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 z₀

The behaviour of this series determines whether or not the initial z₀ is part of the julia set or not. More precisely, if the series tends to infinity then z₀ 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 z₀ 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