The monodromy theorem states that if a complex function 
is analytic
in a disk contained in a simply
connected domain 
and admits analytic continuation
along every path in 
starting in that disk, then it extends
to a single-valued function that is analytic on all of 
.
The more general homotopy form assumes that an initial function element at 
can be continued along every path
in a domain 
starting at 
. If two paths with the same endpoints
are homotopic in 
while their endpoints remain fixed, they produce the same
final function element (Honda n.d., p. 71).
In a simply connected domain,
every pair of such paths is homotopic,
giving the first statement. The proof subdivides a homotopy
into small pieces on which local uniqueness of analytic
continuation applies.
Both hypotheses matter. The complex plane is simply connected, but a local series
expansion of 
cannot be continued through its pole at 
. On the punctured plane,
a local branch of 
can be continued along every path,
but continuation around zero changes its sign. The latter is an example of nontrivial
monodromy.
