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Complex Plot Information |
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Development Information Complex Plot was developed on a number of different platforms and compilers. The current Windows 95 version was compiled with Microsoft's Visual C++ using the Windows version of the V GUI library, a cross-platform user-interface design framework. This allows Cplot to be ported from system to system using exactly the same source code; for information on how to do this, see the General Notes in the Complex Plot Download section. A Macintosh version of Cplot has also been released; although the function of the program is essentially the same, the Mac version does not use V (there is currently no version of V available for MacOS) and therefore its source code is significantly different. In addition, there are a few interface differences that affect a few of the program features (e.g., zooming). These changes are reflected in the Mac version of the Cplot Manual. For the purposes of this site, the version of Cplot developed using V and currently compiled for Windows 95 will be refered to the 'Windows' version and its source code the 'Windows' source code (even though the code should be platfrom-independent with V). The version created only for the MacOS will of course be referred to as the Mac version. Operation Information Complex Plot is a mathematical application that plots the images of a specified domain (an annular or rectangular region) under any kind of common complex-valued function of a complex variable. The program utilizes an inline parser so that the user can dynamically interact with the plotting environment and plot multiple functions at the same time. However, Cplot is unique because of its ability to plot several types of intertwining maps. For an example we will discuss one of these intertwining maps, called the Koenigs eigenfunction after the mathematician Gabriel Koenigs. Koenigs proved that for an analytic(complex-differentiable) function f mapping the unit disc into itself, having 0 as an attractive fixed point, and having f'(0) not equal 0, that there exists a function g that satisfies the equation g°f = cf, where c is a complex scalar. In addition, if we define f[n] to be the n-th iterate of f (e.g., f ° f ° f would be the 3rd iterate of f), Koenigs proved that sequence f [n](z) / f '(0)n converges uniformly to g. We term g the Koenigs eigenfunction of f. Thus Koenigs has given us a numerical method of plotting the output points under these kinds of eigenfunctions, and that is exactly what Cplot does. Similar iterative processes exist for the three other types of intertwining maps that Cplot handles, and Cplot uses those iteration formulas in a similar manner. For more complete information, see the Complex Plot manual or several of the papers cited under Related Stuff. |