If $\mathbb F$ is some field, the group $PSL(2, \mathbb F)$ consists of the mappings $$ x \mapsto \frac{ax + b}{cx + d} $$ with $a,b,c,d \in \mathbb F$ and $ad – bc = 1$. These mappings are defined over the extended field $\mathbb F_{\infty} := \mathbb F \cup \{\infty\}$ (or the so called projective […]

I’m trying to show that a specific Möbius transformation exists, where I have some points that map to some other points. (I don’t wanna be too specific here about what goes where, as I don’t wanna run the risk of having the insight in solving this myself spoiled). How does one go about proving the […]

Given that $\Gamma_0(6)$ is generated by matrices in the set $$G = \left\{\left(\begin{array}{cc} 1 & 1\\ 0 & 1 \end{array}\right), \left(\begin{array}{cc} 7 & -3\\ 12 & -5 \end{array}\right),\left(\begin{array}{cc} 5 & -1\\ 6 & -1 \end{array}\right),\left(\begin{array}{cc} -1 & 0\\ 0 & -1 \end{array}\right) \right\}$$ Show that $$ \eta(\tau) := e^{\frac{\pi iz}{12}}\prod_{n=1}^{\infty} (1-q^n) $$ where $q=e^{2\pi iz}$, […]

Let $S$ denote the Riemann Sphere. Recall that a Mobius transformation is a function $f:S \to S$ defines as $z \to \frac {az+b}{cz+d}$ where $a,b,c,d \in \mathbb C$ with $ad-bc=1$. What is the motivation to study Mobius Transformation? Why should one look at the map defined in the above way?

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 transformations of the form $z\mapsto \frac{az+b}{-\overline{b}z+\overline{a}}$, with $|a|^2+|b|^2=1$. I want to see the proof of the following theorem: Every finite group of Möbius transformations is conjugate to a subgroup of $PSU(2,\mathbb{C})$. […]

What’s the image of the first quadrant $Rez\ge0$ and $Imz\ge0$ under transformation $f(z)=(i-z)/(i+z)$? I know that real axis is mapped to the unit circle, $f(0+i*0)=1$ and $f(\infty)=-1 $.

[Note that this is a reference request; I already know a couple of routine ways to prove the identity.] In April I posted this answer. Then yesterday I had occasion to conjecture that in general $$ \left(\frac{p\sin x + q\cos x}{r\sin x + s\cos x} = \frac{p\tan x + q}{r\tan x + s} \right) = […]

This previous question had me thinking about something I’ve taken for granted. Consider $\mathrm{SL}_2(\mathbb{C})$ acting on $\mathbb{C}^2$, which descends to an action on the complex projective line $\mathbb{CP}^1$, which we may think of as the Riemann sphere $\widehat{\mathbb{C}}=\mathbb{C}\cup\{\infty\}$. It acts by Möbius transformations. By stereographic projection, $\widehat{\mathbb{C}}$ may be identified with a literal sphere $\mathbb{S}^2\subset […]

Say I wish to prove that every möbius transformation of the unit disk onto itself can be written in the form $A(z) = e^{i\theta}\frac{z+a}{1+\bar{a}z}$, where $\theta$ is a real number and $a$ is a point in the unit disk which is defined to be $\mathbb{D} = \{z \in \mathbb{C} : |z|<1\}$. Now I already know […]

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