A note on complex functions and the graphing

When we deal with real numbers, we can easily represent them on a 1-dimensional line. When we deal with pairs of real numbers (like coordinates on a plane), we can represent them on a 2-dimensional plane.

However, complex numbers are a bit trickier. Each complex number is made up of two real numbers (the real part and the imaginary part), so we would need two dimensions just to represent the input to the function. The output of the function is also a complex number, so we would need another two dimensions to represent the output. That's a total of four dimensions, which are difficult to represent on a flat screen or piece of paper.

The way we get around this is by using colors to represent the output of the function. We use one dimension to represent the input to the function (two if we're dealing with a function of a complex variable, like in this case), and we use colors to represent the output.

For complex numbers, we can think of the magnitude (or absolute value) of the number as being represented by the brightness of the color, and the phase (or angle) of the number as being represented by the hue of the color.

In these plots:

  • Brightness corresponds to the magnitude (absolute value) of the function's output. Higher magnitudes are brighter, and lower magnitudes are darker.
  • Hue (color) corresponds to the phase (angle) of the function's output. Different phases are represented by different colors.

This way, we can visualize complex functions, which involve four dimensions, in a two-dimensional image, with the help of colors.