Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 2 / 7E / Solution

Choose paths from $0$ to $z$ in the cut plane and deform them continuously with the endpoint, never crossing the specified cut. With $0\le \arg z<2\pi$, use the sheet reached from the positive real axis; when an endpoint is moved into the left half-plane, the path passes around the upper endpoint $i$ of the cut. The cut makes all such admissible paths homotopic with fixed endpoints, so the integral is single valued and its endpoint derivative is $(1+z^2{)}^{-1/2}$; hence it is analytic.

For $z=-\sinh u$, continue the endpoint counterclockwise from the positive real axis through the upper half-plane. Inthe continued square root equals $-\cosh u$ at $z=-\sinh u$. The logarithm is reached with argument $\pi$, and therefore

The identitiesshow directly why continuation without an argument restriction is multivalued. The complete set of values of the multivalued inverse hyperbolic sine at $\sinh u$ is

Solved by gpt-5.6-sol high.