Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

I'm looking for some software that my help me to graph some complex functions on unit circle. I.e. let say if I have $\ f(z)=1/(1-z)$ I want to see to give an input an image with unit circle and want to get the transformed image of unit circle with $\ f(z)$ function.

Can anybody suggest some grapher for this, or something similar ?

share|improve this question
    
Try Sage, sagemath.org/doc/reference/sage/plot/complex_plot.html –  user31373 Sep 12 '12 at 4:30
add comment

2 Answers

up vote 0 down vote accepted

Follow this guide to Sage: to using Sage Online if you don't want to install Sage on your computer.

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

enter image description here

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

enter image description here

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 surface that is desired we see a color-graph. I have asked that question.

share|improve this answer
add comment

In the case of Möbius transformations, you don't need any software. Consider the map $f : \overline{\mathbb{C}}_z \to \overline{\mathbb{C}}_w$ given by $w = (1-z)^{-1}$. It follows that $z = (w-1)w^{-1}$. If $|z| = 1$ then $|(w-1)w^{-1}| = 1$ and so $|w| = |w-1|$. The image of the unit circle is the perpendicular bisector of $w=0$ and $w=1$, i.e. the line parallel to the imaginary axis that passes through $w = \frac{1}{2}$.

In general, if $f : \overline{\mathbb{C}}_z \to \overline{\mathbb{C}}_w$ is given by

$$w = \frac{az + b}{cz + d} \, . $$

where $(a:b:c:d) \in \mathbb{CP}^3$ then

$$z = \frac{dw-b}{cw-a} \, . $$

The image of the unit circle is given by setting $|z| = 1$ and so $|cw-a| = |dw-b|$ is the equation of the image in the $w$-sphere.

share|improve this answer
    
the example function in my question is just a sample. I need visually see the transformation of unit circle with any complex function. So that why I need software –  deimus Sep 11 '12 at 9:55
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.