Definition:Branch Point

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.