multifunction

It is common practice among complex analysts to speak of multiple valued functions in contexts of “functions” such as $\sqrt{z}.$ This somewhat informal notion can be made very precise when the “function” has finitely many values (as the $\sqrt{z}$ does).

Let $X$ and $Y$ be sets and denote by$Y_{sym}^m$ the $m^{\text{th}}$ symmetric power of $Y.$

Definition.

A function $f:X\to Y_{sym}^m$ is called a multifunction,or an $m$-function from $X$ to $Y$, where $m$ is the multiplicity.

We can think of the value of $f$ at any point as a set of $m$ (or fewer) elements.Let $Y$ be a topological space (resp. $ℂ$)A multifunction is said to be continuous (resp. holomorphic) if all the elementary symmetric polynomials ofthe elements of $f$ are continuous (resp. holomorphic). Equivalently, $f$ is continuous (resp. holomorphic)if it is continuous (resp. holomorphic) as functions to $Y_{sym}^m\cong Y^m$(resp. ${ℂ}_{sym}^m\cong {ℂ}^m$).

With this definition $\sqrt{z}$ is a holomorphic multifunction (or a 2-function), into ${ℂ}_{sym}^2.$

Define the multigraph of $f$ to be the set:

$$\{(x,y)\mid X\times Y\mid y\in f(x)\}.$$

The multigraph of $\sqrt{z}$ is the corresponding Riemann surface imbedded in ${ℂ}^2.$ In general, with the aid of the Weierstrass preparation theorem we can realize any codimension 1 analytic set in ${ℂ}^n$ as a multigraph over ${ℂ}^{n-1}.$The roots of any Weierstrass polynomial (or in general of any monic polynomial with holomorphic coefficients) are a holomorphic multifunction.