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In complex analysis, a complex domain (or simply domain) is any connected open subset of the complex plane C. 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.
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).
is a function from domain X to codomain Y. The yellow oval inside Y is the image of . Sometimes "range" refers to the image and sometimes to the codomain. In mathematics, the range of a function may refer to either of two closely related concepts: the codomain of the function, or; the image of the function.
The set of points in the red oval X is the domain of f. Graph of the real-valued square root function, f(x) = √ x, whose domain consists of all nonnegative real numbers. In mathematics, the domain of a function is the set of inputs accepted by the function.
A codomain is part of a function f if f is defined as a triple (X, Y, G) where X is called the domain of f, Y its codomain, and G its graph. [1] The set of all elements of the form f(x), where x ranges over the elements of the domain X, is called the image of f. The image of a function is a subset of its codomain so it might not coincide with it.
For instance, Picard's theorem asserts that the range of an entire function can take only three possible forms: , {}, or {} for some . In other words, if two distinct complex numbers z {\displaystyle z} and w {\displaystyle w} are not in the range of an entire function f {\displaystyle f} , then f {\displaystyle f} is a constant function.
A commutative ring (not necessarily a domain) with unity satisfying this condition is called a containment-division ring (CDR). [2] Thus a Dedekind domain is a domain that either is a field, or satisfies any one, and hence all five, of (DD1) through (DD5). Which of these conditions one takes as the definition is therefore merely a matter of taste.
A bijection f with domain X (indicated by f: X → Y in functional notation) also defines a converse relation starting in Y and going to X (by turning the arrows around). The process of "turning the arrows around" for an arbitrary function does not, in general , yield a function, but properties (3) and (4) of a bijection say that this inverse ...