Neighborhood (mathematics) in the context of Complex function

Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of a complex variable of complex numbers. It is helpful in many branches of mathematics, including real analysis, algebraic geometry, number theory, analytic combinatorics, and applied mathematics, as well as in physics, including the branches of hydrodynamics, thermodynamics, quantum mechanics, and twistor theory. By extension, use of complex analysis also has applications in engineering fields such as nuclear, aerospace, mechanical and electrical engineering.

At first glance, complex analysis is the study of holomorphic functions that are the differentiable functions of a complex variable. By contrast with the real case, a holomorphic functions is always infinitely differentiable and equal to the sum of its Taylor series in some neighborhood of each point of its domain.This makes methods and results of complex analysis significantly different from that of real analysis. In particular, contrarily, with the real case, the domain of every holomorphic function can be uniquely extended to almost the whole complex plane. This implies that the study of real analytic functions needs often the power of complex analysis. This is, in particular, the case in analytic combinatorics.

Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of a complex variable of complex numbers. It is helpful in many branches of mathematics, including real analysis, algebraic geometry, number theory, analytic combinatorics, and applied mathematics, as well as in physics, including the branches of hydrodynamics, thermodynamics, quantum mechanics, and twistor theory. By extension, use of complex analysis also has applications in engineering fields such as nuclear, aerospace, mechanical and electrical engineering.

At first glance, complex analysis is the study of holomorphic functions that are the differentiable functions of a complex variable. By contrast with the real case, a holomorphic function is always infinitely differentiable and equal to the sum of its Taylor series in some neighborhood of each point of its domain.This makes methods and results of complex analysis significantly different from that of real analysis. In particular, contrarily, with the real case, the domain of every holomorphic function can be uniquely extended to almost the whole complex plane. This implies that the study of real analytic functions needs often the power of complex analysis. This is, in particular, the case in analytic combinatorics.

In mathematics, a point x is called an isolated point of a subset S (in a topological space X) if x is an element of S and there exists a neighborhood of x that does not contain any other points of S. This is equivalent to saying that the singleton {x} is an open set in the topological space S (considered as a subspace of X). Another equivalent formulation is: an element x of S is an isolated point of S if and only if it is not a limit point of S.

If the space X is a metric space, for example a Euclidean space, then an element x of S is an isolated point of S if there exists an open ball around x that contains only finitely many elements of S.A point set that is made up only of isolated points is called a discrete set or discrete point set (see also discrete space).