Editing Multivalued function

{{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}}

[[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 }}</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 
:<math> \Gamma_f\ \subseteq \ X\times Y.</math>
Write ''f(x)'' for the set of those ''y'' ∈ ''Y'' with (''x,y'') ∈ ''Γ<sub>f</sub>''. If ''f'' is an ordinary function, it is a multivalued function by taking its [[Graph of a function|graph]]
:<math> \Gamma_f\ =\ \{(x,f(x))\ :\ x\in X\}.</math>
They are called '''single-valued functions''' to distinguish them.

== Motivation ==
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}}—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==

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<sub>f</sub>'', 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<sup>z</sup>'' : '''C''' → '''C'''<sup>×</sup>, 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>.
*[[Inverse trigonometric function]]s are multiple-valued because trigonometric functions are periodic. We have <math display="block">
\tan\left(\tfrac{\pi}{4}\right) = \tan\left(\tfrac{5\pi}{4}\right)
= \tan\left({\tfrac{-3\pi}{4}}\right) = \tan\left({\tfrac{(2n+1)\pi}{4}}\right) = \cdots = 1.
</math> As a consequence, arctan(1) is intuitively related to several values: {{pi}}/4, 5{{pi}}/4, −3{{pi}}/4, and so on. We can treat arctan as a single-valued function by restricting the domain of tan ''x'' to {{nowrap|−{{pi}}/2 < ''x'' < {{pi}}/2}} – a domain over which tan ''x'' is monotonically increasing. Thus, the range of arctan(''x'') becomes {{nowrap|−{{pi}}/2 < ''y'' < {{pi}}/2}}. These values from a restricted domain are called ''[[principal value]]s''.
* The [[antiderivative]] can be considered as a multivalued function. The antiderivative of a function is the set of functions whose derivative is that function. The [[constant of integration]] follows from the fact that the derivative of a constant function is 0.
*[[Inverse hyperbolic functions]] over the complex domain are multiple-valued because hyperbolic functions are periodic along the imaginary axis. Over the reals, they are single-valued, except for arcosh and arsech.

These are all examples of multivalued functions that come about from non-[[injective function]]s. Since the original functions do not preserve all the information of their inputs, they are not reversible. Often, the restriction of a multivalued function is a [[partial inverse]] of the original function.

== Branch points ==
{{Main articles|Branch point}}
Multivalued functions of a complex variable have [[branch point]]s. For example, for the ''n''th root and logarithm functions, 0 is a branch point; for the arctangent function, the imaginary units ''i'' and &minus;''i'' are branch points. Using the branch points, these functions may be redefined to be single-valued functions, by restricting the range. A suitable interval may be found through use of a [[branch cut]], a kind of curve that connects pairs of branch points, thus reducing the multilayered [[Riemann surface]] of the function to a single layer. As in the case with real functions, the restricted range may be called the ''principal branch'' of the function.

==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 ==
{{Reflist}}

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