Monodromy reflection at a square-root branch point

Letwhere $P$ has a simple zero at $a$, and let $A$ be the limiting value of $W$ at $a$ on one sheet. Analytic continuation once around $a$ changes the sign of the square root and hence of $W^{\prime }$. The continued primitive agrees at $a$ with the original one, so it is

If continuation around two square-root branch points acts on a primitive as $R_A(W)=2A-W$ and $R_B(W)=2B-W$, thenThus pairs of branch-point loops generate additive periods of an inverse function.