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

Let X and Y be sets and denote by Ysymm the mth symmetric power of Y.


A function f:XYsymm 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 spaceMathworldPlanetmath (resp. ) 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 YsymmYm (resp. symmm).

With this definition z is a holomorphic multifunction (or a 2-function), into sym2.

Define the multigraphMathworldPlanetmath of f to be the set:


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


  • 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