it that we throw all other singularities into this category? Do we not care about essential singularities to classify them further? That is, are removable singularities and poles (of order $k$) the only isolated singularities we care about?

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An important point is that a "pole" is actually the same thing as a removable singularity, if we think of our function as a map that takes values on the Riemann sphere (which is the complex plane with a point at $\infty$ added; the complex structure near $\infty$ comes from the map $z\mapsto 1/z$).

So a function that has a removable singularity or a pole at $z_0$ doesn't have a "real" singularity there at all; rather, we can extend the function to an analytic or meromorphic function in $z_0$. If we cannot extend the function in this way, the singularity is indeed "essential"; i.e., we cannot get rid of it. Thus the terminology is not one that is merely used for convenience or pedagogical purposes; rather, it is extremely natural.

As has already been mentioned, the magic of complex numbers results in many beautiful facts about essential singularities: functions with these singularities are very far from extending continuously.

The simplest of these facts is the Casorati-Weierstraß Theorem: The image of a neighborhood of an essential singularity is dense in the complex plane.

This is just a consequence of the removable singularities theorem. (If $f$ omitted a neighborhood of $a$, we could postcompose $f$ with a Möbius transformation that takes $a$ to infinity and see that the resulting function has a removable singularity.)

The most well-known result of this type is Picard's theorem which was already mentioned.

There are various beautiful strengthenings of Picard's theorem that arise from Nevanlinna theory, and Ahlfors's theory of covering surfaces.

So all essential singularities have some things in common, but on the other hand this should not lead us to believe that they are all the same. What they have in common is complicated behaviour, but they can be complicated in very different ways! Indeed, different transcendental entire functions (those that have an essential singularity at infinity; i.e. are not polynomials) can vary very much with respect to their behavior near infinity. Just for example, for some such functions, such as $z\mapsto e^z$, there exist curves tending to infinity on which the function is bounded, while for others this is not the case.

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This was very useful. Thank you! The difference between isolated singularities in perspective of analytic continuation has made the distinction much clearer. Then again, I would just define essential singularities in this manner, but I see the purpose beyond pedagogy now. –  Dustin Tran Oct 24 '11 at 21:37
"while for others this is not the case": According to this article: ams.org/journals/tran/1968-131-01/S0002-9947-1968-0220936-0/…, there is a "well known" theorem of Wiman that says that "any entire function of order $\frac12$ must be unbounded on every curve going to infinity." So for example, $\cos\sqrt z$. –  Jonas Meyer May 28 '13 at 5:25
Jonas - I believe he means to say that any function of order less than 1/2 must have this property (which follows from the $\cos\pi\rho$ theorem). That is indeed what I had in mind in my answer. However, clearly $cos\sqrt{z}$ is not unbounded along the positive real axis ... :) –  mathstribble Jun 5 '13 at 9:46

It turns out that essential singularities have their own 'regular' behavior as well - it's just very different then the behavior that we care about for removable and pole singularities.

In short, there is something called the Big Picard Theorem (wiki) that says that an analytic function with an essential singularity has the quality that in any open region containing that singularity, the function takes on every value of the complex plane infinitely often with at most one exception.

But this is not as elementary, and not as immediately useful as things like the residue theorems or Cauchy's integrals theorems or the various things that come from Laurent expansions (unlike the lighter singularities). So it will take a while for most complex classes or books to arrive at Little and Big Picard (Picard has a little one, too).

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Thanks for the information. As an aspiring mathematician, however, it annoys me if the only reason we use this definition is for pedagogical purposes. I'd much rather define an essential singularity using (Big) Picard's theorem or anything similar, which actually gives insight to its structure. –  Dustin Tran Oct 23 '11 at 6:47