Plotting Complex functions in Mathematica

In summary, the conversation discusses how complex functions can be seen as mappings from one plane to another. The question is raised if Mathematica can plot these transformations. An example is given of how the function e^z maps a square region in the z-plane to a washer region in the w-plane. The code is then modified to map the annulus 1\leq r\leq 5 in the z-plane under the transformation 1/z\to w into the w-plane.
  • #1
Storm Butler
78
0
In a few of my books on Complex variables they show how you can look at a complex function as essentially a mapping from what plane to another.

Does anyone know if there would be a way to have mathematica plot how a complex function would transform one plane into the other?

Thanks for any help.
 
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  • #2
Storm Butler said:
In a few of my books on Complex variables they show how you can look at a complex function as essentially a mapping from what plane to another.

Does anyone know if there would be a way to have mathematica plot how a complex function would transform one plane into the other?

Thanks for any help.

You can plot a parametric region. For example, the mapping [itex]e^z\to w[/itex] maps the square region in the z-plane to the washer region in the w-plane:

Code:
w[z_] := Exp[z]; 
p1 = ParametricPlot[{x, y}, {x, 1/10, 1}, {y, -Pi, Pi}, AspectRatio -> 1]; 
p2 = ParametricPlot[{Re[w[z]], Im[w[z]]} /. z -> x + I*y, {x, 1/10, 1}, {y, -Pi, Pi}, 
    PlotRange -> All]; 
myarrow = Show[Graphics[{{Arrow[{{-0.5, 0}, {0.5, 0}}]}, 
      Text[Style["w[z]=\!\(\*SuperscriptBox[\(E\), \(z\)]\)", 20], {0, 0.1}]}]]; 
GraphicsGrid[{{p1, myarrow, p2}}]

attachment.php?attachmentid=63414&d=1383077503.jpg


Ok, now modify my code to map the annulus [itex]1\leq r\leq 5[/itex] in the z-plane under the transformation [itex]1/z\to w[/itex] into the w-plane.
 

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1. What is Mathematica and how is it used for plotting complex functions?

Mathematica is a powerful computational software program that can be used for a variety of tasks, including plotting complex functions. It uses mathematical algorithms and equations to generate visual representations of complex functions, allowing scientists to explore and analyze these functions in a more intuitive way.

2. Can Mathematica plot any type of complex function?

Yes, Mathematica has a wide range of built-in functions and features that allow it to plot almost any type of complex function. These include trigonometric, exponential, logarithmic, and polynomial functions, as well as more advanced functions such as Bessel functions and hypergeometric functions.

3. How do I input a complex function into Mathematica for plotting?

To input a complex function into Mathematica, you can use the built-in function "Plot" and specify the function using mathematical notation. For example, to plot the function f(z) = z^2 + 1, you would type "Plot[z^2 + 1, {z, -5, 5}]" into the Mathematica input bar.

4. Can Mathematica plot 3D complex functions?

Yes, Mathematica has the capability to plot 3D complex functions using the function "Plot3D". This allows for a more comprehensive visualization of complex functions, as it includes a third axis for the complex values.

5. Are there any limitations to plotting complex functions in Mathematica?

While Mathematica is a powerful tool for plotting complex functions, there are some limitations. For example, some very large or highly oscillatory functions may not plot accurately due to computational limitations. Additionally, certain functions may require additional parameters or options to be specified in order to plot correctly.

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