I saw solving y=xc^{x}+x+1 on m.se today, and decided that it was ripe for a Newton’s method fractal, so, here you go:

It uses the following python code:


from mpmath import *
import pylab

def g(z,x):
return x*(z**x) + x + 1

def newt(z):
x = z
for i in range(1,30):
x = x - g(z,x)/((z**x)*(1.0+x*fp.log(z))+1.0)
return x

fp.cplot(lambda z: newt(z), [-5.0,5.0], [-5.0,5.0], verbose=True, points=800000)