 Branch cut

A branch cut is a removed curve or collection of curves used to choose a single-valued branch of a multivalued function of a multivalued complex expression. On $\mathbb{C}\setminus (-\infty ,0]$, choosing $-\pi <\arg z<\pi$ gives the principal logarithm $\mathrm{Log}z=\log |z|+i\arg z$ and hence $\sqrt{z}=\exp (\mathrm{Log}z/2)$.

The limiting values on opposite sides of the negative real axis differ: for $r>0$, the two square-root values approach $i\sqrt{r}$ and $-i\sqrt{r}$. The original algebraic relation $w^2=z$ has no discontinuity; the discontinuity belongs to the selected branch of a multivalued function. Tracking that distinction prevents sign errors in contour integration and motivates a Riemann surface.