# Conformal map from $U_r = \mathbb{C} \backslash (-\infty,r]$ to the unit disc

Let $U_r = \mathbb{C} \backslash (-\infty,r]$. Find a conformal map from $U_r$ to the unit disc.

Idea: I can conformally map $U_r$ to $U_0$ with the transformation $z\mapsto z-r$. So we can write every element in the form $z= r e^{i\theta}$ with $|\theta| < \pi$ (branch cut). Then I can apply the map $z \mapsto z^{1/2}$. This maps $U_0$ to right half plane which can be conformally mapped (via $z\mapsto e^{i\pi/2}z$) to the upper half plane. Then I can use the conformal map $F(z) = \frac{i-z}{i+z}$ to map the upper half plane to the disc. Is my idea correct or is there any mistake?