Inverse to (z+1/z)/2

The inverse of (z+1/z)/2

If we let the computer use the principal branch of the square root, here is the inverse of (z+1/z)/2:

z+sqrt(z^2-1)

What is going on near zero? i.e. why did the computer draw it this way? However, it is possible to choose the branch of the square root so that the inverse is analytic on C\[-1,1]. The trick is to rewrite the inverse as

z(1+sqrt(1-1/z^2))

Here is the other branch of the inverse:

z(1-sqrt(1-1/z^2))

Can you explain why this branch is darker than the first branch? What will the picture look like if we choose the inverse to be analytic on C\((-inf,-1] U [1,+inf))? How can you rewrite the function so that the colors change continuously in this domain? Answer.