What is the principle branch of the complex function $\sqrt{1+z}+\sqrt{1-z}$? I know that it can be written as follows:
$$e^{\frac{1}{2}(\ln{|1+z|}+i\arg{(1+z)})} + e^{\frac{1}{2}(\ln{|1-z|}+i\arg{(1-z)})}$$
But now I'm confused how to find the principal branch from here; I think I have to find the values of $z$ such that neither $\arg$ is multivalued, but I'm not sure. How do you find the principal branch from here?