Complex Infinity: Exploring Directional Infinity in Complex Variables

[Dragonfall]

I'm trying to learn some elementary complex variables, and I was reading this book on it when I came upon this

In dealing with complex numbers we also speak of infinity, which we call "the complex number infinity." It is designated by the usual symbol. We do not give a sign to the complex infinity nor define its argument. Its modulus, however, is larger than any preassigned real number.
We can imagine that the complex number infinity is represented graphically by a point in the Argand plane[...]

Consider the function $$f(x)=\left\{\begin{array}{cc}0\Leftrightarrow Im(x)\neq 0\\e^x\Leftrightarrow Im(x)=0\end{array}\right$$. Wouldn't it make more sense if we had a concept of "directional infinity"? If there were only 1 point at infinity on the complex plane, does f converge at infinity?

 

[masudr]

Depends on which direction you approach the function. An analytical function will converge to the same value regardless of which direction that point is approached.

 

[TD]

The problem with C is that there is no order, which is why we only have one 'unsigned' infinity. When you consider C as a plane, which is classically done (cfr R²), then I assume it's possible to talk about a direction.

However, some (important) functions such as the square root and logarithm are multivalued in C and require what we call a 'branch cut' when studied in this plane. Unfortunately, these branch cuts aren't always desirable, they may form some problems. If I recall correctly, some of these mathematical 'technicalities' as the branch cut can be resolved when you do not work in a plane, but in what is called a Riemann Surface. This is a one-dimensional complex manifold and I'm assuming 'direction' wouldn't have the same use/application here as in a plane.

Anyway, this is how I vaguely remember it from my complex analysis course; mathematicians will correct if necessary I assume

 

[Hurkyl]

Wouldn't it make more sense if we had a concept of "directional infinity"?

No: it makes different sense.

There are many different ways to compactify the complex numbers, and different ways have different advantages.

Projective complex space is a name for what you get when you add a single point at infinity. The reason it is commonly used is because it has extremely nice algebraic and analytical properties. For example, the map z --> 1/z shows us that, in projective space, the region "near infinity" looks just like the region near the origin.

Topologically, projective complex space looks like a sphere. (I assume you've seen this already in your class)


However, the directional infinity that you describe doesn't look so nice. And even then, just what do you mean? Which of these rays go off to the same point at infinity?

(A) Re(z) = 0, Im(z) > 0
(B) Re(z) = 1, Im(z) > 0
(C) Re(z) = 0, Im(z) < 0

Different formulations might say that all three go to the same point, or maybe just A and B, or even none of them! And I imagine all three would be useful for different problems.

The problem with C is that there is no order, which is why we only have one 'unsigned' infinity.

That's slightly misleading -- it's true that you cannot order C, but you can use complex conjugation to do all of the same things.

But, of course, complex conjugation is not analytic!

 

[Dragonfall]

I see. Well, it is perhaps misleading for the book to present the projective complex plane as the only method of defining a point at infinity.

By the same method we can define a point at infinity on R with a unit circle and it would be the only "point at infinity" on R. I'm not familiar with projective geometry so I don't know if that would be useful.

 

[Hurkyl]

When studying the real line, it is more common to define the extended real numbers by adding two endpoints: $\pm \infty$.

But the projective reals are still useful. For example, functions like 1/x or tan x are maps from the reals to the projective reals... but they fail to be maps from the reals to the reals, or from the reals to the extended reals.


Each method of study has its own sorts of constructions. In analytic or algebraic geometry, the set of analytic or algebraic functions suggest that there ought to be a single point at infinity.


For something completely different, the study of real algebraic geometry suggests an entirely different way of adding things to the reals: the real spectrum of the real numbers.

This space contains, for each real number x, three points:
x itself
x₊: a point infinitessimally larger than x
x_-: a point infinitessimally smaller than x
as well as the two points at infinity.

For example, tan x function is still not a function on this space, but we do have that $\tan (\pi / 2)_+ = -\infty$, and such.

Of course, tan is not the sort of thing considered in real algebraic geometry, but I've seen this idea used in other circumstances too.