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In $\mathbb R$ a sequence can diverge to infinity in two directions: $+\infty$ and $-\infty$. These two cases of divergence are quite different from a sequence that diverges to "nowhere", like $\{(-1)^n\}$. One can be interested, for example, in the behavior of a function in one of these two directions of infinity.

Is there a similar notion for sequences in the complex plane? (Or other spaces) Does it arise naturally in some field of study? What are possible uses for it?

I imagine, for example, that the sequence $\{ni\}$ could be said to diverge to infinity in the direction $i$, whereas the sequence $\{ne^{in\sqrt2}\}$ could be said to diverge to infinity in "every direction", or in a "divergent direction", or in a "set of directions".

Edit: Added two geometry tags, following the comment of Moishe Cohen

dafinguzman
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  • On the Reimann sphere there is only one infinity and all sequences which "diverge to infinity" (as opposed to "diverging to nowhere") can be thought of on the sphere as instead converging to infinity instead using the Reimannian metric. – JMoravitz Aug 09 '17 at 22:10
  • The most common way to proceed in the complex plane is to consider absolute values and to just consider divergence to infinity with respect to absolute value. Yet it is possible to do what you propose. Right now I do not know a good reference or application though. – quid Aug 09 '17 at 22:10
  • Sure, why not? To the divergent infinity in all directions, we'd probably shoot this down what WolframAlpha calls complex infinity, and to the other, directed infinity. – Simply Beautiful Art Aug 09 '17 at 22:11
  • @quid WolframAlpha has almost everything lol – Simply Beautiful Art Aug 09 '17 at 22:17
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    Yes, this field of study is called geometry (metric geometry and Riemannian geometry to be more precise) and describing different ways to diverge to infinity (different directions of divergence), is quite important. Consider for instance: Gromov boundary. – Moishe Kohan Aug 09 '17 at 22:23
  • @MoisheCohen first time I hear about that! Thank you – dafinguzman Aug 09 '17 at 22:26
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    The real projective plane has a whole line of points at infinity. In this setting the lines $y=0$ and $y=1$ can be said to meet at a point at infinity, but a different point than the one the lines $x=0$ and $x=1$ meet at. –  Aug 10 '17 at 00:47
  • @MoisheCohen The OP probably wanted the idea of compactification of metric and topological spaces ? $\mathbb{C}$ can be compactified in many different ways. – reuns Aug 10 '17 at 17:20

1 Answers1

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Yes, this field of study is called geometry (metric geometry and Riemannian geometry to be more precise) and describing different ways to diverge to infinity (different directions of divergence), is quite important. Consider for instance: Gromov boundary. In their book "Encyclopedia of Distances" (pp. 116-117), M.Deza and E.Deza list 5 different ideal boundaries (and some of these 5 items have several subitems), which all capture different aspects of diverging to infinity in different directions in different spaces. (Actually, there are more boundaries than they list, once one realizes that probabilists also have ways to capture divergence to infinity, resulting in, say, Poisson, Furstenberg and Martin boundaries.) As for the complex plane, one gets the circle as the ideal boundary (in item 1 on the Deza-Deza list). This corresponds to divergence to infinity along one of the rays starting at the origin. (Each ray gives one direction.) One can start at different points in the plane, then parallel rays yield the same direction of divergence.

Moishe Kohan
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  • This is the kind of thing I was looking for :). Can you please elaborate a little on why these different aspects of divergence are important or what they are used for? – dafinguzman Aug 10 '17 at 17:34
  • @dafinguzman: I will, but it takes a bit of time. It also depends on your background: Are you taking (already took) a complex analysis class? A topology class? A differential geometry class? (Or self-read, does not matter.) – Moishe Kohan Aug 10 '17 at 23:16
  • I took a complex analysis class once, and I have a basic knowledge of topology in normed spaces – dafinguzman Aug 10 '17 at 23:25