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 of the elements of $f$ are continuous (resp. holomorphic). Equivalently, $f$ is continuous (resp. holomorphic) if it is continuous (resp. holomorphic) as functions to $Y_{sym}^m\cong Y^m$ (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.

Title	multifunction
Canonical name	Multifunction
Date of creation	2013-03-22 17:42:08
Last modified on	2013-03-22 17:42:08
Owner	jirka (4157)
Last modified by	jirka (4157)
Numerical id	4
Author	jirka (4157)
Entry type	Definition
Classification	msc 32A12
Synonym	m-function
Related topic	SymmetricPower
Related topic	WeierstrassPolynomial
Related topic	MultivaluedFunction
Defines	multigraph
Defines	multiple valued function