Monodromy reflection at a square-root branch point
= Monodromy reflection at a square-root branch point
Let
$$
W(z)=\int_{z_0}^z\frac{q(t)}{\sqrt{P(t)}}\,dt,
$$
where $P$ 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 $W'$. The continued primitive agrees at $a$ with the original one, so it is
$$
W\longmapsto 2A-W.
$$