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.

458 questions
59 views

73 views

Prove: If f is a bijective meromorphic function from the Riemann sphere to itself, then f is a Mobius transformation. [duplicate]

Knowing that a meromorphic function f requires f to be either analytic or has a pole at every point in its domain, what does bijective meromorphic mean?
70 views

If points of mobius transformation are given, then how to determine the mapping?

A Mobius transformation is a map $$f(x)=\frac{rx+s}{tx+u}$$ where $ru-st \neq 0$. Suppose we have $f(a)=c, f(b)=d, f(c)=a, f(d)=b$, where $a,b,c,d \in \mathbb{R}$. Then from here, the answer given by ...
41 views

Constructing a Fractional Linear Map

I am working on a practice prelim question: "Construct a nonlinear fractional map $\phi(z) = \frac{az+b}{cz+d}$ with $c \ne 0$ such that $\phi(\phi(\phi(z))) = z$. I feel like I just need to take ...
270 views

Constructing a Mobius transformation that acts on any two points of the upper half complex plane:

I would like to construct a Mobius transformation that sends any two points $z_1$ and $z_2$ from the upper half of the complex plane to i and to $iR^+$, i.e., given any two points $z_1$ and $z_2$, ...
148 views

350 views

Finite groups of Möbius Transformations

Let $M_2(\mathbb{C})$ be the group of all Möbius transformations $z\mapsto \frac{az+b}{cz+d}$ from $\mathbb{C}\cup\{ \infty\}$ to itself. Let $PSU(2,\mathbb{C})$ be the group of all Möbius ...
96 views

The image of a specific Mobius transformation

Let $f:D\rightarrow \mathbb{C} :f(z)=\frac{z}{z-1}$ and $D= \{ z:|z|=1\}$ ,what is the image of $f$, $f(D)$? Can one elaborate on some general methods of dealing with these kind of questions?
2k views

what is a Möbius transformation with no fixed points?

is there a Mobius transformation with no fixed points? I have the equation $$\frac{az+b}{cz+d}=z\implies cz^2+dz-az-b=0$$ given the fixed points when this is true. So if we set $c\ne 0$ we get two ...
233 views

Is this a Möbius transformation? Line $\mapsto$ hyperbola

I was trying to make a Möbius transformation $M:\Bbb C \to \Bbb C$ with $2$ fixed points. What I did was: Make a degree 2 formula $\frac{az+b}{cz+d}=cz^2+dz-az-b=0$ so $c\ne 0$ and choose $a,b,c,d$...
46 views

Mobius transformation always defined on $f:\Bbb C \to \Bbb C$ or always $f:\hat {\Bbb C} \to \hat {\Bbb C}$

Is a mobius transformation ever defined on $f:\Bbb C \to \Bbb C$ or is it always $f:\hat {\Bbb C} \to \hat {\Bbb C}$? The wikipedia page makes me believe it is the latter, but my assignment has the ...