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 $Y^m_{sym}$ the $m^{{th}}$ symmetric power of $Y.$

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. ${\mathbb{C}}$ A multifunction is said to be continuous (resp. holomorphic) if all the elementary symmetric polynomials of the elements of $f$ are continuous (resp. holomorphic). Equivalently, $f$ is continuous (resp. holomorphic) if it is continuous (resp. holomorphic) as functions to $Y^m_{sym} \cong Y^m$ (resp. ${\mathbb{C}}^m_{sym} \cong {\mathbb{C}}^m$ .

With this definition $\sqrt{z}$ is a holomorphic multifunction (or a 2-function), into ${\mathbb{C}}^2_{sym} .$ Define the multigraph of $f$ to be the set: \begin{equation*} \{ (x,y) \mid X \times Y \mid y \in f(x) \} . \end{equation*} The multigraph of $\sqrt{z}$ is the corresponding Riemann surface imbedded in ${\mathbb{C}}^2.$ In general, with the aid of the Weierstrass preparation theorem we can realize any codimension 1 analytic set in ${\mathbb{C}}^n$ as a multigraph over ${\mathbb{C}}^{n-1} .$ The roots of any Weierstrass polynomial (or in general of any monic polynomial with holomorphic coefficients) are a holomorphic multifunction.

Bibliography

1

Hassler Whitney. Complex Analytic Varieties. Addison-Wesley, Philippines, 1972.