= Branch point
{wiki}
A branch point is a point around which a sufficiently small loop can change a <germ of a holomorphic function> under <analytic continuation>. This is a local condition: a large loop around a regular point might also enclose a distant <branch point>, so that large loop alone cannot classify the regular point. For the <complex logarithm> and square root, every nonzero point has a local <holomorphic> branch, but a loop once around $0$ changes the chosen branch. <Branch Points and Branch Cuts by Rodolfo R. Rosales>[Rosales, section 1, Definition 1.1 and its clarifications]{ref}
For $z=re^{it}$ with fixed $r>0$, start at $t=0$ with values $\sqrt r$ and $\log r$. Their continuations are
$$
w(t)=\sqrt r\,e^{it/2},\qquad L(t)=\log r+it.
$$
One counterclockwise turn gives $w(2\pi)=-\sqrt r$ and $L(2\pi)=\log r+2\pi i$. At $r=4$, the square root therefore changes from $2$ to $-2$, while the logarithm changes from $\log4$ to $\log4+2\pi i$. Two turns restore the square root to $2$; no nonzero number of turns restores the logarithm. This distinguishes a two-sheeted <algebraic branch point> from a <logarithmic singularity>.
A <pole> need not be a <branch point>: $1/z$ is unbounded near $0$, yet $1/(re^{it})=r^{-1}e^{-it}$ returns to its starting value after one turn. Conversely, the square-root values tend to $0$ at their <branch point>. The issue is whether a single <holomorphic> choice survives going around the point, rather than whether the values become large. On the <Riemann sphere>, infinity is also a <branch point> of both $\sqrt z$ and $\log z$: in the local coordinate $\zeta=1/z$, these are $\zeta^{-1/2}$ and $-\log\zeta$, whose continuations around $\zeta=0$ change sign and subtract $2\pi i$, respectively. <Branch Points and Branch Cuts by Rodolfo R. Rosales>[Rosales, section 1, equation 1.3, and Example 2.4]{ref}
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