It is common practice among complex analysts to speak of multiple valued functions in contexts of “functions” such as This somewhat informal notion can be made very precise when the “function” has finitely many values (as the does).
Let and be sets and denote by the symmetric power of
A function is called a multifunction, or an -function from to , where is the multiplicity.
We can think of the value of at any point as a set of (or fewer) elements. Let be a topological space (resp. ) A multifunction is said to be continuous (resp. holomorphic) if all the elementary symmetric polynomials of the elements of are continuous (resp. holomorphic). Equivalently, is continuous (resp. holomorphic) if it is continuous (resp. holomorphic) as functions to (resp. ).
With this definition is a holomorphic multifunction (or a 2-function), into
Define the multigraph of to be the set:
The multigraph of is the corresponding Riemann surface imbedded in In general, with the aid of the Weierstrass preparation theorem we can realize any codimension 1 analytic set in as a multigraph over 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.
|Date of creation||2013-03-22 17:42:08|
|Last modified on||2013-03-22 17:42:08|
|Last modified by||jirka (4157)|
|Defines||multiple valued function|