Plotting complex maps as z-plane and w-plane

Question

I have seen many plots of complex maps as colors, such as $w = sin(z) = 0$:

However, I am looking for more involved plotting capabilities. For example I would like the ability to see the z-plane and w-plane side by side (for the map $w=f(z)$. Then I would like to be able to select a region of the z-plane and see what it got mapped to in the w-plane.

In my example above, I know that in the semi-strip: $\dfrac{-\pi}{2} \leq x \leq \dfrac{\pi}{2}, y \geq 0$ that the vertical lines get mapped to hyperbolas and the horizontal lines get mapped to elipses. It would be nice to plot this and see the semi-strip in the z-plane and its image in the w-plane.

1. What software/tools can I look into for these capabilities?

Note, that I use Sage and know a little about many programming languages (I am not afraid to customize an existing tool)

Thanks for all the help.

Answer 1

The conformal command of Maple is it. Also complexplot and complexplot3d and transform may be useful.

PS. For example, the command $$plots:-conformal(sin(z), z = -(1/2)*Pi .. (1/2)*Pi+2*I, grid = [8, 8], numxy = [50, 50]) $$ produces

Answer 2

https://github.com/search?l=C%2B%2B&q=complex+functions&ref=cmdform&type=Repositories

http://www.codeproject.com/Articles/80641/Visualizing-Complex-Functions

http://webs.um.es/gvb/AC/FC.zip

Maybe you find what you're looking in the first two links, the third is in Spanish and need Ms-Dos to run, I remember that was possible choosing strips.

But without doubt the best method is programming directly with help of some graphics library and feel the freedom of making what you need at any moment. I use to work with C and OpenGL and works really fast. You can insert parameters, modify them in real time and make what you want. That's pretty fun.