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Multivalued function

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Generalized mathematical function

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This article is about multivalued functions as they are considered in mathematical analysis. For set-valued functions as considered in variational analysis, seeset-valued function.

Not to be confused withMultivariate function.

Multivalued function {1,2,3} → {a,b,c,d}.

Inmathematics, amultivalued function,^[1]multiple-valued function,^[2]many-valued function,^[3] ormultifunction,^[4] is a function that has two or more values in its range for at least one point in its domain.^[5] It is aset-valued function with additional properties depending on context; though some authors do not distinguish between set-valued functions and multifunctions.^[6]

Amultivalued function of setsf : X → Y is a subset

\Gamma _{f}\ \subseteq \ X\times Y.

Writef(x) for the set of thosey ∈Y with (x,y) ∈Γ_f. Iff is an ordinary function, it is a multivalued function by taking itsgraph

\Gamma _{f}\ =\ \{(x,f(x))\ :\ x\in X\}.

They are calledsingle-valued functions to distinguish them.

Motivation

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Main article:Global analytic function

The term multivalued function originated in complex analysis, fromanalytic continuation. It often occurs that one knows the value of a complexanalytic function $f(z)$ in someneighbourhood of a point $z=a$ . This is the case for functions defined by theimplicit function theorem or by aTaylor series around $z=a$ . In such a situation, one may extend the domain of the single-valued function $f(z)$ along curves in the complex plane starting at $a {\displaystyle a}$ . In doing so, one finds that the value of the extended function at a point $z=b$ depends on the chosen curve from $a {\displaystyle a}$ to $b {\displaystyle b}$ ; since none of the new values is more natural than the others, all of them are incorporated into a multivalued function.

For example, let $f(z)={\sqrt {z}}\,$ be the usualsquare root function on positive real numbers. One may extend its domain to a neighbourhood of $z=1$ in the complex plane, and then further along curves starting at $z=1$ , so that the values along a given curve vary continuously from ${\sqrt {1}}=1$ . Extending to negative real numbers, one gets two opposite values for the square root—for example±i for−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 fornth roots,logarithms, andinverse trigonometric functions.

To define a single-valued function from a complex multivalued function, one may distinguish one of the multiple values as theprincipal 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 ofRiemann surfaces: to consider a multivalued function $f(z)$ as an ordinary function without discarding any values, one multiplies the domain into a many-layeredcovering space, amanifold which is the Riemann surface associated to $f(z)$ .

Inverses of functions

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Iff : X → Y is an ordinary function, then its inverse is the multivalued function

\Gamma _{f^{-1}}\ \subseteq \ Y\times X

defined asΓ_f, viewed as a subset ofX ×Y. Whenf is adifferentiable function betweenmanifolds, theinverse function theorem gives conditions for this to be single-valued locally inX.

For example, thecomplex logarithmlog(z) is the multivalued inverse of the exponential functione^z :C →C^×, with graph

\Gamma _{\log(z)}\ =\ \{(z,w)\ :\ w=\log(z)\}\ \subseteq \ \mathbf {C} \times \mathbf {C} ^{\times }.

It is not single valued, given a singlew withw = log(z), we have

\log(z)\ =\ w\ +\ 2\pi i\mathbf {Z} .

Given anyholomorphic function on an open subset of thecomplex planeC, itsanalytic continuation is always a multivalued function.

Concrete examples

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Everyreal number greater than zero has two realsquare roots, so that square root may be considered a multivalued function. For example, we may write ${\sqrt {4}}=\pm 2=\{2,-2\}$ ; although zero has only one square root, ${\sqrt {0}}=\{0\}$ . Note that ${\sqrt {x}}$ usually denotes only the principal square root of $x {\displaystyle x}$ .
Each nonzerocomplex number has two square roots, threecube roots, and in generalnnth roots. The onlynth root of 0 is 0.
Thecomplex logarithm function is multiple-valued. The values assumed by $\log(a+bi)$ for real numbers $a {\displaystyle a}$ and $b {\displaystyle b}$ are $\log {\sqrt {a^{2}+b^{2}}}+i\arg(a+bi)+2\pi ni$ for allintegers $n {\displaystyle n}$ .
Inverse trigonometric functions are multiple-valued because trigonometric functions are periodic. We have $\tan \left({\tfrac {\pi }{4}}\right)=\tan \left({\tfrac {5\pi }{4}}\right)=\tan \left({\tfrac {-3\pi }{4}}\right)=\tan \left({\tfrac {(2n+1)\pi }{4}}\right)=\cdots =1.$ As a consequence, arctan(1) is intuitively related to several values:π/4, 5π/4, −3π/4, and so on. We can treat arctan as a single-valued function by restricting the domain of tanx to−π/2 <x <π/2 – a domain over which tanx is monotonically increasing. Thus, the range of arctan(x) becomes−π/2 <y <π/2. These values from a restricted domain are calledprincipal values.
Theantiderivative can be considered as a multivalued function. The antiderivative of a function is the set of functions whose derivative is that function. Theconstant 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 functions. 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 apartial inverse of the original function.

Branch points

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Main article:Branch point

Multivalued functions of a complex variable havebranch points. For example, for thenth root and logarithm functions, 0 is a branch point; for the arctangent function, the imaginary unitsi and −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 abranch cut, a kind of curve that connects pairs of branch points, thus reducing the multilayeredRiemann surface of the function to a single layer. As in the case with real functions, the restricted range may be called theprincipal branch of the function.

Applications

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In physics, multivalued functions play an increasingly important role. They form the mathematical basis forDirac'smagnetic monopoles, for the theory ofdefects in crystals and the resultingplasticity of materials, forvortices insuperfluids andsuperconductors, and forphase transitions in these systems, for instancemelting andquark confinement. They are the origin ofgauge field structures in many branches of physics.^{[citation needed]}

References

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^"Multivalued Function".archive.lib.msu.edu. Retrieved2024-10-25.
^"Multiple Valued Functions | Complex Variables with Applications | Mathematics".MIT OpenCourseWare. Retrieved2024-10-25.
^Al-Rabadi, Anas; Zwick, Martin (2004-01-01)."Modified Reconstructability Analysis for Many-Valued Functions and Relations".Kybernetes.33 (5/6):906–920.doi:10.1108/03684920410533967.
^Ledyaev, Yuri; Zhu, Qiji (1999-09-01)."Implicit Multifunction Theorems".Set-Valued Analysis Volume.7 (3):209–238.doi:10.1023/A:1008775413250.
^"Multivalued Function".Wolfram MathWorld. Retrieved10 February 2024.
^Repovš, Dušan (1998).Continuous selections of multivalued mappings. Pavel Vladimirovič. Semenov. Dordrecht: Kluwer Academic.ISBN 0-7923-5277-7.OCLC 39739641.

v t e Function
History List of specific functions
Types by domain and codomain	X → 𝔹 𝔹 → X 𝔹ⁿ → 𝔹 X → ℤ ℤ → X X → ℝ ℝ → X ℝⁿ → X X → ℂ ℂ → X ℂⁿ → X
Classes/properties	Constant Identity Linear Polynomial Rational Algebraic Analytic Smooth Continuous Measurable Injective Surjective Bijective
Constructions	Restriction Composition λ Inverse
Generalizations	Relation (Binary relation) Set-valued Multivalued Partial Implicit Space Higher-order Morphism Functor
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