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For example, the entire complex plane is a domain, as is the open unit disk, the open upper half-plane, and so forth. Often, a complex domain serves as the domain of definition for a holomorphic function. In the study of several complex variables, the definition of a domain is extended to include any connected open subset of C n.
In mathematics, the domain of a function is the set of inputs accepted by the function. It is sometimes denoted by dom ( f ) {\displaystyle \operatorname {dom} (f)} or dom f {\displaystyle \operatorname {dom} f} , where f is the function.
The domain of definition of such a function is the set of inputs for which the algorithm does not run forever. A fundamental theorem of computability theory is that there cannot exist an algorithm that takes an arbitrary general recursive function as input and tests whether 0 belongs to its domain of definition (see Halting problem).
In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of definition of a given analytic function.Analytic continuation often succeeds in defining further values of a function, for example in a new region where the infinite series representation which initially defined the function becomes divergent.
In mathematics, the restriction of a function is a new function, denoted | or , obtained by choosing a smaller domain for the original function . The function f {\displaystyle f} is then said to extend f | A . {\displaystyle f\vert _{A}.}
In algebra, a domain is a nonzero ring in which ab = 0 implies a = 0 or b = 0. [1] (Sometimes such a ring is said to "have the zero-product property".) Equivalently, a domain is a ring in which 0 is the only left zero divisor (or equivalently, the only right zero divisor). A commutative domain is called an integral domain.
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A is a GCD domain satisfying ACCP. A is a Schreier domain, [6] and atomic. A is a pre-Schreier domain and atomic. A has a divisor theory in which every divisor is principal. A is a Krull domain in which every divisorial ideal is principal (in fact, this is the definition of UFD in Bourbaki.) A is a Krull domain and every prime ideal of height 1 ...