# 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 $Y^{m}_{sym}$ the $m^{\text{th}}$ symmetric power of $Y.$

###### Definition.

A function $f\colon X\to Y^{m}_{sym}$ 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. ${\mathbb{C}}$) 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 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 ${\mathbb{C}}^{2}.$ 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.

## References

• 1 Hassler Whitney. . Addison-Wesley, Philippines, 1972.
 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