Questions tagged [mobius-transformation]

For questions about the geometry, complex-analytic, and group-theoretic properties of the Möbius transformations (linear fractional transformations) $z \mapsto \frac{az + b}{cz + d}$ of the complex plane, which can be identified with the group $PGL(2, \mathbb{C})$, or certain subgroups thereof.

476 questions
Filter by
Sorted by
Tagged with
25 views

64 views

Transform Name for $(1-z)/(1+z)$

I am trying to find the actual name of the transform $f(z) = \frac{1-z}{1+z}$; the transform from the open unit disk to the right half plane. Another variant; the transform from the upper half plane ...
102 views

Möbius transformations and groups

I have two questions regarding Möbius transformations and groups. In my notes there are two statements, which I can't prove/understand why they hold. The subgroup of Möbius Transformations which ...
85 views

Homeomorphism of unit disc onto itself interchanging two points. [duplicate]

I've searched this topic already and found similar question. But it was not fully answered. Actually I know the Möbius transformation of form $\frac{z-a}{1-\bar a z}$ is the automorphism of unit ...
31 views

Why is the map analytic?

The map $$z\mapsto \frac{z-i}{z+i}$$ is an analytic isomorphism of the upper half-plane $\mathbb{H}$ and the unit disc $D=\lbrace w \in \mathbb{C}||w|<1 \rbrace$ . So , what I do not undertand is ...
60 views

Can one replace the Riemann sphere with other objects? What happens?

When doing one complex variable I think we talked about the Riemann Sphere, and how it relates to Möbius transformations, the complex "point at infinity" and other things. Is it possible (and useful) ...
50 views

97 views

Find Möbius transformation for half-plane to unit disk $|w|<1$?

Consider the half-plane depicted in the following figure How can a Möbius transformation that takes that half-plane onto the unit disk $|w|<1$ be found? What are the steps and things to think ...
Find the image under the Mobius transformation $f(z) = 1/z$ of $|z-3i|=1$ [closed]
I am given $|z-3i|=1$ which is a circle with radius $1$ and centre $(0,3i)$ on the complex plane. I want to find the image (to sketch it) under the transformation $1/z$ WITHOUT taking points and ...