This diagram does not represent a "true" function, because the element 3 in X is associated with two elements, b and c, in Y. The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function

In mathematics, a multivalued function is a total relation; i. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Mathematics, a Binary relation R over a set X is total if it holds for all a and b in X that e. every input is associated with one or more outputs. Input is the term denoting either an entrance or changes which are inserted into a System and which activate/modify a Process. Output is the term denoting either an exit or changes which exit a System and which activate/modify a Process. Strictly speaking, a "well-defined" function associates one, and only one, output to any particular input. The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function Output is the term denoting either an exit or changes which exit a System and which activate/modify a Process. Input is the term denoting either an entrance or changes which are inserted into a System and which activate/modify a Process. The term "multivalued function" is, therefore, a misnomer: true functions are single-valued. A misnomer is a term which suggests an interpretation that is known to be untrue However, a multivalued function from A to B can be represented as a single-valued function from A to the set of nonempty subsets of B.

Examples

• Each real or complex number except 0 has two square roots. In Mathematics, the real numbers may be described informally in several different ways Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted In Mathematics, a square root of a number x is a number r such that r 2 = x, or in words a number r whose The square roots of 4 are in the set {+2,−2}. The square roots of 0 are described by the multiset {0,0}, because 0 is a root of multiplicity 2 of the polynomial x². In Mathematics, a multiset (or bag) is a generalization of a set.

• Each complex number has three cube roots. In Mathematics, a cube root of a number denoted \sqrt{x} or x1/3 is a number a such that a 3 =  x

• The complex logarithm function is multiple-valued. In Complex analysis, the complex logarithm is the extension of the Natural logarithm function ln( x) &ndash originally defined for Real numbers The values assumed by log(1) are 2πni for all integers n. The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French

• Inverse trigonometric functions are multiple-valued because trigonometric functions are periodic. We have

Consequently arctan(1) may be thought of as having multiple values: π/4, 5π/4, −3π/4, and so on. This can be overcome by limiting the domain of tan(x) to -π/2 < x < π/2. Thus, the range of arctan(y) becomes -π/2 < y < π/2. These values from a limited domain are called principal values. In considering complex Multiple-valued functions in Complex analysis, the principal values of a function are the values along one chosen branch of that

• The indefinite integral is a multivalued function of another function f. In Calculus, an antiderivative, primitive or indefinite integral of a function f is a function F whose Derivative Its domain X is a set of functions. For any input f, it yields infinitely many possible solutions (the antiderivatives of f). In Calculus, an antiderivative, primitive or indefinite integral of a function f is a function F whose Derivative

Notice that all of these examples refer to quasi-inverses of information-losing functions (i. e. imperfect inverses of non-injective functions). In Mathematics, if &fnof is a function from A to B then an inverse function for &fnof is a function in the opposite direction from B

Multivalued functions of a complex variable have branch points. In the mathematical field of Complex analysis, a branch point may be informally thought of as a point z 0 at which a " multi-valued For example the nth root and logarithm functions, 0 is a branch point; for the arctangent function, the imaginary units i 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 a branch cut, a kind of curve which connects pairs of branch points, thus reducing the multilayered Riemann surface of the function to a single layer. In the mathematical field of Complex analysis, a branch point may be informally thought of as a point z 0 at which a " multi-valued As in the the case with real functions the restricted range may be called principal branch of the function.

Riemann surfaces

A more sophisticated viewpoint replaces "multivalued functions" with functions whose domain is a Riemann surface (so named in honor of Bernhard Riemann). In Mathematics, particularly in Complex analysis, a Riemann surface, first studied by and named after Bernhard Riemann, is a one-dimensional

History

The practice of allowing function in mathematics to mean also multivalued function dropped out of usage at some point in the first half of the twentieth century. The twentieth century of the Common Era began on Some evolution can be seen in different editions of Course of Pure Mathematics by G. H. Hardy, for example. A Course of Pure Mathematics is a classic textbook in introductory Mathematical analysis, written by G Godfrey Harold Hardy FRS ( February 7, 1877 Cranleigh, Surrey, England &ndash December 1, 1947 It probably persisted longest in the theory of special functions, for its occasional convenience. Special functions are particular mathematical functions which have more or less established names and notations due to their importance for the Mathematical analysis

In physics, multivalued functions play an increasingly important role. They form the mathematical basis for Dirac's magnetic monopoles, for the theory of defects in crystal and the resulting plasticity of materials, for vortices in superfluids and superconductors, and for phase transitions in these systems, for instance melting and quark confinement. In Physics, a magnetic monopole is a hypothetical particle that is a Magnet with only one pole (see Maxwell's equations for more on magnetic Crystalline solids have a very regular atomic structure that is the local positions of atoms with respect to each other are repeated at the atomic scale V erification of the O rigins of R otation in T ornadoes Ex periment or VORTEX, is a field project that seeks to understand how a Superfluidity is a phase of matter or description of Heat capacity in which unusual effects are observed when Liquids, typically of Helium-4 Superconductivity is a phenomenon occurring in certain Materials generally at very low Temperatures characterized by exactly zero electrical resistance In Thermodynamics, phase transition or phase change is the transformation of a thermodynamic system from one phase to another Melting is a process that results in the phase change of a substance from a Solid to a Liquid. Color confinement, often called just confinement, is the Physics phenomenon that Color charged particles (such as Quarks cannot be isolated singularly They are the origin of gauge field structures in many branches of physics. Gauge theory is a peculiar Quantum field theory where the Lagrangian is invariant under certain transformations

References

• Kleinert, Hagen, Multivalued Fields in in Condensed Matter, Electrodynamics, and Gravitation, World Scientific (Singapore, 2008) (also available online)
• Kleinert, Hagen, Gauge Fields in Condensed Matter, Vol. Hagen Kleinert (born 1941 is Professor of Theoretical Physics at the Free University of Berlin, Germany (since 1968 Honorary Professor at the Kyrgyz-Russian Hagen Kleinert (born 1941 is Professor of Theoretical Physics at the Free University of Berlin, Germany (since 1968 Honorary Professor at the Kyrgyz-Russian I, "SUPERFLOW AND VORTEX LINES", pp. 1--742, Vol. II, "STRESSES AND DEFECTS", pp. 743-1456, World Scientific (Singapore, 1989); Paperback ISBN 9971-5-0210-0 (also available online: Vol. I and Vol. II)

See also

• partial function
• correspondence