3D Complex Function Graphs

To fully graph a complex function w = f(z), which maps from ℂ → ℂ, it requires four dimensions. The input z has a real and imaginary part, and so does the output w.

This approach is to slice through the real axis of the input, graphing the transformed imaginary lines of each slice. The real axis of the input becomes the X axis of the 3D graph, and the real axis of the output becomes the Y axis of the 3D graph. (The XY plane is the familiar real function graph.) The vertical axis is the imaginary axis of the output, and the color hue represents the imaginary axis of the input. The animation at the top shows a graph of the function w = z²

Mathematically, we expect to see quadratic curves:

z² = (Re(z) + iIm(z))² = Re(z)² - Im(z)² + i(2Re(z)Im(z))

If you fix the real component for a specific slice, and then vary the imaginary component, you'd expect to see -Im(z)² in the real output, and a linear change in the imaginary output, which describes a quadratic curving left. You can see these quadratics on each slice in the animation.

Or, if you fix the imaginary component and vary the real component, you'd expect to see Re(z)² in the real output, and again a linear change in the imaginary output, which describes a quadratic curving right. You can see this in the colors of the 3D graph. The lines of a constant color follow quadratic curves to the right.

And, of course, the sanity check is the familiar graph of y = x², which should happen when z is real. To see it, rotate the graph below to look straight down with the red axis (X) pointing right and green (Y) pointing up.

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Here are interactive graphs of some interesting functions. You can make your own with the 3D Complex Function Grapher webpage.