singlevalued

Singlevalued is an adjective used in mathematics to describe a mapping that assigns exactly one value to each input in its domain. In standard usage, a function is singlevalued by definition; the term is commonly used to contrast a true function with multivalued mappings that, for a given input, may yield several outputs. In practical contexts, a multivalued function is treated as a relation or as a family of branches.

Examples help distinguish the concept. The real-valued square root on the nonnegative real axis is singlevalued: sqrt(x) gives one nonnegative root for each x ≥ 0. The real cube root on all real numbers is also singlevalued. In contrast, the complex logarithm log z and complex roots are multivalued: log z = ln|z| + i Arg(z), where Arg(z) can differ by multiples of 2π.

To work with a singlevalued version of a multivalued function on a larger domain, one selects a branch cut or works on a simply connected domain that avoids looping around branch points. For example, a principal branch of log z is defined on C minus the non-positive real axis, giving a singlevalued Log z; similarly, z^a can be made singlevalued on a chosen branch by fixing a branch of Log z. In complex analysis, singlevalued holomorphic functions are those without monodromy when analytically continued around loops; multivaluedness is captured by the monodromy representation on covering spaces or Riemann surfaces.

See also: multivalued function, branch cut, branch of a function, monodromy, Riemann surface.