Branch point

Remarks

If $n$ turns around $a$ bring each element back to itself for some $n \ge 2$, the branch point is algebraic, and the least such $n$ is its order. Otherwise $a$ is a transcendental branch point; a logarithmic branch point, where no number of turns brings any element back, is the main example. Some authors call $n - 1$ the order instead of $n$. Example: for $F(z) = \sqrt{z}$, continuing the element $e^{(\log z)/2}$ once around $\abs{z} = 1$ multiplies it by $e^{i\pi} = -1$. So $0$ is an algebraic branch point of order $2$, and so is $\infty$, because $\sqrt{1/w}$ also changes sign around $w = 0$. The complex logarithm has logarithmic branch points at $0$ and $\infty$, since each turn adds $2\pi i$. Non-example: $0$ is not a branch point of $1/z$, which is single-valued on $\C \setminus \set{0}$; it is a pole. For a holomorphic map between Riemann surfaces, a point where the map is not locally injective is also called a branch point or ramification point.