Section3.10\(z/2 - 1/(2z)\text{:}\) Newton's method applied to a quadratic
Suppose we apply Newton's method to the function \(f(z)=z^2+1\text{.}\) We get
\begin{equation*}
N(z) = z - \frac{z^2+1}{2z} = \frac{z}{2}-\frac{1}{2z}.
\end{equation*}
It's easy to show that \(\pm i\) are both super-attractive fixed points. It seems easy to believe that any point in the upper half plane should converge to \(i\text{,}\) while any point in the lower half plane should converge to \(-i\) - and, that's exactly what happens. Hardly seems worth a picture. We'll do better - we'll prove it!
Claim3.10.1
Let \(f(z)=z^2+1\) so that \(N(z)=z/2-1/(2z)\) is the corresponding Newton's method iteration function. Let \(z_0\) be an initial seed for iteration under \(N\text{.}\) If \(\text{Im}(z_0) \gt 0\text{,}\) then the orbit of \(z_0\) converges to \(i\text{;}\) if \(\text{Im}(z_0) \gt 0\text{,}\) then the orbit of \(z_0\) converges to \(-i\text{.}\) Furthermore, the restriction of \(N\) to the real line is conjugate to \(g(z)=z^2\) on the unit circle and, thus, chaotic there.
For the conjugacy function, we choose the Mobius transformation
\begin{equation*}
\varphi(z) = \frac{z-i}{z+i}.
\end{equation*}
Note that \(\varphi(\infty)=1\text{,}\) \(\varphi(0)=-1\text{,}\) and \(\varphi(1)=-i\text{.}\) Thus, the image of the real axis is exactly the unit circle. To show that \(\varphi\) conjugates \(g\) to \(N\text{,}\) we simply compute: \begin{align*} \varphi(N(z)) & = \frac{\frac{z}{2}-\frac{1}{2z}-i}{\frac{z}{2}-\frac{1}{2z}+i} = \frac{2 z^2 - 2 - 4iz}{2 z^2 - 2 - 4iz} \\ & = \frac{2(z-i)^2}{2(z+i)^2} = \left(\frac{z-i}{z+i}\right)^2 = g(\varphi(z)). \end{align*}
Furthermore, \(\varphi(i)=0\) and \(\varphi(-i)=\infty\text{.}\) Thus the super-attractive fixed point \(i\) for \(N\) corresponds to the super-attractive fixed point at the origin for \(g\) and the top half of the plane maps to the interior of the unit circle. As a result, everything in the top half of the plane converges to \(i\) under iteration of \(N\text{,}\) just as everything in the interior of the unit circle converges to the origin under iteration of \(g\text{.}\) The correspondence between the bottom half of the plane and the exterior of the unit circle is similar.
Claim Claim 1 can be generalized. Let \(f\) be a complex quadratic with distinct roots \(z_1\) and \(z_2\) and let \(N\) be the corresponding Newton's method iteration function. It can be shown that the the line of points equidistant from \(z_1\) and \(z_2\) is invariant under \(N\) and the dynamics of \(N\) on that line are conjugate to \(z\to z^2\) on the unit circle. Furthermore, the orbit of an initial seed off of that line will converge to the closer of the the two roots. This result was established independently by Ernst Schroder and Arthur Cayley in the 1870s. This is probably the first theorem of complex dynamics. Both authors explored the same idea for cubic polynomials and commented that there were significant difficulties. It was exactly this challenge that gave rise to complex dynamics as a field of study.