complexification of $SO(2)$ While computing the complexification of Lie group $SO(2)$, I get the result is all the matrix of the following form 
$$\left(\begin{array}{cc}
\frac{e^{t-\sqrt{-1}\theta}+e^{-t+\sqrt{-1}\theta}}{2} & \frac{e^{-t+\sqrt{-1}\theta}-e^{t-\sqrt{-1}\theta}}{2\sqrt{-1}} \\
-\frac{e^{-t+\sqrt{-1}\theta}-e^{t-\sqrt{-1}\theta}}{2\sqrt{-1}}& \frac{e^{t-\sqrt{-1}\theta}+e^{-t+\sqrt{-1}\theta}}{2} \\
\end{array}
\right),$$
for $t$ and $\theta$ are both real numbers. I want to know what are this matrices like, or there is another description. Specially $SO(2)$ can act on $\mathbb{C}$ as a rotation? How could its complexification acts on $\mathbb{C}$ naturally?

Consider the map $\mathbb{C}^{\ast}\rightarrow SO_2(\mathbb{C})$ given by
$$
t\mapsto \begin{pmatrix}
\frac{t+t^{-1}}{2} & \frac{i(t-t^{-1})}{2} \cr
-\frac{i(t-t^{-1})}{2} &  \frac{t+t^{-1}}{2} 
\end{pmatrix}
$$
This is a group isomorphism. So we have a simple description.
