Multivalued function

Summary

A multivalued function from a set $X$ to a set $Y$ assigns to each $x \in X$ a subset $F(x)$ of $Y$, and $F(x)$ has two or more elements for at least one $x$. In complex analysis, the main examples come from the inverses of functions that are not injective. For example, each $z \neq 0$ has two square roots, and it has infinitely many logarithms and arguments.

Remarks

A branch of a multivalued function $F$ on a set $U$ is a function $f$ with $f(x) \in F(x)$ for each $x \in U$, usually continuous or holomorphic. A principal branch is one branch that a standard convention chooses. Riemann surfaces give a way to treat a multivalued function as one single-valued function.

Explanations

L1 human

The equation $w^2 = 4$ has two solutions, $2$ and $-2$, so the square root of $4$ has two values. A multivalued function keeps all the values together, and a branch picks one value at each point.