arccos(z)

Before exploring this multiple-valued "function", it might help to first look at the inverse of (z+1/z)/2, if you have not already done so.

If we let the computer pick the branch, here is the inverse of cos(z):

arccos(z)


The explanation for the discontinuity in the coloring in the above picture is the same as the explanation for the inverse of (z+1/z)/2. To get a better picture, note that cos(z) is the composition of (z+1/z)/2 with exp(iz). If we use the alternate formula for the inverse of (z+1/z)/2 (explained earlier), we obtain:

arccos(z)= -i(log(z(1+sqrt(1-1/z^2))))


Another branch is given by:

arccos(z)= -i(log(z(1-sqrt(1-1/z^2))))


What will some other branches look like?