Graphing Compex Functions 3D (x,y,i axes) Instead Of Color-Coded (SAGE).

Question

Following this guide to Sage: and using Sage Online produced the following graphs:

Graphing $\frac{1}{1-z}$ that way yeilds:

Graphing $\frac{1}{1-z^2}$ that way yields:

It would be nice to see it in 3D instead of merely color coded. The y-axis is coming out of the picture toward us and instead of seeing the 3D surface (in x,y,i coordinates) we see a color-graph on the x-i plane.

Answer 1

You can understand complex functions better by expanding in terms of real variables. For example, let's take a look at the function $$\frac{1}{1+z^2}$$ by letting $z=x+iy$, where $x$ and $y$ are real. Then $$\frac{1}{1+(x+iy)^2}=\frac{1}{1+x^2-y^2+2xyi}=\frac{1+x^2-y^2-2xyi}{(1+x^2-y^2)^2+4x^2y^2},$$ where we have multiplied numerator and denominator of the middle expression by the complex conjugate of the denominator to make the denominator purely real. You can then separate to obtain $$\frac{1}{1+z^2}=\frac{1+x^2-y^2}{(1+x^2-y^2)^2+4x^2y^2}-i\frac{2xy}{(1+x^2-y^2)^2+4x^2y^2}.$$

Notice how the complex function is broken into real and imaginary components.

$$\text{Re}\left(\frac{1}{1+z^2}\right)=\frac{1+x^2-y^2}{(1+x^2-y^2)^2+4x^2y^2}$$

$$\text{Im}\left(\frac{1}{1+z^2}\right)=\frac{-2xy}{(1+x^2-y^2)^2+4x^2y^2}$$

ADDED BY OTHER USER: AntonioVargas shared the following plot from Mathematica of $\text{Re}(1/(1+z^2))$ colored according to $\text{Im}(1/(1+z^2))$:

However it is just as easy to graph it in Wolfram Alpha

The graph the Re(1/(1+z^2) on Wolfram Alpha!

Also the 3D plot of the imaginary component.

Answer 2

This is the codes in which you can visualize the complex functions in Maple's environment:

  [> with(plots):
  [> f := z-> 1/(1-z):
     g:=z-> 1/(1-z^2):
  [> complexplot3d(f, -2-2*I .. 2+2*I);
     complexplot3d(g, -2-2*I .. 2+2*I);

Answer 3

To the best of my knowledge, Sage at present does not offer 3d plots with color determined by a function. (But I'm not a Sage expert like those who inhabit ask Sage Q&A site.) As Babak S. pointed out, Maple does the job, for those who have access to it. However, I think the following picture (made with a different Maple command) looks nicer:

f:=1/(1-(x+I*y)^2):  
plot3d(abs(f), x=-3..3, y=-3..3, color=argument(f), grid=[50,50]);

Answer 4

The simplest way is to merely replace z with "x+iy" and hit enter in wolfram alpha:

$$ 1/(1+(x+iy)^2)$$

or for 1/(1-(x+iy)^2)

1/(1-x^2+y^2-2xyi) times