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

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]

You might be interested in taking a look at

http://www.math.osu.edu/~fowler.291/phase/

which Steve Gubkin and I will be using to build an online complex analysis course, along the lines of our https://mooculus.osu.edu/