Multivalued function

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This diagram does not represent a "true" function, because the element 3 in X is associated with two elements, b and c, in Y.

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, technically, a misnomer: true functions are single-valued. However, a multivalued function from A to B can be represented as a function from A to the set of nonempty subsets of B.

1 Examples
2 Riemann surfaces
3 History
4 Books
5 See also

[edit] Examples

Each real or complex number except 0 has two square roots. Each complex number has three cube roots.

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

$\tan(\pi/4) = \tan(5\pi/4) = \tan(-3\pi/4) = \cdots = 1.$

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.

The natural logarithm function from the positive reals to the reals is single-valued, but its generalization to complex numbers (excluding 0) is multiple-valued, because the natural exponential function exp(z) (evaluated at complex arguments z) is periodic with period 2πi. Denoting this multi-valued function by "Log", with a capital "L" to distinguish it from its single-valued counterpart defined only for positive real arguments, the values assumed by Log(e) are 1 + 2πin for all integers n.

Multivalued functions of a complex variable have branch points. For the nth root and logarithm functions, 0 is a branch point; for the arctangent functions, the imaginary units i and −i are branch points.

[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. The are the origin of gauge field structures in many branches of physics.

[edit] Books

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 9971-50-210-0 (also available online: Vol. I and Vol. II)