Graphing the complex function

Question

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 ?

Answer 1

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.