Fractal-like phase plots

Posted on by John

Define f(z) = iz*exp(-z/sin(iz)) + 1 and g(z) = f(f(z)) + 2 for a complex argument z. Here’s what the phase plots of g look like.

The first image lets the real and imaginary parts of z range from −10 to 10.

This close-up plots real parts between −1 and 0, and imaginary part between −3 and −2.

The plots were produced with this Python code:

from mpmath import cplot, sin, exp

def f(z): return 1j*z*exp(-z/sin(1j*z)) + 1

def g(z): return f(f(z)) + 2

cplot(g, [-1,0], [-3,-2], points=100000)

The function g came from Visual Complex Functions.

Read more: Applied complex analysis

2 thoughts on “Fractal-like phase plots”

  1. I’ve been using mpmath for a while to generate Newton’s method fractals of various functions, e.g. this entry

    I goofed a little on the title, it’s Newton’s method with
    z = z – [exp(cos(z)) – cos(exp(z) -z]/[-sin(z)exp(cos(z)) + exp(z)sin(exp(z)) -1.0]

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