2.4: The Point at Infinity

By definition the extended complex plane \(= C \cup \{\infty\}\). That is, we have one point at infinity to be thought of in a limiting sense described as follows.

A sequence of points \(\{z_n\}\) goes to infinity if \(|z_n|\) goes to infinity. This “point at infinity” is approached in any direction we go. All of the sequences shown in Figure \(\PageIndex{1}\) are growing, so they all go to the (same) “point at infinity”.

Figure \(\PageIndex{1}\): Various sequences all going to infinity. (CC BY-NC; Ümit Kaya)

If we draw a large circle around 0 in the plane, then we call the region outside this circle a neighborhood of infinity (Figure \(\PageIndex{2}\)).

Figure \(\PageIndex{2}\): The shaded region outside the circle of radius \(R\) is a neighborhood of infinity.

Limits involving infinity

The key idea is \(1/\infty = 0\). By this we mean

\[\lim_{z \to \infty} \dfrac{1}{z} = 0 \nonumber \]

We then have the following facts:

• \(\lim_{z \to z_0} f(z) = \infty \Leftrightarrow \lim_{z \to z_0} 1/f(z) = 0\)
• \(\lim_{z \to \infty} = w_0 \Leftrightarrow \lim_{z \to 0} f(1/z) = w_0\)
• \(\lim_{z \to \infty} = \infty \Leftrightarrow \lim_{z \to 0} \dfrac{1}{f(1/z)} = 0\)

Example \(\PageIndex{1}\)

\(\lim_{z \to \infty} e^z\) is not defined because it has different values if we go to infinity in different directions, e.g. we have \(e^z = e^x e^{iy}\) and

\(\lim_{x \to -\infty} e^x e^{iy} = 0\)
\(\lim_{x \to +\infty} e^x e^{iy} = \infty\)
\(\lim_{y \to +\infty} e^x e^{iy}\) is not defined, since \(x\) is constant, so \(e^x e^{iy}\) loops in a circle indefinitely.

Example \(\PageIndex{2}\)

Show \(\lim_{z \to \infty} z^n = \infty\) (for \(n\) a positive integer).

Solution

We need to show that \(|z^n|\) gets large as \(|z|\) gets large. Write \(z = Re^{i \theta}\), then

\[|z^n| = |R^n e^{in \theta}| = R^n = |z|^n \nonumber \]

Stereographic Projection from the Riemann Sphere

One way to visualize the point at \(\infty\) is by using a (unit) Riemann sphere and the associated stereo-graphic projection. Figure \(\PageIndex{4}\) shows a sphere whose equator is the unit circle in the complex plane.

Figure \(\PageIndex{3}\): Stereographic projection from the sphere to the plane. (CC BY-NC; Ümit Kaya)

Stereographic projection from the sphere to the plane is accomplished by drawing the secant line from the north pole \(N\) through a point on the sphere and seeing where it intersects the plane. This gives a 1-1 correspondence between a point on the sphere \(P\) and a point in the complex plane \(z\). It is easy to see show that the formula for stereographic projection is

\[P = (a, b, c) \mapsto z = \dfrac{a}{1 - c} + i \dfrac{b}{1 - c}. \nonumber \]

The point \(N = (0, 0, 1)\) is special, the secant lines from \(N\) through \(P\) become tangent lines to the sphere at \(N\) which never intersect the plane. We consider \(N\) the point at infinity.

In the figure above, the region outside the large circle through the point \(z\) is a neighborhood of infinity. It corresponds to the small circular cap around \(N\) on the sphere. That is, the small cap around \(N\) is a neighborhood of the point at infinity on the sphere!

Figure \(\PageIndex{4}\) shows another common version of stereographic projection. In this figure the sphere sits with its south pole at the origin. We still project using secant lines from the north pole.

Figure \(\PageIndex{4}\): Alternative stereographic projection from the sphere to the plane. (CC BY-NC; Ümit Kaya)