I was trying to plot some complex functions with branch cuts on mathematica but I have two problems.
1) The function is with z-Sqrt[z-1]*Sqrt[z+1] with z complex. Now Mathematica says that the standard branch cut for the square root is chosen to be ]-inf, 0]. In this case I would expect to see the branch cut only between -1 and 1, but instead the branch cuts do not "cancel out" (mathematicians please don't kill me) before -1 (see picture, I am contour-plotting real and imaginary part of the function). What is happening here?
2) If I try to plot z-Sqrt[z^2-1] I get a very strange function (see picture). I also have a branch cut on the imaginary axis, but this is due to the fact that if z=i, mathematica is forced to calculate the square root of -2, which is on the branch cut for the standard definition of the square root.
Everything kind of fixes if I rotate the branch cuts by multiplying by -1 in the square root and by i outside. Of course now the branch cuts are ]-inf, -1] and [1, +inf[ but the function behaves like it should, but that is not what I want.
This boils down to understand how mathematica treats branch cuts, but I can't figure out why I have these strange behaviors.
Thanks everyone in advance!