# Graphing the complex function

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 ?

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–  user31373 Sep 12 '12 at 4:30

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:

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

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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.

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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