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{SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 59 "The Maple Commands confor
mal, complexplot and complexplot3d" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT 256 25 "Mth 417 Complex Analysis " }}{PARA 0 "" 
0 "" {TEXT 257 30 "Spring 1998 - Bent E. Petersen" }}{PARA 0 "" 0 "" 
{TEXT 258 13 "April 6, 1998" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 286 "Maple has a rich selection of plot commands. T
hree of them, conformal, complexplot, complexplot3d, are designed spec
ifically for dealing with complex numbers and functions. Many of the o
thers could be pressed into service with some ingenuity, for example, \+
contourplot and contourplot3d." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 243 "We must load the \+
plots package in order to have all plot features enabled. We use a col
on to suppress the output, though if you want to see what commands are
 enabled by the plots package, you can terminate the command with a se
micolon instead." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT 259 21 "The conformal command" }
}{PARA 0 "" 0 "" {TEXT -1 85 "The conformal command plots the image of
 a rectangular grid under a complex function." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 56 "Let's start with somethin
g really simple - the identity:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "conformal(z,z=1-2*I..3+2*I, \+
grid=[21,41]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 291 "We see Maple plotted the values of the identity fun
ction in a square with lower left corner 1-2I and upper right corner 3
+2I along 21 vertical lines and 41 horizontal lines. The scales along \+
the two axes are different. We can have the same horizontal and vertic
al scales by adding an option:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "conformal(z,z=1-2*I..3+2*I, \+
grid=[21,41], scaling=constrained,title=`Constrained`);" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 33 "There are
 numerous other options." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 267 "Well, lets plot something a bit more interesting \+
than the identity. We will plot the polynomial z^2 evaluated on the li
nes indicated above. What will result will be a family of curves which
 we think of as the image under the squaring map of the lines indicate
d above." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 79 "conformal(z^2,z=1-2*I..3+2*I, grid=[21,41], scaling=`
constrained`,title=`z^2`);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 222 "Here we have used constrained scaling so
 we can see that the contours are orthogonal. We can see quite well ho
w the real part of z^2 becomes increasingly negative as the imaginary \+
part of z tends to plus or minus infinity." }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 444 "It is unfortunate that conform
al is restricted to plotting the images of vertical and horizontal lin
es, since often we want to consider are the images of other curves, fr
equently cirles. One way to do this is to use complexplot (see next se
ction) to plot individual curves. Another way is to plot a suitable co
mposition, where the first function maps the rectangular grid to the d
esired grid and the second one is the function we want to study." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 143 "Let's lo
ok at the images of circles for example. First we define a family of c
ircles with center at the origin - the exponential is convenient:" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 13 "h:=w->exp(w);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "conf
ormal(h,-2-Pi*I..1+Pi*I,grid=[11,21], scaling=`constrained`,numxy=[15,
40]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 273 "The numxy command specifies the number of points to be p
lotted in each direction, that is, it is a measure of the resolution. \+
If you do not specify it then you may get some circles with a lot of f
lat parts (the default is numxy=[15,15]). For example, let's try [5,5]
 above:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 77 "conformal(h,-2-Pi*I..1+Pi*I,grid=[11,21], scaling=`co
nstrained`,numxy=[5,5]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 109 "Ouch! That is bad - but no worse than on
e might expect. One should usually set numxy to an appropriate value.
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 43 "Note \+
a point style graph is also available:" }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 93 "conformal(h,-2-Pi*I..1+
Pi*I,grid=[11,21], scaling=`constrained`,numxy=[15,40],style=`point`);
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
70 "Now let's look at the values of  sin(z) on circles as explained ab
ove:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 101 "conformal(sin(h(w)),w=-2-Pi*I..1+Pi*I,grid=[11,21], \+
scaling=`constrained`,numxy=[80,80],thickness=2);" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 321 "If you experiment
 you will see that we need to set numxy quite high to get an acceptabl
e image. Thus the command will be very slow, but at least we are no lo
nger limited to looking at conformal images of boring grid lines! Stil
l, it might be interesting to look at what the sine function looks lik
e on a rectangular grid:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "conformal(sin,-2-2*I..2+2*I, grid=[
21,21], scaling=`constrained`, thickness=2);" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 9 "Note that" }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "Re(Sin
(x+I*y))=evalc(Re(sin(x+I*y))); Im(Sin(x+I*y))=evalc(Im(sin(x+I*y)));
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
77 "Thus the closed curves above are ellipses and the open curves are \+
hyperbolae." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "
" {TEXT 260 23 "The complexplot command" }}{PARA 0 "" 0 "" {TEXT -1 
99 "The complexplot command plots complex valued functions of a real v
ariable, that is, complex curves." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "complexplot(sin(x+I),x=-Pi..
Pi/2,thickness=3);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 40 "Let's look \+
at some other complex curves:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "g:=t->(2+I*t)^2;" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "complexplot(g(t),t=-2..4, scaling=`
constrained`,thickness=3);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 126 "It is not difficult to see this curve is
 a parabola. Suppose we wanted to look at the exponential along this p
arabola. We have" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 77 "complexplot(exp(g(t)),t=-2..4, scaling=`const
rained`,color=blue,thickness=3);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 "Very nice!" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "
" {TEXT 261 25 "The complexplot3d command" }}{PARA 0 "" 0 "" {TEXT -1 
105 "The complexplot3d command draws 3 dimensional representations of \+
complex functions of a complex variable." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "complex
plot3d(z^2, z = 1 - 2*I .. 3 + 2*I,axes=boxed,title=`z^2`);" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 383 "W
hat is being plotted here is the modulus (absolute value or magnitude)
 of z^2, but with an interesting twist - the graph is colored accordin
g to the values of the argument of z^2. It's hard to interpret, but ma
y be useful sometimes. This command has numerous options and is polymo
rphic, that is, alters its behavior when called with different types o
f arguments. See the Maple help." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "37" 0 }
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