Julia set of sin(z)
Written by Paul Bourke
May 1998, updated examples in 2019
Python generator: julia_set.py by Tim Meehan.
The following illustrates the Julia set for functions of the form
c sin(z)
where z is a complex number x + iy.
To do this we form the series
z_{k+1} = c sin(z_{k})
starting with some initial z_{0}
The behaviour of this series determines whether or not the initial
z_{0} is part of the julia set or not. More precisely, if the
series tends to infinity then z_{0} is part of the Julia set,
otherwise it isn't. In the examples shown on the right the white regions are
in the Julia set, black points are outside the Julia set.
To create images of the Julia set we map pixels in the image onto
values of z_{0} and colour the pixel dependent on the behaviour of
the series. In the examples on the right the image is mapped onto the range
+2pi in both the real and imaginary axes. The series is tested after
50 terms, it is decided that it tends to infinity if the absolute value
of the imaginary part of z_{k} is greater than 50.
Footnote
If you are wondering how to compute the sine of a complex number, you
can use the following relationships:
x_{k+1} = sin(x_{k}) cosh(y_{k})
y_{k+1} = cos(x_{k}) sinh(y_{k})
where z_{k} = x_{k} + i y_{k}
