# 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. $\u2102$)
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. ${\u2102}_{sym}^{m}\cong {\u2102}^{m}$).

With this definition $\sqrt{z}$ is a holomorphic multifunction (or a 2-function), into ${\u2102}_{sym}^{2}.$

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 ${\u2102}^{2}.$ In general, with the aid of the Weierstrass preparation theorem we can realize any codimension 1 analytic set in ${\u2102}^{n}$ as a multigraph over ${\u2102}^{n-1}.$ 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 |