All the complex analysis graphics displayed were made using Mathematica. There is a package called ComplexMap.m among the standard Mathematica packages which shows the image of a Cartesian or polar grid under a complex mapping. However, I chose to write my own package, complexcurves.m. This shows the (static) image under a complex mapping of an arbitrary collection of curves in the plane, the curves being specified parametrically. Unlike ComplexMap.m, complexcurves.m uses Mathematica's adaptive plotting routines, and therefore curves come out looking curved, rather than as polygonal approximations. It also includes a variety predefined curves and curve families. On the other hand, it does not include as sophisticated handling of singularities as ComplexMap.m. Instructions for complexcurves.m are in the comments at the top of the file.
Once we can display the image of curve collection, animating it is easy. We simply create a frame showing the image of the curves under various maps, save these frames to files (I use GIF format). To display a sequence of GIF files as an animation I use the VarAnimator java class, which is Dave Torok's extension of Herb Jellinek's Animator class. Alternatively, we can merge the files into one big animated GIF (e.g., with GIFMerge, or use any of a variety of programs which can animate sequences of GIF files or convert them into other animation formats.
A simple example of the process is the Mathematica package homotopyframes.m, which makes the GIF files for a linear homotopy animation.
The following Mathematica files were used, in conjunction with the packages above, to make the animations: