Limits involving points at infinity in the extended complex plane

Question

Im trying to understand the idea of infinity in imaginary numbers.

In the book "Complex variables and applications" (8 edition, brown/churchill, section 17) to explain this concept the authors use the idea of a complex plane passing through the equator of a unit sphere centered at origin.

Each point of the sphere P is projected onto the complex plane at one specific point Z. All the point in the sphere has a unique representation in the complex plane except the highest point N which goes to infinity.

This correspondence is called stereographic projection.

All good up to here.

Then they said that some points at 1/epsilon distance from origin correspond to points close to N in the sphere. They call those points "neighbourhood of infinity".

All good.

And then...they replace the expression of limits (lim of z as z goes to z(sub 0) of f(z) is equal to w(sub 0)) and then they describe 3 cases where they replace both z(sub 0) and w(sub 0) with infinity.

I just don't understand this.

Later they try to prove it using the delta epsilon definition when, for example, they never specify what delta would be in the stereographic projection representation.

Apart of that it doesn't have sense algebraicly all this ideas...

Any way if you can give an intuition about this without using abstract mathematical symbols id really appreciated.

Thanks

Answer 1

Definition of complex infinity

You have the complex plane $\mathbb C$. Then, we add an arbitrary point $\infty$ to $\mathbb C$ and say that if you go away from the origin (from any direction or path) you are going towards $\infty$. That is, if the magnitude $|z|$ of a complex number $z$ increases, it is going towards $\infty$.

Limits involving infinity

The theorem you reference in the textbook is quoted below.

Theorem. If $z_0$ and $w_0$ are points in the $z$ and $w$ planes, respectively, then

1. $\lim_{z \to z_0} f(z) = \infty\quad$ if and only if $\quad\lim_{z \to z_0} \frac{1}{f(z)} = 0$, and
2. $\lim_{z \to \infty} f(z) = w_0\quad$ if and only if $\quad\lim_{z \to 0} f\left(\frac{1}{z}\right) = w_0$.

Moreover,

3. $\lim_{z \to \infty} f(z) = \infty \quad$ if and only if $\quad\lim_{z \to 0} \frac{1}{f(1/z)} = 0$.

As we mentioned, if $f(z)$ is going towards $\infty$ as $z$ goes to $z_0$, then the magnitude of $f(z)$ is increasing. Now, if the magnitude of $f(z)$ is increasing, then the magnitude of $\frac{1}{f(z)}$ is decreasing and note that the smallest possible magnitude is $0$. The opposite (converse) is also true. If the magnitude of $\frac{1}{f(z)}$ decreases, then the magnitude of $f(z)$ increases. So, $f(z)$ goes to $\infty$ if and only if $\frac{1}{f(z)}$ goes to $0$. This gives part 1 of the theorem.

By the same argument, $z$ goes to $\infty$ if and only if $\frac{1}{z}$ goes to $0$. This gives us part 2 of the theorem.

Notice that part 3 of the theorem simply combines parts 1 and 2.

Epsilon-delta definition of limits involving infinity

Recall that the geometrical meaning of a limit $\lim_{z \to z_0} f(z) = w_{0}$ is that for each $\epsilon$-neighbourhood $\{w \in \mathbb C \mid |w - w_{0}| < \epsilon\}$ ($\epsilon > 0$) of $w_0$, there is a deleted $\delta$-neighbourhood $\{z \in \mathbb C \mid 0 < |z - z_0| < \delta\}$ ($\delta > 0$) of $z_0$ such that every point $z$ in it has an image $w$ lying in the $\epsilon$-neighbourhood.

Also recall that the book defines an $\epsilon$-neighbourhood of $\infty$ to be $\{z \in \mathbb C \mid |z| > \frac{1}{\epsilon}\}$.

So, we can naturally extend the definition of a limit $\lim_{z \to z_0} f(z) = w_{0}$ when $z_{0}$, $w_{0}$ or both are the point at infinity by simply replacing the neighbourhood with the neighbourhood of $\infty$.

When the book says "We start the proof by noting that the first of limits (1) means that for each positive number $\epsilon$, there is a positive number $\delta$ such that $|f(z)| > 1/\epsilon$ whenever $0 < |z - z_0| < \delta$," it is simply writing the definition of $\lim_{z \to z_{0}} f(z) = \infty$.

Answer 2

The simplest way to justify the point at infinity in (complex) projective geometry is in terms of homogeneous coordinates. Thus, the ordinary points $z\in \mathbb C$ correspond to the pair $[z,1]$ in homogeneous coordinates (alternatively, $[\lambda z, \lambda]$ for any nonzero $\lambda$). Meanwhile, the point at infinity corresponds to the pair $[1,0]$ (alternatively, $[\lambda, 0]$ for any nonzero $\lambda$). All of the "mysterious" properties of the point at infinity can be easily read off from this description. In fact, the point of view of homogeneous coordinates is the "right" point of view to adopt if one wants to understand what happens in higher dimensions (hint: one has to add more than one point there).