Definition:Branch Point
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Definition
Let $U \subseteq \C$ be an open set.
Let $f : U \to \C$ be a complex multifunction.
A branch point of $f$ is a point $a$ in $U$ such that:
- $f$ has more than one value at one or more points in every neighborhood of $a$
- $f$ has exactly one value at $a$ itself.
Examples
Cube Root of $z - a$
Let $f: \C \to \C$ be the complex function defined as:
- $\forall z \in \C: \map f z = \paren {z - a}^{1/3}$
for some $a \in \C$.
Then $a$ is a branch point of $f$.
Also see
- Results about branch points can be found here.
Sources
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): branch point
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): singular point (singularity): 1.
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): branch point
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): singular point (singularity): 1.