What is the real and imaginary part of complex infinity?

Question

I know that complex infinity is a pole on the Riemann-sphere, but what is actually the real and imaginary part of it? Is it $\pm\infty \pm \infty i$? And how is it different from simple non-complex infinity?

score 6 · Accepted Answer · answered Jul 04 '18 at 18:43

Complex infinity is just that: $\infty$. In particular, as far as we can make sense of this at all, we have $\infty=-\infty=i\infty$ (or in fact $\infty=z\infty$ for any finite non-zero $z\in\Bbb C$). It makes no sense to speak of its real or imaginary part or its argument.

In spite of the same symbol being used, the $\infty$ that is added to $\Bbb C$ in its one-point compactification is not directly related to that $\infty$ that is (together with $-\infty$) added to $\Bbb R$ in its two-point compactification. (There is also a one-point compactification of $\Bbb R$, and the $\infty$ used for that may perhaps be identified with the complex infinity if we view the - compactified - real line as a great circle of the Riemann sphere). I'd even say that the uses of the symbol $\infty$ in things like $\sum_{n=1}^\infty a_n$ or $\|f\|_\infty$ are only loosely related to these as well. (And we haven't even started with infinities occurring as cardinalities and being written with a whole different bunch of symbols)

And if $\mathrm{Re}: \mathbb{C}\to\mathbb{R}$ denotes the real-part function (continuous but not holomorphic), then on every (punctured) neighborhood of $\infty$ in the Riemann sphere, $\mathrm{Re}$ takes any real value (also numerically small values). So there is no relevant way to extend $\mathrm{Re}$ to be defined in $\infty$. And it is the same for the imaginary part, of course. — Jeppe Stig Nielsen, Jul 04 '18 at 20:33

score 5 · Answer 2 · edited Dec 01 '21 at 19:37

According to the definition from Wolfram MathWorld "Complex infinity is an infinite number in the complex plane whose complex argument is unknown or undefined".

Basically, it's a number in the complex plane whose magnitude is infinite, but whose phase is unknown. Unlike with a 1-dimensional number where we have only one direction to go to positive or negative infinity, we can approach complex infinity from an infinite number of directions depending on the phase.

What is the real and imaginary part of complex infinity?

2 Answers2