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]], '''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.

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

Multivalued functions arise commonly in applications of [[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. For example, the [[complex logarithm]] is a multivalued function, as the inverse of the exponential function. It cannot be considerd as an ordinary function, since, when one follow 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]].

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 |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

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''. If ''f'' is an ordinary function, it is a multivalued function by taking its [[Graph of a function|graph]]

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

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

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

== 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>.

==Inverses of functions==

==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>.

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]] 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]] '''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>.

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==Applications==

In physics, multivalued functions play an increasingly important role. They form the mathematical basis for [[Paul Dirac|Dirac]]'s [[magnetic monopole]]s, for the theory of [[Crystallographic defect|defect]]s in crystals and the resulting [[Plasticity (physics)|plasticity]] of materials, for [[vortex|vortices]] in [[superfluid]]s and [[superconductor]]s, and for [[phase transition]]s in these systems, for instance [[melting]] and [[quark confinement]]. They are the origin of [[gauge field]] structures in many branches of physics.{{Citation needed|reason=reliable source needed for the paragraph|date=July 2013}}

== See also ==

* [[Relation (mathematics)]]

* [[Function (mathematics)]]

* [[Binary relation]]

* [[Set-valued function]]

==Further reading==

* [[Hagen Kleinert|H. Kleinert]], ''Multivalued Fields in Condensed Matter, Electrodynamics, and Gravitation'', [https://web.archive.org/web/20080315225354/http://www.worldscibooks.com/physics/6742.html World Scientific (Singapore, 2008)] (also available [http://www.physik.fu-berlin.de/~kleinert/re.html#B9 online])

* [[Hagen Kleinert|H. Kleinert]], ''Gauge Fields in Condensed Matter'', Vol. I: Superflow and Vortex Lines, 1–742, Vol. II: Stresses and Defects, 743–1456, World Scientific, Singapore, 1989 (also available online: [http://users.physik.fu-berlin.de/~kleinert/kleiner_reb1/contents1.html Vol. I] and [http://users.physik.fu-berlin.de/~kleinert/kleiner_reb1/contents2.html Vol. II])

== References ==

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