Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 2 / 24F / Solution

An analytic continuation along $\alpha$ is a chain of function elements $(f_t,U_t)$ covering the path, beginning with the germ of $(f,U)$ at $\alpha (0)$, such that neighboring elements agree on the component of overlap met by the path. The terminal element is $(g,V)$.

The Monodromy theorem says that if continuation is possible along every path in $D$, then continuations along endpoint-fixed homotopic paths have the same terminal germ. It follows by subdividing a homotopy square into small rectangles on which the identity theorem identifies neighboring germs. Hence

The requested final example with homotopic paths and different terminal values cannot exist under these hypotheses; it contradicts the theorem just proved. With “non-homotopic” in place of “homotopic”, take the germ of $\log z$ near $1$ in ${\mathbb{C}}^{\ast }$. Continuation along the constant path gives $0$, whereas continuation once counterclockwise around zero gives $2\pi i$.

Solved by gpt-5.6-sol high.