The Great Picard Theorem

The great Picard theorem states that, if an analytic function contains an essential singularity, then within any punctured neighborhood of the singularity, the function takes on every value with at most a single exception.

OK, that probably looks like complete gibberish.  Let's break it down.

First of all, what are we even talking about?  The phrase "analytic function" is a hint, and "essential singularity" is a dead giveaway.  These terms are used when discussing complex functions, that is, functions whose domains are the complex plane (or some subset thereof, but usually not a subset of the reals).  Just as we can define functions that operate on real numbers, we may also define functions on complex numbers.

For example, consider this function:

f(z) = z²+1

By convention, we use z instead of x when dealing with complex functions.  Otherwise, this is a standard polynomial, and we can just substitute complex numbers into it:

f(i) = i²+1

f(i) = 0

This particular polynomial is apropos because it is central to the definition of complex numbers.  i and −i are defined as the roots of this polynomial, and the rest of the complex plane is then derived from that.  With a little more work, you can show that every nth degree polynomial has n complex roots (though they may not all be distinct).

So now that we have a basic grasp of complex functions, what is an "analytic" function?  The precise definition of analytic function is a function which is everywhere equal to its Taylor series, but once you do some math it can be shown that the analytic functions are all and only those functions whose derivatives exist at every point in their respective domains (more formally, holomorphic functions are analytic).  An "entire" function is an analytic function whose domain is exactly the complex plane (as opposed to the complex plane minus some points).  The above polynomial is entire because its derivative exists everywhere (equivalently, it is its own Taylor series, which converges everywhere).

If we have some more complicated function, like the sine function or the exponential function, we can analytically continue it by taking its Taylor series.  Assuming the original function is sufficiently well-behaved on (some interval of) the real number line, the continuation will be analytic.  This is true of the exponential function, sine and cosine, and a number of other functions, as well as some variations on them such as the error function, but not the hyperbolic sine and cosine.  For a more interesting example, the Riemann zeta function's analytic continuation is far more interesting and useful than the original infinite sum definition, which diverges for negative numbers.

Once you know that a complex function is analytic, you can immediately deduce a number of beautiful properties.  I'm not going to cover all of them here, but I do recommend researching this for yourself.

The next stumbling block in the theorem is "essential singularity."  A singularity is a point absent from a function's domain.  An isolated singularity is a singularity which is not "near" any other singularities.  This means there exists a circle with nonzero diameter centered on this singularity which does not contain any other singularities.  More simply, the singularity is a point by itself, and not part of an entire singular line or region.

Isolated singularities can be further broken down into three types:

• Removable singularities: If the function took on a particular value at this point (instead of failing to exist), it would still be analytic.  Intuitively, the function behaves as if its value ought to be a particular value, but for some reason it is instead undefined.
• Poles: Loosely, the function approaches unsigned infinity, like 1/z does.  Intuitively, we would like to say the function actually is infinite at this point.  We can use the Riemann sphere to formalize this notion, and we end up with a so-called meromorphic function, assuming its singularities are all poles.
• Essential singularities: All others.  Intuitively, there is no value, not even complex infinity, which we may assign without losing analyticity.

Essential singularities can be thought of as pathological cases, where a function behaves in a particularly bizarre or irregular fashion.  Of course, even the pathological cases must follow some rules, and the great Picard is one of them.

Now we need to divert into topology to define "punctured neighborhood."  Well, actually, we don't need most of the baggage topology gives this term, so we can just define a punctured neighborhood as a disk centered on a given point, but with that point removed.  It's usually implied the disk's radius is on the small side, and the theorem we're using specifically applies to any punctured neighborhood, no matter how small.

We can finally pull it all together.  Take a function with an essential singularity, say, this one:

f(z) = exp(1/z)

The function fails to exist at z = 0.  What's more, if you take the limit approaching from the left (more rigorously: approaching along the real number line from the left), you get zero, but if you approach from the right you get infinity.  Pick any positive real number ε.  Considering only the points within ε of the origin, the function takes on every complex value infinitely many times, except that it is never equal to zero.  Here's a picture, courtesy of Functor Salad at Wikimedia Commons:

Lighter colors are results farther away from zero, and the hue of the color indicates the direction from zero to the result (positive real results are cyan, and negative real results are red; complex values are other colors).  Zero itself is perfectly black.  Notice how the colors seem to cycle more rapidly as we approach the center from above or below.  You can also clearly see a mixture of light and dark, with the boundary becoming more pronounced towards the center.  But since the only singularity is at z = 0, the boundary is never truly sharp (or else the derivative would fail to exist).  It may appear to turn into black and white close to the origin, but this is an artifact of the rendering.  The image would require an infinite resolution to avoid this kind of bleeding.  At no point is the function exactly equal to zero, though it comes very close.

Unfortunately, some of us are used to looking at simple line charts, and may find the above presentation disorienting.  The notion that a function takes on every value infinitely many times just seems profoundly unintuitive.  It turns out that some real-valued functions do this too.  Consider this one:

f(x) = sin(1/x)/x

Here is the graph.  As we get closer to x = 0, the amplitude and frequency of the waveform both increase without bound.  It takes on every real value infinitely many times, and continues doing so no matter how close we get.  The great Picard theorem just says that in the complex plane, all essential singularities look like that.