zmap: Draw a contour map centered at the origin in a square region 6.3 units high and wide of the analytic function of z defined in the function script fun.m. The map is drawn in figure 1. It is an magnitude map, indicated by color and black level lines; superimposed are white contours of phase, spaced by 30 degrees. Indicated in yellow and red are estimates of the singularities and the branch cuts as positioned by matlab defaults. If the user wishes to displace the cuts, he or she may be able to trick the matlab algorithm. For example, to get the square-root cut to run up the imaginary axis write:
y = (1 - i)*sqrt(i*z/2);
in fun.m. This may not be possible with more complicated functions like the Hankel function.
The command zmap must be the first one issued before any of the others in the catalog can be used. If fun.m needs to be modified, a separate window should be opened, and the script edited there, without having to stop matlab.
solid: In figure 2, draw a fishnet altitude plot of the function magnitude |f(z)| in the same region as the zmap graph. This command draws on arrays computed by zmap. It can be viewed from a variety of angles with the command turn.
zview: In figure 3 draw two altitude plots, one for the real part the other the imaginary part of the function f.
turn: Rotate in 30 degree increments the most recently drawn altitude plot.
zooom: Recompute and redraw the zmap magnitude-phase plot on a larger scale, with center unaltered. The size of the viewing box is reduced by a factor of two-thirds by this command The native matlab command zoom still works, but it does not recompute the function, and so no more detail will be added as the viewer closes in on a feature of interest.
pullback: Reverse the effect of zooom.
pan : Recompute and redraw the zmap plot centered on a new point that is within the current viewing window. The new center is specified by double-clicking on the desired point. If the pointer is left in the viewing window, another center can be chosen without re-issuing the pan command, so that the user can conveniently focus the center of interest again. This command modifies the 2-vector center.
integrate: Perform a complex contour integration numerically over a path drawn on the zmap graph with the mouse. The path may be closed, if the first and last points are close together, or open if the program decides they are far apart. The instructions for mouse input are printed to the terminal and will not be repeated here.
zlab:
A do-nothing script containg a comments that can be accessed by the
matlab help system, as in
help zlab
More detailed information on each command can also be found by
entering the command name in the help call; for exmaple
help integrate
The documentation that you are now reading is the most extensive
however.
fun : This is not a zlab command, but a required function which zmap refers to in order to get the complex function under investigation. An example of such a function might be
function y = fun(z)
% Integrand of a simple Fourier transform
y = exp(i*z) ./ (1 + z.ˆ2); return
The calculations must be written to accept a complex matrix for the variable z and operations must not be those of linear algebra, but element-by-element. Thus, as shown in this example, z squared must be written z .ˆ 2 not as zˆ2 which is matrix multiply in matlab. The "return" instruction here allows the user to maintain a list of alternative functions below these lines in the script, any one of which can be promoted to the top to be executed when needed. A small library of interesting functions is provided in the zlab distribution.