Adding infinity to the upper half plane

Question

I have a weak physicist background in complex analysis and topology. I've been looking at things defined on the upper half complex plane, and it is not clear to me if there are subtleties in "going to infinity from there".

I'm used to work in Riemann sphere and so it's quite intuitive for me that there "is only one infinity" and that I'll end up on the same point by infinitely increasing my distance from the center (or any point) no matter each direction I'm taking. But $\mathbb{R}$ for example admits many compactification: adding two infinities ($+\infty$ and $-\infty$) or only one (ring of infinite radius).

Are there also different compactifications in adding infinity to the upper half plane (two different infinities along the real axis and one for the imaginary direction).

Is this question stupid?

Thank you in advance for clarifying this point to me.

Answer 1

From the viewpoint of differential geometry or complex analysis, the most natural "set of infinitely far points" of the upper half plane is the 1-point compactification of the real line, that is the circle. All in all, the answer depends on what are you going to do with this "infinity".

Answer 2

I'm not sure I have understood your question well, but it depends what your problem is.

There are many more infinities than the three you described : you can "go to infinity" following the direction you want provided it is in the upper half of the complex plane. You are not constrained by a horizontal or vertical direction.

If some text says "going to infinity from there", it isn't clear in the general case if the variable is complex. However, if it is real (like a distance for example), only one direction is admissible : the real (horizontal) axis from left to right (so the limit is to $+\infty$). It doesn't matter if the function uses complex numbers.

If you are dealing with integrals, "going to infinity" simply means that you are covering the entire upper half of the complex plane. This can be done integrating with the axis: $\int_{x=-\infty}^{+\infty}\int_{y=0}^{+\infty}f(x,y)$ or switching to polar coordinates : $\int_{r=0}^{+\infty}\int_{\theta{}=0}^{\pi}f(r,\theta{})$.