Multivalued function
[5] It is a set-valued function with additional properties depending on context; some authors do not distinguish between set-valued functions and multifunctions,[6] but English Wikipedia currently does, having a separate article for each.
[7] Geometrically, this means that the graph of a multivalued function is necessarily a line of zero area that doesn't loop, while the graph of a set-valued relation may contain solid filled areas or loops.
[7] The term multivalued function originated in complex analysis, from analytic continuation.
be the usual square root function on positive real numbers.
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 for nth roots, logarithms, and inverse trigonometric functions.
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 of Riemann surfaces: to consider a multivalued function
as an ordinary function without discarding any values, one multiplies the domain into a many-layered covering space, a manifold which is the Riemann surface associated to
For example, the complex logarithm log(z) is the multivalued inverse of the exponential function ez : C → C×, with graph It is not single valued, given a single w with w = log(z), we have Given any holomorphic function on an open subset of the complex plane C, its analytic continuation is always a multivalued function.
Since the original functions do not preserve all the information of their inputs, they are not reversible.
Multivalued functions of a complex variable have branch points.
A suitable interval may be found through use of a branch cut, a kind of curve that connects pairs of branch points, thus reducing the multilayered Riemann surface of the function to a single layer.
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 crystals 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.
