Multivalued function
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In mathematics, a multivalued function is a total relation; i.e. every input is associated with one or more outputs. Strictly speaking, a "well-defined" function associates one, and only one, output to any particular input. The term "multivalued function" is, therefore, a misnomer: true functions are single-valued. 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.
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[edit] Examples
- Each real or complex number except 0 has two square roots. 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.
- Each complex number has three cube roots.
- The complex logarithm function is multiple-valued. The values assumed by log(1) are 2πni for all integers n.
- 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.
- The indefinite integral is a multivalued function of another function f. Its domain X is a set of functions. For any input f, it yields infinitely many possible solutions (the antiderivatives of f).
Notice that all of these examples refer to quasi-inverses of information-losing functions (i.e. imperfect inverses of non-injective functions).
Multivalued functions of a complex variable have branch points. 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. As in the the case with real functions the restricted range may be called principal branch of the function.
[edit] Riemann surfaces
A more sophisticated viewpoint replaces "multivalued functions" with functions whose domain is a Riemann surface (so named in honor of Bernhard Riemann).
[edit] 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. Some evolution can be seen in different editions of Course of Pure Mathematics by G. H. Hardy, for example. It probably persisted longest in the theory of special functions, for its occasional convenience.
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. They are the origin of gauge field structures in many branches of physics.
[edit] 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. 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)
