COMPLEX ANALYSIS: Maple Worksheets, 2009
(c) John H. Mathews Russell W. Howell
mathews@fullerton.edu howell@westmont.edu

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COMPLEX ANALYSIS: for Mathematics & Engineering, 4th Ed, 2001, ISBN: 0-7637-1425-9
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CHAPTER 4 SEQUENCES, JULIA and MANDELBROT SETS, and Power Series

Section 4.2 Julia and Mandelbrot Sets Julia Sets

Load Maple's "densityplot" procedure.
Make sure this is done only ONCE during a Maple session.

> with(plots):

Warning, the name changecoords has been redefined

The Julia set

The Julia set associated to a complex number c is found by iterating the map .
The set of points that do not escape to infinity is the Julia set.

Here is the Julia set generated by .

This is a fuzzy picture of Color Plate 4.
Increase the number of grid points to get a sharper picture. Also, it will increase the computing
time, and the memory required is 3,557K. It takes quite a bit of time with grid=[50,50].

> juliaC := proc(X,Y)
local Z, ct;
Z := X + I*Y;
for ct from 1 while ct<25 and evalf(abs(Z))<2.0
do
Z := Z^2 + (-1.25 + I*0.0)
od;
-ct;
end:

> densityplot('juliaC'(x,y),
x=-1.5..1.5, y=-1.5..1.5,
grid=[50,50],
scaling=constrained,
style=PATCHNOGRID, axes=NONE);

The Mandelbrot Set

The Mandelbrot set is the set of points c that do not escape to infinity under iteration of the map
. Points in the Mandelbrot set have connected
This modification of the juliaC code can be used to plot the Mandelbrot set.

This is a fuzzy picture of Color Plate 6.
Increase the number of grid points to get a sharper picture. Also, it will increase the computing
time, and the memory required is 3,492K. It takes quite a bit of time with grid=[50,50].

> mandelbrotC := proc(X,Y)
local Z, ct;
Z := X + I*Y;
for ct from 1 while ct<50 and evalf(abs(Z))<2.0
do
Z := Z^2 + (X + I*Y)
od;
-ct;
end:

> densityplot('mandelbrotC'(x,y),
x=-2..0.55, y=-1.15..1.15,
grid=[50,50],
scaling=constrained,
style=PATCHNOGRID, axes=NONE);

End of Section 4.3.