the monodromy theorem
/ muh-NOD-roh-mee /
We have just seen that continuing a function from a to b can give different answers along different paths. So when can we trust that the answer does NOT depend on the path — when does continuation produce an honest single-valued function on a whole region? The monodromy theorem gives the clean condition: if you can continue your function element along every path inside a region, and the region is simply connected (no holes), then continuation is path-independent, and the result is a single, well-defined holomorphic function on the entire region.
The mechanism is homotopy. Two paths from a to b are homotopic if one can be deformed continuously into the other while keeping the endpoints fixed and staying inside the region. The heart of the theorem is: continuation along two HOMOTOPIC paths gives the same endpoint element, provided continuation is possible along every path in the deforming family. Intuitively, as you slide one path slowly toward the other, the chain of disks slides too, and at each tiny step the uniqueness of continuation forces the answer not to jump — so it cannot have changed by the time you finish deforming. Now add 'simply connected': in such a region ANY two paths with the same endpoints are homotopic, because there are no holes to get stuck on. Hence all routes from a to b agree, and the function is single-valued.
This is the theorem that rescues continuation from chaos. It tells you exactly where multivaluedness can hide: only in regions with holes, and only because a path can loop around a hole in a way that cannot be deformed away. The logarithm is single-valued and well-defined on any simply connected region avoiding 0 (for example the plane slit along the negative real axis), but multivalued on the punctured plane, whose single hole at 0 lets a loop pick up an extra 2 pi i. The theorem is also the bridge to covering spaces: the obstruction to single-valuedness is exactly the fundamental group of the region.
On the slit plane (the plane with the ray from 0 to -infinity removed) you cannot loop around 0, so any two paths between two points are homotopic there. The monodromy theorem then guarantees Log z continues to a single-valued holomorphic function on the whole slit plane — this is exactly the principal branch. Remove the slit and the theorem no longer applies: a loop around 0 changes the value by 2 pi i.
No holes means all routes agree. Cut the plane to kill the bad loops and the logarithm becomes single-valued.
Simple connectedness is essential, not decorative. The theorem fails on regions with holes, which is not a defect — it is precisely how genuinely multivalued functions like log and sqrt arise.