Cardano3 Package

Complex Function Graphics

Embellished map of log function

Note: The Cardano3 package is only for pre-Version 6 Mathematica and DrawGraphics. If you are working with Version 6 purchase the Presentations package. It now contains all the complex function graphics and with easier usage.

The Cardano3 package provides routines for many different graphical representations of complex functions. There is a uniform interface for the various plot types and the ability to produce custom complex function graphics. The package contains individual Help pages with examples for every command and many extended examples. The package works with and requires the DrawGraphics package and can tap all the functionality of that package.

There are routines that convert the regular 2D Graphics into equivalent complex forms. For example ComplexLine[{z1,z2,z3...}] takes complex numbers for the point coordinates. There are routines for producing one or two panel plots or animations of complex functions. Each panel may be one of the following plot types. Sin mapped to Riemann sphere

1) Cartesian/PolarSurface - Plots the surface s[f[z]] where f is a complex function and s is a real function.
2) Cartesian/PolarCoded3D - Plots the surface s[f[z]] and then puts a colored contour plot on the surface representing r[f[z]] where s and r are two real functions.
3) Cartesian/PolarGrid - Basically the ComplexMap routines from Mathematica, but you can combine multiple grids of different colors.
4) Cartesian/PolarContour - 2D colored contour plots with labeling of all or selected contours.
5) CodedDensity (Domain Coloring) - A density plot using color to code the complex information. There is one color function provided that codes modulus and argument and a template routine for using different colors. The user can also write and use his own color routines.

Domain coloring for 6th order polynomial with
closeup on one zero
6) ComplexVector - Attaches a scaled vector representing the complex value to each of a set of points. The user supplies whatever set of points he wishes.
7) RiemannSphere - Maps any set of graphics primitives, but usually a complex map grid, through a complex function to the Riemann sphere.
The animations are rotation of 3D surfaces and homotopies between two complex functions. Other custom animations may also be produced. All of the plot types also allow the user to add extra graphics primitives to embellish plots.
The ComplexVector, CartesianGrid, PolarGrid and RiemannSphere plots allow multifunctions.

Murray Eisenberg has provided valuable suggestions in implementing this application.

The package requires the 3 Apr 2005 or later DrawGraphics package. A copy of the installation instructions is also in the downloaded zip file.

Cardano3 Installation Instruction 2KB, 3 Apr 2005.
Purchase Cardano3 Package $20, 202KB Packages and Documentation, 19 Jun 2006.

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