.. status: ok



10.8 Drawing Complex Functions
------------------------------

Here is another way to graph a complex function of complex arguments.
For each complex value z, compute f(z), again expressing the value in
polar coordinates . We draw the complex valued function, again
considering the (x,y)-plane as the complex plane, using r as the height
(or z-coordinate) and θ as the color. This is a standard plot---we
learned how to do this in Chapter
`ugGraph <section-7.0.html#ugGraph>`__ --- but here we write a new
program to illustrate the creation of polygon meshes, or grids.

Call this function drawComplex. It displays the points using the mesh of
points. The function definition is in three parts.


.. spadVerbatim

::

 drawComplex: (C -> C, S, S) -> VIEW3D
 drawComplex(f, realRange, imagRange) ==                The first part

  delReal := (hi(realRange)-lo(realRange))/realSteps   The real step size

  delImag := (hi(imagRange)-lo(imagRange))/imagSteps   The imaginary step size

                                          Initial list of list of points $llp$
   llp:List List Point DFLOAT := []



Variables delReal and delImag give the step sizes along the real and
imaginary directions as computed by the values of the global variables
realSteps and imagSteps. The mesh is represented by a list of lists of
points llp, initially empty. Now [] alone is ambiguous, so to set this
initial value you have to tell FriCAS what type of empty list it is.
Next comes the loop which builds llp.


.. spadVerbatim

::


  real := lo(realRange)                            The initial real value
   for i in 1..realSteps+1 repeat                   Begin real iteration

    imag := lo(imagRange)                          initial imaginary value

    lp := []$(List Point DFLOAT)                  initial list of points $lp$

    for j in 1..imagSteps+1 repeat                 Begin imaginary iteration

      z := f complex(real,imag)                    value of $f$ at the point
       pt := point [real,imag, clipFun sqrt norm z, Create a point
                    argument z]

      lp := cons(pt,lp)                            Add the point to $lp$

      imag := imag + delImag                       The next imaginary value
     real := real + delReal                         The next real value
     llp := cons(lp, llp)                           Add $lp$ to $llp$



The code consists of both an inner and outer loop. Each pass through the
inner loop adds one list lp of points to the list of lists of points
llp. The elements of lp are collected in reverse order.


.. spadVerbatim

::


  makeViewport3D(mesh(llp), "Complex Function")    Create a mesh and display



The operation mesh then creates an object of type
ThreeSpace(DoubleFloat) from the list of lists of points. This is then
passed to makeViewport3D to display the image.

Now add this function directly to your vectors.input file and re-read
the file using read vectors. We try drawComplex using a user-defined
function f.

Read the file.


.. spadInput
::

	)read vectors


.. spadMathAnswer
This one has a pole at z=0.


.. spadInput
::

	f(z) == exp(1/z)


.. spadMathAnswer
Draw it with an odd number of steps to avoid the pole.



drawComplex(f,-2..2,-2..2)





|picture|






.. |picture| image:: ps/complexExp.png