newton’s method to find complex fixed points of cos(z)=z

Starting from f(z)=\cos(z)-z, do Newton’s method on its roots to
determine the complex fixed points of \cos(z):

To give a sense of scale on the first image, the unique real fixed point is the Dottie number, around 0.73 or so, and the place where three lines cross in a hex pattern in red is where the function has the value cos(1)-1 at the point z=0

and magnified:

All of the roots represented in the first image represent complex (or real) fixed points of the cosine map. Red in the Newton’s method fractals corresponds to the unique real root — the Dottie number. If you look at the arrangment of roots in the first image, you can kind of see that they’re in the same location as the large Wada basins.

WRONG: (I mucked up on the sign of the sin in the derivative, but it’s important to keep a record of your mistakes so you know what avenues you’ve explored)


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