0
votes
2answers
49 views

conformal map/Mobius transformation from annulus to $\mathbb{C}\setminus \overline{D(0,1)}$

Does there exist a conformal bijection/Mobius transformation from the open unit disk to the whole complex plane? Does there exist a conformal bijection/Mobius transformation from the annulus $\{z\in ...
0
votes
0answers
42 views

Fundamental Domain for Congruence (mod 2) Group

How can I show that the area between the circles $|z|=1$, $|z+\frac{1}{2}|=\frac{1}{2}$, $|z-\frac{1}{2}|=\frac{1}{2}$ in the upper-half plane (here's a picture) is a fundamental domain for the ...
3
votes
2answers
125 views

Showing the function $f(x,y)$ is one by one

Yesterday, while teaching geometry, I was faced to a problem saying that the function below is an distance function: $$d(P,Q)=\Big|\ln\frac{\frac{x_1-c+r}{y_1}}{\frac{x_2-c+r}{y_2}}\Big|$$ where in ...
3
votes
3answers
150 views

A question about hyperbolic functions

Suppose $(x,y,z),(a,b,c)$ satisfy $$x^2+y^2-z^2=-1, z\ge 1,$$ $$ax+by-cz=0,$$ $$a^2+b^2-c^2=1.$$ Does it follow that $$z\cosh(t)+c\sinh(t)\ge 1$$ for all real number $t$?
4
votes
4answers
248 views

Completeness of Upper Half Plane

I am trying to prove that the upper half plane, defined as $\mathbb{H} = \{z \in \mathbb{C} : \Im(z)>0 \}$, is complete with respect to the hyperbolic metric. First I note that if I have some ...