Visualization of complex functions by animation

Douglas N. Arnold

This is a collection of flash animations for visualizing maps of one complex variable. Each animation shows the image of a domain in the complex plane as it morphed by a given function. In mathematical terms we display the image of the domain under a homotopy of maps ranging from the identity to the given function, so the first frame of the animation shows the domain and the last frame shows its image under the function. The domain is colored with a pattern to make it easier to follow the homotopy.

• function: f(z)=z², domain: unit circle
• function: f(z)=z², domain: two squares
• function: f(z)=z², domain: unit square (colored with my picture)
• function: f(z)=e^z, domain: [-2,2]x[-π,-π]
• function: f(z)=e^z, domain: [-2,2]x[0,2π]
• function: f(z)=e^z, domain: [-2,2]x[-2π,2π]
• function: a cubic polynomial, domain: a square

These animations were made with Mathematica. The Mathematica notebook is available.

This work is licensed under a Creative Commons Attribution-Noncommercial-Share Alike 3.0 License.