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Figure 1 illustrates the complicated patterns of convergence that Newton's method for finding roots can have. Here we have plotted the basins of attraction for the three cubic roots of unity under iterations using Newton's method. Each color corresponds to a root. Notice how complicated the boundary between the regions is!

In fact, the boundary has a self-similar fractal structure: any piece of it that is blown up will display the whole pattern.

(All images created with MATLAB® software)

Click on picture for more information and an image of higher resolution.

• Multiple Valued Functions

• Branch cuts and branches

• Riemann Surfaces: the logarithm

• Riemann Surfaces: the square root

• Riemann Surfaces: f(z)=(z²-1)^1/2

• Riemann Surfaces: f(z)=log((z+1)/(z-1))

• Riemann Surfaces: f(z)=log(z²-1)

• Riemann Surfaces: f(z)=log(1-z^1/2)