# How to find a section of $\mathbb{D}$ in $\text{SU}(1,1)$?

How to find a section of $$\mathbb{D}$$ in $$\text{SU}(1,1)$$? What article or book do you recommend to learn about $$\text{SU}(1,1)$$?

$$\text{SU}(1,1)=\left\{\left( \begin{array}{ccc} \alpha & \beta \\\overline\beta & \overline\alpha \end{array} \right)\mid \alpha,\beta\in \mathbb C,|\alpha|^2-|\beta|^2=1\right\}$$

I think that if $$z\in\mathbb{D}$$, $$z\neq0$$ then a section is: $$\left( \begin{array}{ccc} \frac{z}{|z|} & 0 \\\ 0 & \frac{\bar z}{|z|} \end{array} \right)\left( \begin{array}{ccc} \frac{1}{(1-|z|^2)^{\frac{1}{2}}} & \frac{|z|}{(1-|z|^2)^{\frac{1}{2}}} \\\ \frac{|z|}{(1-|z|^2)^{\frac{1}{2}}} & \frac{1}{(1-|z|^2)^{\frac{1}{2}}} \end{array} \right)$$

I have problems when $$z=0$$, Is this a section? how can i prove it?

I need to find a section of the form:

$$\left( \begin{array}{ccc} e^{i\theta(z)} & 0 \\\ 0 & e^{-i\theta(z)} \end{array} \right)\left( \begin{array}{ccc} C(z) & S(z) \\\ S(z) & C(z) \end{array} \right)$$

Such that: $$C^2(z)-S^2(z)=1$$, $$C(z)\geq 1$$, $$S(z)\geq 0$$