Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 2 / 13E / a

PutThis slit complex plane is a simply connected domain. On $D$ there is one branch of $\sqrt{1-z^2}$ that equals $1$ at zero, and its reciprocal is holomorphic. The primitive of a holomorphic function on a simply connected domain therefore makes the integral from zero path-independent, so $G$ is single-valued.

For unrestricted paths, going once around either branch point changes the sign of the square root. The resulting analytic continuation therefore need not return to the original germ, and the value of $g(z)$ can depend on the path.

The endpoint values of the principal branch areBy the monodromy reflection at a square-root branch point, continuation around $-1$ acts on a value $w$ aswhile continuation around $1$ acts as

The contour $C_1$ makes the first continuation and then follows the principal path to $z$, soThe contour $C_2$ continues successively around $-1$ and $1$. Hence

Every path can change sheets only by applying the two reflectionsTheir products satisfyThus an even number of reflections gives a translation by an integral multiple of $2\pi$, while an odd number gives a translated reflection. All possible values are thereforein agreement with the branches of the complex inverse sine.