What is the concept of 'one infinity' in complex analysis?

[marellasunny]

'In the real case,we can distinguish between the limits (+infinity) and (-infinity),but in the complex case there is only one infinity'-this was given in Lars Ahlfors's book on complex analysis.
Can someone explain what he means by 'one infinity'?
My proffesor asked me to look into the concept of the projective reals and extended reals...but i still don't get it!
Please help.I'm still a novice.

 

[HallsofIvy]

In topology, we can talk about the "compactification" of a non-compact (roughly, either open or not bounded) by adding new points. And there are many different ways to do that- the two most important are the "Stone-Cech compactification" and the "one point compactification". The "Stone-Cech compactification" preserves as much of the topological properties as possible while the "one point compactification", well, just adds one point!

We can, for example, make the open interval, (a, b), compact by adding the two endpoints to get the closed interval, [a, b]- that's the "Stone-Cech compactification". Or we can imagine bending the interval into a circle, adding a single point connecting the two ends so that it now has the topology of a circle- That's the "one-point compactification". We can make the infinite line- all of R- compact by adding "-\infty" and "+\infty", redefining the metric so that the larger positive numbers are "closer" to +\infty and "larger" negative numbers closer to -\infty, the "Stone-Cech" compactification, or we can add a single point, \infty, redefining the metric so that all "larger" numbers, both positive and negative, are closer to \infty, the "one point" compactification.

In the plane, whether R² or the complex plane (we are only talking about the the geometry, not the algebra), we can do the "Stone-Cech" compactification by adding a new "point at infinity" at the "ends" of all straight lines through the origin, forming and entire circle "at infinity", giving it the geometry of a disk, or do the "one point compactification" by adding a single "point at infinity", giving it the topology of a sphere. Because the first now involves adding an infinity of new points, we tend to prefer the second.

This also is equivalent to the "Riemann sphere". Imagine a sphere, of radius r, with center at (0, 0, r) so that it is "sitting" one the xy-plane at (0, 0, 0). For ever point, (x, y), in the xy-plane, draw a line from (0, 0, 2r) to (x, y, 0). The point where that line crosses the sphere is associated with the number x+ iy. The point (0, 0, 2r) becomes the single "point at infinity".

 

[kloptok]

I guess what the author is hinting at is that you can not compare the size of two complex numbers unless you look at the modulus |z|. That is, a statement such as z<3 makes no sense (what would happen if z is purely imaginary?) but a statement such as |z|<3 does. In the same sense, it doesn't make sense to say z\rightarrow +\infty, it only makes sense to say |z| \rightarrow +\infty. And we can not say that |z| \rightarrow -\infty since the modulus is always a non-negative real number. Hence there is only "one infinity" and not two like along the real line. Do you see what I mean?

That said, I don't know much about projective/extended reals and how this relates, but I guess this is what HallsofIvy tried to explain. I would assume that his/her post is a fancier way of saying what I said above.

 

[Mandlebra]

Look at how C is mapped to the riemann sphere. In this mapping there is only one point that can't quite be reached, the north pole.

 

[micromass]

We can, for example, make the open interval, (a, b), compact by adding the two endpoints to get the closed interval, [a, b]- that's the "Stone-Cech compactification". Or we can imagine bending the interval into a circle, adding a single point connecting the two ends so that it now has the topology of a circle- That's the "one-point compactification". We can make the infinite line- all of R- compact by adding "-\infty" and "+\infty", redefining the metric so that the larger positive numbers are "closer" to +\infty and "larger" negative numbers closer to -\infty, the "Stone-Cech" compactification, or we can add a single point, \infty, redefining the metric so that all "larger" numbers, both positive and negative, are closer to \infty, the "one point" compactification.

Uuuh, what you describe here isn't the Stone-Cech compactification at all. The Stone-Cech compactification is much more complicated and is in general not metrizable (unless the original space is compact)

 

[A. Bahat]

This heuristic may (or may not) be helpful: I usually think of the Riemann sphere as the complex plane "folded up" with one point (the point at infinity) added at the north pole.

 

[Ben M]

Perhaps counter-intuitively, analytic functions of a complex variable have a well-defined "behaviour at infinity" which is just the limit as z tends to zero of f(1/z). This is unique provided that the function universally satisfies the C-R relations.

 

[marellasunny]

Perhaps counter-intuitively, analytic functions of a complex variable have a well-defined "behaviour at infinity" which is just the limit as z tends to zero of f(1/z). This is unique provided that the function universally satisfies the C-R relations.

What does "well-defined behaviour" mean?Can you give an example of a function that satisfies C-R and no have a "well-defined behaviour"?
Thanks!