Complex Functions

Here are some visualizations of complex functions that I generated with a program a wrote in C++ using the SFML library. (The interested reader can check out this Wikipedia page for technical stuff).

Screenshot from 2016-02-25 14:32:19
This is the starting point, the untransformed complex plane. The rings indicate when the modulus of that point approaches an integer value.

 

Screenshot from 2016-02-25 14:34:16
Here we have z = f(z) = z^2. Notice how the colors cycle twice as you circle the origin. Also notice how the distance between the rings is no longer uniform.

 

Screenshot from 2016-02-25 14:46:10
This is z = f(z) = sin(z). Things are starting to get more interesting.
Screenshot from 2016-02-25 14:47:59
z = f(z) = tan(z)

 

Screenshot from 2016-02-25 14:57:24
z = f(z) = 1/z. Notice how the shading is now on the inside of the rings, whereas before it was on the outside.
Screenshot from 2016-02-25 15:20:55
z = sin(tan(z))
Screenshot from 2016-02-25 19:38:37
z = f(z) = tan(cos(z)). What I find most interesting about this is the patterns made by the shading.
Screenshot from 2016-02-25 15:22:47
z = f(z) = tan(e^z)
Screenshot from 2016-02-25 15:24:35
z = f(z) = e^(-tan(z))
Screenshot from 2016-02-25 15:23:45
z = f(z) = e^(-z)
Screenshot from 2016-02-25 14:53:56
z = f(z) = tan(sin(z))

 

Screenshot from 2016-02-25 14:55:51
z = f(z) = tan(log(z))
Screenshot from 2016-02-25 14:51:49
z = f(z) = tan(z + e^z)
Screenshot from 2016-02-25 14:50:34
z = f(z) = tan(z + sin(z))

Of course we could play around with more and more combinations of functions, as well as different coloring schemes, and different shading schemes, but now it’s time to move on to bigger and better things.

Note: The image that shows at the top of the page is z = f(z) = sin(z + tan(e^z)).

 

 

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