Einzelwerts

Einzelwerts is a term used in German-language mathematics to denote a single value associated with each input, i.e., the property of being single-valued. A function or mapping is called single-valued when, for every input, there is exactly one output. This stands in contrast to multi-valued constructs that may assign several outputs to a single input, a situation that occurs in various branches of mathematics.

In complex analysis, many expressions are inherently multi-valued, such as roots and logarithms. To work with a single-valued object, one defines a branch by restricting the domain, for example by removing a branch cut and thereby obtaining a single-valued logarithm on that domain. The concept of single-valuedness is central to ensuring well-defined computations and analyses.

Common single-valued functions include polynomials, the exponential function, and the real-valued absolute value. By contrast, multi-valued examples include the complex square root function sqrt(z) (which yields two values for most z) and the complex logarithm prior to choosing a branch.

In German mathematical writing, Einzelwerts may appear in contexts describing single-valuedness, though the standard noun is Einzelwert and the genitive form is des Einzelwerts. The term serves to distinguish single-valued quantities from multi-valued ones in definitions, theorems, and discussions across analysis and algebra.

See also: single-valued function, branch cut, multi-valued function.