Multivalued function: Difference between revisions

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[[File:Multivalued_function.svg|thumb|Multivalued function {1,2,3} → {a,b,c,d}.]]

In [[mathematics]], a '''multivalued function''',<ref>{{Cite web |title=Multivalued Function |url=https://archive.lib.msu.edu/crcmath/math/math/m/m450.htm |access-date=2024-10-25 |website=archive.lib.msu.edu}}</ref> '''multiple-valued function''',<ref>{{Cite web |title=Multiple Valued Functions {{!}} Complex Variables with Applications {{!}} Mathematics |url=https://ocw.mit.edu/courses/18-04-complex-variables-with-applications-fall-1999/pages/study-materials/multiple-valued-functions/ |access-date=2024-10-25 |website=MIT OpenCourseWare |language=en}}</ref> '''many-valued function''',<ref>{{Cite journal |last1=Al-Rabadi |first1=Anas |last2=Zwick |first2=Martin |date=2004-01-01 |title=Modified Reconstructability Analysis for Many-Valued Functions and Relations |url=https://pdxscholar.library.pdx.edu/sysc_fac/30/ |journal=Kybernetes |volume=33 |issue=5/6 |pages=906–920 |doi=10.1108/03684920410533967}}</ref> or '''multifunction''',<ref>{{Cite journal |last1=Ledyaev |first1=Yuri |last2=Zhu |first2=Qiji |date=1999-09-01 |title=Implicit Multifunction Theorems |url=https://scholarworks.wmich.edu/math_pubs/22/ |journal=Set-Valued Analysis Volume |volume=7 |issue=3 |pages=209–238|doi=10.1023/A:1008775413250 |url-access=subscription }}</ref> is a function that has two or more values in its range for at least one point in its ___domain.<ref>{{cite web |title=Multivalued Function |url=https://mathworld.wolfram.com/MultivaluedFunction.html |website=Wolfram MathWorld |access-date=10 February 2024}}</ref> It is a [[set-valued function]] with additional properties depending on context; some authors do not distinguish between set-valued functions and multifunctions,<ref>{{Cite book |last=Repovš |first=Dušan |url=https://www.worldcat.org/oclc/39739641 |title=Continuous selections of multivalued mappings |date=1998 |publisher=Kluwer Academic |others=Pavel Vladimirovič. Semenov |isbn=0-7923-5277-7 |___location=Dordrecht |oclc=39739641}}</ref> but English Wikipedia currently does, having a separate article for each.

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

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:<math> \Gamma_f\ =\ \{(x,f(x))\ :\ x\in X\}.</math>

They are called '''single-valued functions''' to distinguish them.

== Distinction from set-valued relations ==

[[File:Multivalued_functions_illustration.svg|thumb|right|600px|Illustration distinguishing multivalued functions from set-valued relations according to the criterion in page 29 of ''New Developments in Contact Problems'' by Wriggers and Panatiotopoulos (2014).]]

Although other authors may distinguish them differently (or not at all), Wriggers and Panatiotopoulos (2014) distinguish multivalued functions from set-valued relations (also called [[Set-valued function|set-valued functions]]) by the fact that multivalued functions only take multiple values at finitely (or denumerably) many points, and otherwise behave like a [[Function (mathematics)|function]].<ref name=":0">{{Cite book |last1=Wriggers |first1=Peter |url=https://books.google.com/books?id=R4lqCQAAQBAJ |title=New Developments in Contact Problems |last2=Panatiotopoulos |first2=Panagiotis |date=2014-05-04 |publisher=Springer |isbn=978-3-7091-2496-3 |pages=29 |language=en}}</ref> Geometrically, this means that the graph of a multivalued function is necessarily a line of zero area that doesn't loop, while the graph of a set-valued relation may contain solid filled areas or loops.<ref name=":0" />

== Motivation ==

{{Main|Global analytic function}}

The term multivalued function originated in complex analysis, from [[analytic continuation]]. It often occurs that one knows the value of a complex [[analytic function]] <math>f(z)</math> in some [[neighbourhood (mathematics)|neighbourhood]] of a point <math>z=a</math>. This is the case for functions defined by the [[implicit function theorem]] or by a [[Taylor series]] around <math>z=a</math>. In such a situation, one may extend the ___domain of the single-valued function <math>f(z)</math> along curves in the complex plane starting at <math>a</math>. In doing so, one finds that the value of the extended function at a point <math>z=b</math> depends on the chosen curve from <math>a</math> to <math>b</math>; since none of the new values is more natural than the others, all of them are incorporated into a multivalued function.

For example, let <math>f(z)=\sqrt{z}\,</math> be the usual [[square root]] function on positive real numbers. One may extend its ___domain to a neighbourhood of <math>z=1</math> in the complex plane, and then further along curves starting at <math>z=1</math>, so that the values along a given curve vary continuously from <math>\sqrt{1}=1</math>. Extending to negative real numbers, one gets two opposite values for the square root—for example {{math|±''i''}} for {{math|–1−1}}—depending on whether the ___domain has been extended through the upper or the lower half of the complex plane. This phenomenon is very frequent, occurring for [[nth root|{{mvar|n}}th roots]], [[logarithm]]s, and [[inverse trigonometric function]]s.

To define a single-valued function from a complex multivalued function, one may distinguish one of the multiple values as the [[principal value]], producing a single-valued function on the whole plane which is discontinuous along certain boundary curves. Alternatively, dealing with the multivalued function allows having something that is everywhere continuous, at the cost of possible value changes when one follows a closed path ([[monodromy]]). These problems are resolved in the theory of [[Riemann surface]]s: to consider a multivalued function <math>f(z)</math> as an ordinary function without discarding any values, one multiplies the ___domain into a many-layered [[Branched covering|covering space]], a [[manifold]] which is the Riemann surface associated to <math>f(z)</math>.

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

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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.

==Concrete examples==

*Every [[real number]] greater than zero has two real [[square root]]s, so that square root may be considered a multivalued function. For example, we may write <math>\sqrt{4}=\pm 2=\{2,-2\}</math>; although zero has only one square root, <math>\sqrt{0} =\{0\}</math>. Note that <math>\sqrt{x}</math> usually denotes only the principal square root of <math>x</math>.

*Each nonzero [[complex number]] has two square roots, three [[cube root]]s, and in general ''n'' [[nth root|''n''th roots]]. The only ''n''th root of 0 is 0.

*The [[complex logarithm]] function is multiple-valued. The values assumed by <math>\log(a+bi)</math> for real numbers <math>a</math> and <math>b</math> are <math>\log{\sqrt{a^2 + b^2}} + i\arg (a+bi) + 2 \pi n i</math> for all [[integer]]s <math>n</math>.