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→Concrete examples: Using sqrt(x) on real numbers to denote a multivalued function contradicts common terminology, I think it's reasonable to point this out |
Tito Omburo (talk | contribs) →Motivation: {{Main|Global analytic function}} |
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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
A ''multivalued function'' of sets ''f : X → Y'' is a subset
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== 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|
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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