Multivalued function: Difference between revisions

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{{Short description|Generalized mathematical function}}

{{More footnotes needed|date=January 2020}}

{{About|multivalued functions as they are considered in mathematical analysis|set-valued functions as considered in variational analysis|set-valued function}}{{distinguish|Multivariate function}}In [[mathematics]], a '''multivalued function''', also called '''multifunction''' and '''many-valued function''', is a [[set-valued function]] with continuity properties that allow considering it locally as an ordinary function.

In [[mathematics]], a '''multivalued function''' is a [[set-valued function]] with additional properties depending on context.

Multivalued functions arise commonly in applications of the [[implicit function theorem]], since this theorem can be viewed as asserting the existence of a multivalued function. In particular, the [[inverse function]] of a [[differentiable function]] is a multivalued function, and is single-valued only when the original function is [[monotonic]]. For example, the [[complex logarithm]] is a multivalued function, as the inverse of the exponential function. It cannot be considered as an ordinary function, since, when one follows one value of the logarithm along a circle centered at {{math|0}}, one gets another value than the starting one after a complete turn. This phenomenon is called [[monodromy]].

A ''multivalued function'' of sets ''f : X → Y'' is a subset

Another common way for defining a multivalued function is [[analytic continuation]], which generates commonly some monodromy: analytic continuation along a closed curve may generate a final value that differs from the starting value.

:<math> \Gamma_f\ \subseteq \ X\times Y.</math>

Write ''f(x)'' for the set of those ''y'' ∈ ''Y'' with (''x,y'') ∈ ''Γ_f''. Ordinary functions are multivalued functions by taking their [[Graph of a function|graph]]. They are called '''single-valued functions''' to distinguish them.

If ''f : X → Y'' is an ordinary function, then its inverse the multivalued function

:<math> \Gamma_{f^{-1}}\ \subseteq \ Y\times X</math>

defined as ''Γ_f'', viewed as a subset of ''X'' × ''Y''. When ''f'' is a [[differentiable function]] between [[Manifold|manifolds]], the [[Inverse function theorem|inverse function theorem]] gives conditions for this to be single-valued locally in ''X''.

For example, the [[complex logarithm]] ''log(z)'' is the multivalued inverse of the exponential function ''e^z'' : '''C''' → '''C'''^×, with graph

:<math> \Gamma_{\log(z)}\ =\ \{(z,w)\ :\ w=\log (z)\}\ \subseteq\ \mathbf{C}\times\mathbf{C}^\times.</math>

It is not single valued, given a single ''w'' with ''w = log(z)'', we have

:<math>\log(z)\ =\ w\ +\ 2\pi i \mathbf{Z}.</math>

Given any [[Holomorphic function|holomorphic]] function on an open subset of the [[Complex plane|complex plane]] '''C''', its [[analytic continuation]] is always a multivalued function.

Multivalued functions arise also as solutions of [[differential equation]]s, where the different values are parametrized by [[initial condition]]s.

The terms '''multifunction''' and '''many-valued function''' are sometimes also used.

== Motivation ==