# Complex Region using Presentations Package

Just purchased the Presentations package. First sketched this region:

<< Presentations

Draw2D[
{Opacity[0.6],
ComplexRegionDraw[0 <= Re[z] <= 1, {z, -2 - 2 I, 2 + 2 I}]},
Frame -> True,
Axes -> True,
PlotRange -> 2,
FrameLabel -> {Re, Im},
PlotLabel -> z,
ImageSize -> 300
]


And I got the image I wanted.

Next, I want to sketch the image of this region under the exponential function $w=e^z$. I tried this:

Draw2D[
{Opacity[0.6],
ComplexRegionDraw[Exp[z], {z, -2 I, 1 + 2 I}]},
Frame -> True,
Axes -> True,
PlotRange -> 2,
FrameLabel -> {Re, Im},
PlotLabel -> z,
ImageSize -> 300
]


But I got a blank image. Any suggestions?

Looks like I might have an idea.

f[z_] := Exp[z]

Draw2D[
{Opacity[0.6],
ComplexRegionDraw[0 <= Re[z] <= 1, {z, -2 - 2 I, 1 + 2 I},
PlotPoints -> 30] // ComplexMap[f]},
Frame -> True,
Axes -> True,
PlotRange -> All,
FrameLabel -> {Re, Im},
PlotLabel -> z,
ImageSize -> 300
]


And the resulting image:

Of course, what I really wanted was the image of the the strip with $0\le Re[z]\le1$ and $\infty\le Im[z]\le \infty$.

Draw2D[
{Opacity[0.6],
ComplexRegionDraw[0 <= Re[z] <= 1, {z, 0, 1 + 2 Pi I},
PlotPoints -> 30] // ComplexMap[f]},
Frame -> True,
Axes -> True,
PlotRange -> All,
FrameLabel -> {Re, Im},
PlotLabel -> z,
ImageSize -> 300
]


And the resulting image.

• I don't have the package (heard only good things about it!) but from looking at your syntax my guess would be: the first argument in ComplexRegionDraw[Exp[z]... has to be a boolean condition, not a complex function. Therefore, you should replace that by Abs[Exp[z]]<1, or Re[Exp[z]]>0, or some condition along those lines. – Jens Jan 19 '13 at 5:33

I think you really want $-\pi\leq \Im(z) \leq \pi$. Then let the Presentations function ComplexMap do the work for you!

<< Presentations

strip = ComplexRegionDraw[0 <= Re[z] <= 1, {z, -2 - Pi I, 2 + 4 Pi I}];
domain = Draw2D[{Opacity[0.6], strip},
Frame->True, Axes->True, PlotRange->2, FrameLabel->{Re, Im}, PlotLabel->z];
image = Draw2D[{Opacity[0.6], strip // ComplexMap[Exp]},
Frame->True, Axes->True,PlotRange->4, FrameLabel->{Re, Im}, PlotLabel->w];
GraphicsRow[{domain, image}, PlotLabel -> "Mapping vertical strip by exp"]


Note the use of ComplexMap[Exp] in the definition of image; that's how the image is found.

• Nice! And it shows me how to write good code. Thanks. – David Jan 19 '13 at 17:10
• Question: I notice a lot of people paste their Row image. How do you copy that image so that you can import it to this site? – David Jan 19 '13 at 17:50
• I finding this a very helpful introduction to the Complex part of the Presentations package: mathematica-journal.com/data/uploads/2011/10/EisenbergPark.pdf – David Jan 19 '13 at 17:58
• @David, for the row image, first I had downloaded and installed the "SE Uploader" palette from this site -- search the "meta" items from the link at the top of this page for the palette -- then highlighted the entire graphic row, clicked a button on that palette to upload the image (which also places the URL of the upload onto the clipboard), then pasted that URL into my answer here; the image appears! – murray Jan 19 '13 at 19:05
• Of course I have been using the ComplexGraphics part of Presentations to handle complex regions and more general plotting about complex functions. But I even find it expedient to use the ComplexGraphics routines to construct plane figures having nothing explicit to do with complex numbers. Thus if I have to use a bunch of {x, y} points, I find it often easier to write them in x + I y form and use that form in the graphics code. – murray Jan 19 '13 at 19:08