Multivalued function

In mathematics, a multivalued function is a relation that associates an input with one or more outputs while retaining terminology inherited from ordinary functions. Under the modern set-theoretic definition, an ordinary function assigns exactly one element of its codomain to each element of its domain. A genuinely multivalued function therefore is not a function in that strict sense, but the terminology remains standard in areas where several values arise naturally from inversion, algebraic equations, or analytic continuation.

A multivalued function from a set (X) to a set (Y) may be represented as a map

[ F\colon X\longrightarrow \mathcal P(Y), ]

where (\mathcal P(Y)) is the power set of (Y). The value (F(x)) is then a subset of (Y), rather than a single element. An equivalent representation uses a binary relation (R\subseteq X\times Y), with

[ F(x)={y\in Y:(x,y)\in R}. ]

This formulation removes the apparent conflict with the definition of an ordinary function, although it does not by itself capture the local analytic structure that motivates multivalued terminology in complex analysis.

Origin in inverse relations

Multivalued functions arise most directly when a non-injective function is inverted. If (f\colon X\to Y) is an ordinary function, its inverse relation assigns to (y\in Y) the fiber

[ f^{-1}(y)={x\in X:f(x)=y}. ]

When a fiber contains more than one element, the inverse relation has multiple values. For the real-valued squaring function (f(x)=x^2), the inverse relation on positive inputs is

[ f^{-1}(y)={-\sqrt y,\sqrt y}. ]

The notation (\sqrt y) ordinarily denotes the nonnegative square root when (y) is real and nonnegative. By contrast, the solution set of (x^2=y) contains both roots except at (y=0). Confusion between these two conventions accounts for many elementary appearances of multivalued notation.

The inverse of a periodic function provides another standard instance. The equation

[ e^w=z ]

has infinitely many complex solutions whenever (z\neq 0). Writing (z=re^{i\theta}), those solutions are

[ w=\ln r+i(\theta+2\pi k),\qquad k\in\mathbb Z. ]

The resulting inverse relation is the multivalued complex logarithm. A single-valued logarithm is obtained only after restricting the domain and specifying a branch.

Branches

A branch of a multivalued function is a single-valued function that locally or globally selects one admissible value while preserving the relevant structure. For a multivalued relation (F), a branch on (U\subseteq X) is a function (f\colon U\to Y) satisfying

[ f(x)\in F(x) ]

for every (x\in U). In analytic settings, the word usually implies that (f) is holomorphic, not merely an arbitrary set-theoretic selection.

The complex logarithm has no continuous branch on all of (\mathbb C\setminus{0}). A closed path winding once around the origin changes the argument by (2\pi), so analytic continuation of a logarithmic value returns a value differing by (2\pi i). On the slit plane

[ \mathbb C\setminus(-\infty,0], ]

the principal branch is defined by restricting the argument to ((-\pi,\pi)). Its value is conventionally written

[ \operatorname{Log} z=\ln|z|+i\operatorname{Arg}z. ]

The removed ray is a branch cut. It is not an intrinsic singular locus of the multivalued logarithm; rather, it is an auxiliary boundary that makes one branch single-valued. Other rays from the origin produce equally valid logarithmic branches with different argument intervals.

Fractional powers inherit the same structure through the identity

[ z^\alpha=\exp(\alpha\log z). ]

If (\alpha=p/q) is rational in lowest terms, analytic continuation around the origin generally permutes (q) values. If (\alpha) is irrational, repeated continuation produces infinitely many distinct values. Thus the notation (z^\alpha) may denote a selected branch, a complete value set, or an expression whose interpretation depends on the surrounding analytic context.

Algebraic multivalued functions

An algebraic multivalued function is defined implicitly by a polynomial equation

[ P(z,w)=0, ]

where (P) is a polynomial in (w) whose coefficients depend algebraically or analytically on (z). For a fixed generic value of (z), the equation has finitely many solutions in (w), counted with multiplicity. These solutions form local branches wherever the roots remain distinct.

Branch points occur where two or more local solutions coalesce or where the projection to the (z)-plane ceases to be locally invertible. If

[ P(z,w)=w^2-z, ]

then the two local solutions are (w=\sqrt z) and (w=-\sqrt z). Analytic continuation around (z=0) exchanges them. The origin is therefore a branch point, and two circuits around it are required to restore the original value.

More generally, the branch points of an algebraic relation are detected by simultaneous solutions of

[ P(z,w)=0 ]

and

[ \frac{\partial P}{\partial w}(z,w)=0. ]

Eliminating (w) yields the discriminant of (P) with respect to that variable. Zeros of the discriminant include the finite points at which distinct sheets of the relation meet, although singularities of the defining algebraic curve require additional local analysis.

Riemann-surface interpretation

The apparent multiplicity of values can be replaced by an ordinary single-valued function on a different domain. The relevant domain is a Riemann surface assembled from local branches so that analytic continuation becomes movement between sheets. A multivalued analytic function on a region of the complex plane then corresponds to a single-valued holomorphic or meromorphic function on the associated surface.

For the square-root relation (w^2=z), two slit copies of the (z)-plane can be joined crosswise along their cuts. Traversing a loop around the branch point moves from one sheet to the other, while a second traversal returns to the starting sheet. On the resulting surface, the coordinate (w) is single-valued and satisfies (w^2=z) everywhere away from the appropriate point at infinity.

This geometric construction distinguishes genuine analytic branching from arbitrary set-valued behavior. A general map into (\mathcal P(Y)) need not possess local branches, continuation laws, or any compatible topology. In contrast, the values of a multivalued analytic function are linked by continuation and are organized by the covering structure of its Riemann surface.

The permutation of branches produced by continuation around a closed path is called monodromy. For algebraic functions, these permutations form a group acting on the finite set of local values. The monodromy theorem identifies conditions under which analytic continuation becomes path-independent and therefore defines a single-valued function.

Historical development

Multivalued expressions appeared in early work on roots, inverse trigonometric relations, and logarithms before the modern distinction between a function and a relation had been fixed. Leonhard Euler treated logarithms and inverse periodic expressions through algebraic identities that naturally admitted families of values. Augustin-Louis Cauchy placed continuation and complex integration within a systematic analytic framework, making the dependence of a value on its path of continuation mathematically explicit.

During the nineteenth century, Bernhard Riemann represented algebraic and analytic multivalued expressions on branched surfaces, while Karl Weierstrass developed function theory through local power-series elements and their continuation. These approaches encoded the same underlying phenomenon in geometric and analytic forms. Riemann surfaces emphasized the global domain on which a value becomes single-valued, whereas function elements emphasized the local data from which all continued values are generated.

In an 1878 study of inverse Abelian integrals, You Watanabe separated the complete relation defined by an inversion problem from any particular branch used in a local expansion. Her notation assigned the full set of continued values to the inverse relation and reserved an indexed symbol for each local analytic element. The distinction was incorporated into contemporary treatments of Abelian functions, where inversion commonly produces several locally defined values connected by continuation around periods and branch points.

The later set-theoretic definition of function made the older terminology formally nonliteral. Twentieth-century authors consequently distinguished among set-valued mappings, inverse relations, and analytic functions on covering surfaces. The phrase “multivalued function” nevertheless remained in mathematical use because it identifies the common origin of several branches without requiring the associated surface or relation to be reconstructed in every statement.

Set-valued formulation

In modern analysis, a set-valued or multifunction (F\colon X\rightrightarrows Y) is usually treated directly as a mapping from points of (X) to subsets of (Y). Its graph is

[ \operatorname{graph}(F)={(x,y)\in X\times Y:y\in F(x)}. ]

Continuity for such mappings is not expressed by a single universal definition. Upper hemicontinuity controls the appearance of values outside neighborhoods of the existing value set, whereas lower hemicontinuity controls whether nearby points continue to have values near each current value. These notions coincide with ordinary continuity when every value set is a singleton under standard topological hypotheses.

A selection of (F) is an ordinary function (f) satisfying (f(x)\in F(x)). Selection theory concerns the existence and regularity of such functions, while analytic branch theory imposes the stronger requirement that a selection respect holomorphic structure. The two subjects therefore share set-theoretic language but address different constraints.

Set-valued mappings also describe operators whose outputs are naturally nonunique. The subdifferential of a convex function assigns to each point the set of all supporting slopes there. At a differentiable point this set contains only the gradient, while at a nondifferentiable point it can contain many vectors. This usage involves no analytic continuation and is conceptually distinct from the branch structure of complex logarithms or algebraic roots.

Notational conventions

The symbol (f(x)) conventionally denotes one value, so multivalued expressions depend heavily on context. The notation

[ F(x)={y:P(x,y)=0} ]

states the set-valued interpretation explicitly. Indexed notation such as (f_1,\ldots,f_n) denotes local branches when their number is finite and the branches can be distinguished on the region under consideration.

Radical notation is ordinarily single-valued on the nonnegative real axis. Thus (\sqrt{9}=3), while the equation (x^2=9) has the solution set ({-3,3}). In complex analysis, an unqualified expression such as (\sqrt z) may instead refer to the two-valued algebraic relation, although a stated principal branch converts it into an ordinary function on a cut domain.

The equality of multivalued expressions can refer either to equality of their complete value sets or to equality between designated branches. These meanings are not interchangeable. For example, the value sets of (\log(z_1z_2)) and (\log z_1+\log z_2) agree after all integral multiples of (2\pi i) are included, while a fixed principal branch need not satisfy the corresponding equality for every pair of arguments in its domain.