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A First Course in Complex Analysis

Matthias Beck, Gerald Marchesi, Dennis Pixton, Lucas Sabalka

Section 3.2 Infinity and the Cross Ratio

In the context of Möbius transformations, it is useful to introduce a formal way of saying that a function \(f\) “blows up” in absolute value, and this gives rise to a notion of infinity.
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Definition 3.2.1.

Suppose \(f: G \to \C\text{.}\)
  1. \(\lim_{z\to z_0}f(z)=\infty\) means that for every \(M>0\) we can find \(\delta>0\) so that, for all \(z\in G\) satisfying \(0\lt \abs{z-z_0}\lt \delta\text{,}\) we have \(\abs{f(z)}>M\text{.}\)
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  2. \(\lim_{z\to \infty}f(z)=L\) means that for every \(\epsilon>0\) we can find \(N>0\) so that, for all \(z\in G\) satisfying \(\abs{z}>N\text{,}\) we have \(\abs{f(z)-L}\lt \epsilon\text{.}\)
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  3. \(\lim_{z\to \infty}f(z)=\infty\) means that for every \(M>0\) we can find \(N>0\) so that, for all \(z\in G\) satisfying \(\abs{z}>N\text{,}\) we have \(\abs{f(z)}>M\text{.}\)
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In the first definition we require that \(z_0\) be an accumulation point of \(G\) while in the second and third we require that \(\infty\) be an extended accumulation point of \(G\text{,}\) in the sense that for every \(B>0\) there is some \(z\in G\) with \(\abs z>B\text{.}\)
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As in Section 2.1, the limit, in any of these senses, is unique if it exists.
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Example 3.2.2.

We claim that \(\lim_{ z \to 0 } \frac 1 { z^2 } = \infty\text{:}\) Given \(M > 0\text{,}\) let \(\delta := \frac 1 { \sqrt M }\text{.}\) Then \(0 \lt |z| \lt \delta\) implies
\begin{equation*} |f(z)| \ = \ \left| \frac 1 { z^2 } \right| \ > \ \frac 1 { \delta^2 } \ = \ M \, \text{.} \end{equation*}
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Example 3.2.3.

Let \(f(z) = \frac{ az+b }{ cz+d }\) be a Möbius transformation with \(c \ne 0\text{.}\) Then \(\lim_{ z \to \infty } f(z) = \frac a c\text{.}\)
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To simplify the notation, introduce the constant \(L := |ad -bc|\text{.}\) Given \(\epsilon > 0\text{,}\) let \(N := \frac L { |c|^2 \epsilon } + \left| \frac d c \right|\text{.}\) Then \(|z| > N\) implies, with the reverse triangle inequality (Corollary 1.3.5(b)), that
\begin{equation*} |cz+d| \ \ge \ \bigl| |c||z| - |d| \bigr| \ \ge \ |c||z| - |d| \ > \ \frac L { |c| \epsilon } \end{equation*}
and so
\begin{equation*} \left| f(z) - \frac a c \right| \ = \ \left| \frac{ c(az+b) - a(cz+d) }{ c(cz+d) } \right| \ = \ \frac L { |c| \, |cz+d| } \ \lt \ \epsilon \, \text{.} \end{equation*}
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We stress that \(\infty\) is not a number in \(\C\text{,}\) just as \(\pm \infty\) are not numbers in \(\R\text{.}\) However, we can extend \(\C\) by adding on \(\infty\text{,}\) if we are careful. We do so by realizing that we are always talking about a limit when handling \(\infty\text{.}\) It turns out (see Exercise 3.6.11) that the usual limit rules behave well when we mix complex numbers and \(\infty\text{.}\) For example, if \(\lim_{z\to z_0}f(z)=\infty\) and \(\lim_{z\to z_0}g(z)=a\) is finite then the usual limit of sum = sum of limits rule still holds and gives \(\lim_{z\to z_0}(f(z)+g(z))=\infty\text{.}\) The following definition captures the philosophy of this paragraph.
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Definition 3.2.4.

The extended complex plane is the set \(\Chat := \C\cup\listset\infty\text{,}\) together with the following algebraic properties: For any \(a \in \C\text{,}\)
  1. \(\displaystyle \infty+a = a+\infty=\infty\)
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  2. if \(a \neq 0\) then \(\infty\cdot a = a\cdot\infty = \infty\)
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  3. \(\displaystyle \infty\cdot\infty =\infty\)
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  4. \(\displaystyle \frac a \infty = 0\)
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  5. if \(a \neq 0\) then \(\frac a 0 = \infty \text{.}\)
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The extended complex plane is also called the Riemann sphere or the complex projective line, denoted \(\mathbb{CP}^1\text{.}\)
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If a calculation involving \(\infty\) is not covered by the rules above then we must investigate the limit more carefully. For example, it may seem strange that \(\infty+\infty\) is not defined, but if we take the limit of \(z+(-z)=0\) as \(z\to\infty\) we will get \(0\text{,}\) even though the individual limits of \(z\) and \(-z\) are both \(\infty\text{.}\)
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Now we reconsider Möbius transformations with \(\infty\) in mind. For example, \(f(z)= \frac 1 z\) is now defined for \(z=0\) and \(z=\infty\text{,}\) with \(f(0)=\infty\) and \(f(\infty)=0\text{,}\) and so we might argue the proper domain for \(f(z)\) is actually \(\Chat\text{.}\) Let’s consider the other basic types of Möbius transformations. A translation \(f(z) = z+b\) is now defined for \(z=\infty\text{,}\) with \(f(\infty) = \infty+b = \infty\text{,}\) and a dilation \(f(z) = az\) (with \(a\ne0\)) is also defined for \(z=\infty\text{,}\) with \(f(\infty) = a\cdot\infty=\infty\text{.}\) Since every Möbius transformation can be expressed as a composition of translations, dilations and inversions (Proposition 3.1.4), we see that every Möbius transformation may be interpreted as a transformation of \(\Chat\) onto \(\Chat\text{.}\) This general case is summarized in the following extension of Proposition 3.1.2.
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This corollary also holds for \(c = 0\) if we then define \(f(\infty) = \infty\text{.}\)
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Example 3.2.6.

Continuing Example 3.1.3 and Example 3.1.6, consider once more the Möbius transformation \(f(z) = \frac{ z-1 }{ iz+i }\text{.}\) With the definitions \(f(-1) = \infty\) and \(f(\infty) = -i\text{,}\) we can extend \(f\) to a function \(\Chat \to \Chat\text{.}\)
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With \(\infty\) on our mind we can also add some insight to Theorem 3.1.5. We recall that in Example 3.1.6, we proved that \(f(z) = \frac{ z-1 }{ iz+i }\) maps the unit circle to the real line. Essentially the same proof shows that, more generally, any circle passing through \(-1\) gets mapped to a line (see Exercise 3.6.4). The original domain of \(f\) was \(\C \setminus \{ -1 \}\text{,}\) so the point \(z = -1\) must be excluded from these circles. However, by thinking of \(f\) as function from \(\Chat\) to \(\Chat\text{,}\) we can put \(z = -1\) back into the picture, and so \(f\) transforms circles that pass through \(-1\) to straight lines plus \(\infty\text{.}\) If we remember that \(\infty\) corresponds to being arbitrarily far away from any point in \(\C\text{,}\) we can visualize a line plus \(\infty\) as a circle passing through \(\infty\text{.}\) If we make this a definition, then Theorem 3.1.5 can be expressed as: any Möbius transformation of \(\Chat\) transforms circles to circles.
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We can take this remark a step further, based on the idea that three distinct points in \(\Chat\) determine a unique circle passing through them: If the three points are in \(\C\) and nonlinear, this fact comes straight from Euclidean geometry; if the three points are on a straight line or if one of the points is \(\infty\text{,}\) then the circle is a straight line plus \(\infty\text{.}\)
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Example 3.2.7.

The Möbius transformation \(f: \Chat \to \Chat\) given by \(f(z) = \frac{ z-1 }{ iz+i }\) maps
\begin{equation*} 1 \mapsto 0 \, , \qquad \qquad i \mapsto 1 \, , \qquad \text{ and } \qquad -1 \mapsto \infty \, \text{.} \end{equation*}
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The points \(1\text{,}\) \(i\text{,}\) and \(-1\) uniquely determine the unit circle and the points 0, 1, and \(\infty\) uniquely determine the real line, viewed as a circle in \(\Chat\text{.}\) Thus Corollary 3.2.5 implies that \(f\) maps the unit circle to \(\R\text{,}\) which we already concluded in Example 3.1.6.
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Conversely, if we know where three distinct points in \(\Chat\) are transformed by a Möbius transformation then we should be able to figure out everything about the transformation. There is a computational device that makes this easier to see.
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Definition 3.2.8.

If \(z\text{,}\) \(z_1\text{,}\) \(z_2\text{,}\) and \(z_3\) are any four points in \(\Chat\) with \(z_1\text{,}\) \(z_2\text{,}\) and \(z_3\) distinct, then their cross ratio is defined as
\begin{equation*} [z,z_1,z_2,z_3] \ := \ \frac{(z-z_1)(z_2-z_3)}{(z-z_3)(z_2-z_1)} \,\text{.} \end{equation*}
This includes the implicit definitions \([z_3,z_1,z_2,z_3] = \infty\) and, if one of \(z\text{,}\) \(z_1\text{,}\) \(z_2\text{,}\) or \(z_3\) is \(\infty\text{,}\) then the two terms containing \(\infty\) are canceled; e.g., \([z,\infty, z_2, z_3] = \frac{ z_2 - z_3 }{ z - z_3 }\text{.}\)
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Example 3.2.9.

Our running example \(f(z) = \frac{ z-1 }{ iz+i }\) can be written as \(f(z) = [z,1,i,-1]\text{.}\)
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Proof.

Most of this follows from our definition of \(\infty\text{,}\) but we need to prove the uniqueness statement. By Proposition 3.1.2, the inverse \(f^{-1}\) is a Möbius transformation and, by Exercise 3.6.10, the composition \(h := g\circ f^{-1}\) is again a Möbius transformation. Note that \(h(0) = g(f^{-1}(0)) = g(z_1) = 0\) and, similarly, \(h(1) = 1\) and \(h(\infty) = \infty\text{.}\) If we write \(h(z)=\frac{az+b}{cz+d}\) then
\begin{align*} 0 \ = \ h(0) \ = \ \frac bd \amp \implies b=0\\ \infty \ = \ h(\infty) \ = \ \frac ac \amp \implies c=0\\ 1 \ = \ h(1) \ = \ \frac{a+b}{c+d} \ = \ \frac {a+0}{0+d} \ = \ \frac ad \amp \implies a=d \end{align*}
and so
\begin{equation*} h(z)\ = \\frac{az+b}{cz+d} \ = \ \frac{az+0}{0+d} \ = \ \frac ad \, z \ = \ z \, \text{,} \end{equation*}
the identity function. But since \(h = g\circ f^{-1}\text{,}\) this means that \(f\) and \(g\) are identical.
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So if we want to map three given points of \(\Chat\) to \(0\text{,}\) \(1\) and \(\infty\) by a Möbius transformation, then the cross ratio gives us the only way to do it. What if we have three points \(z_1\text{,}\) \(z_2\) and \(z_3\) and we want to map them to three other points \(w_1\text{,}\) \(w_2\) and \(w_3\text{?}\)
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Proof.

Let \(h=g^{-1}\circ f\) where \(f(z)=[z,z_1,z_2,z_3]\) and \(g(w)=[w,w_1,w_2,w_3]\text{.}\) Uniqueness follows as in the proof of Proposition 3.2.10.
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This theorem gives an explicit way to determine \(h\) from the points \(z_j\) and \(w_j\) but, in practice, it is often easier to determine \(h\) directly from the conditions \(f(z_j)=w_j\) (by solving for \(a\text{,}\) \(b\text{,}\) \(c\) and \(d\)).
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