In mathematical analysis, the word * region* usually refers to a subset of or that is open (in the standard Euclidean topology), simply connected and non-empty. A

Regions and closed regions are often used as domains of functions or differential equations.

According to Kreyszig,

- A region is a set consisting of a domain plus, perhaps, some or all of its boundary points. (The reader is warned that some authors use the term "region" for what we call a domain [following standard terminology], and others make no distinction between the two terms.)
^{ [1] }

According to Yue Kuen Kwok,

- An open connected set is called an
*open region*or*domain*. ...to an open region we may add none, some, or all its limit points, and simply call the new set a*region*.^{ [2] }

- ↑ Erwin Kreyszig (1993)
*Advanced Engineering Mathematics*, 7th edition, p. 720, John Wiley & Sons, ISBN 0-471-55380-8 - ↑ Yue Kuen Kwok (2002)
*Applied Complex Variables for Scientists and Engineers*, § 1.4 Some topological definitions, p 23, Cambridge University Press, ISBN 0-521-00462-4

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

In mathematics, a **holomorphic function** is a complex-valued function of one or more complex variables that is, at every point of its domain, complex differentiable in a neighborhood of the point. The existence of a complex derivative in a neighbourhood is a very strong condition, for it implies that any holomorphic function is actually infinitely differentiable and equal, locally, to its own Taylor series (*analytic*). Holomorphic functions are the central objects of study in complex analysis.

In complex analysis, the **Riemann mapping theorem** states that if *U* is a non-empty simply connected open subset of the complex number plane **C** which is not all of **C**, then there exists a biholomorphic mapping *f* from *U* onto the open unit disk

In mathematics, **real analysis** is the branch of mathematical analysis that studies the behavior of real numbers, sequences and series of real numbers, and real functions. Some particular properties of real-valued sequences and functions that real analysis studies include convergence, limits, continuity, smoothness, differentiability and integrability.

In mathematics, a **conformal map** is a function that locally preserves angles, but not necessarily lengths.

In mathematics, the **Cauchy integral theorem** in complex analysis, named after Augustin-Louis Cauchy, is an important statement about line integrals for holomorphic functions in the complex plane. Essentially, it says that if two different paths connect the same two points, and a function is holomorphic everywhere in between the two paths, then the two path integrals of the function will be the same.

In mathematics, an **analytic function** is a function that is locally given by a convergent power series. There exist both **real analytic functions** and **complex analytic functions**. Functions of each type are infinitely differentiable, but complex analytic functions exhibit properties that do not generally hold for real analytic functions. A function is analytic if and only if its Taylor series about *x*_{0} converges to the function in some neighborhood for every *x*_{0} in its domain.

In mathematics, the **winding number** or **winding index** of a closed curve in the plane around a given point is an integer representing the total number of times that curve travels counterclockwise around the point. The winding number depends on the orientation of the curve, and is negative if the curve travels around the point clockwise.

In mathematics, a **curve** is an object similar to a line, but that does not have to be straight.

In mathematics, **complex geometry** is the study of complex manifolds, complex algebraic varieties, and functions of several complex variables. Application of transcendental methods to algebraic geometry falls in this category, together with more geometric aspects of complex analysis.

In mathematics, particularly in algebraic geometry, complex analysis and algebraic number theory, an **abelian variety** is a projective algebraic variety that is also an algebraic group, i.e., has a group law that can be defined by regular functions. Abelian varieties are at the same time among the most studied objects in algebraic geometry and indispensable tools for much research on other topics in algebraic geometry and number theory.

The theory of **functions of several complex variables** is the branch of mathematics dealing with complex-valued functions. The function on the complex coordinate space of n-tuples of complex numbers.

In mathematical analysis, and applications in geometry, applied mathematics, engineering, and natural sciences, a **function of a real variable** is a function whose domain is the real numbers , or a subset of that contains an interval of positive length. Most real functions that are considered and studied are differentiable in some interval. The most widely considered such functions are the **real functions**, which are the real-valued functions of a real variable, that is, the functions of a real variable whose codomain is the set of real numbers.

In mathematics, a **manifold** is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or *n-manifold* for short, is a topological space with the property that each point has a neighborhood that is homeomorphic to the Euclidean space of dimension n.

In the branch of mathematics known as complex analysis, a **complex logarithm** is an analogue for nonzero complex numbers of the logarithm of a positive real number. The term refers to one of the following:

**Geometric function theory** is the study of geometric properties of analytic functions. A fundamental result in the theory is the Riemann mapping theorem.

In complex analysis, the **monodromy theorem** is an important result about analytic continuation of a complex-analytic function to a larger set. The idea is that one can extend a complex-analytic function along curves starting in the original domain of the function and ending in the larger set. A potential problem of this **analytic continuation along a curve** strategy is there are usually many curves which end up at the same point in the larger set. The monodromy theorem gives sufficient conditions for analytic continuation to give the same value at a given point regardless of the curve used to get there, so that the resulting extended analytic function is well-defined and single-valued.

In mathematical analysis, a **domain** is any connected open subset of a finite-dimensional vector space. This is a different concept than the domain of a function, though it is often used for that purpose, for example in partial differential equations and Sobolev spaces.

In mathematics, an **ordinary differential equation** (**ODE**) is a differential equation containing one or more functions of one independent variable and the derivatives of those functions. The term *ordinary* is used in contrast with the term partial differential equation which may be with respect to *more than* one independent variable.

In mathematics, a **planar Riemann surface** is a Riemann surface sharing the topological properties of a connected open subset of the Riemann sphere. They are characterized by the topological property that the complement of every closed Jordan curve in the Riemann surface has two connected components. An equivalent characterization is the differential geometric property that every closed differential 1-form of compact support is exact. Every simply connected Riemann surface is planar. The class of planar Riemann surfaces was studied by Koebe who proved in 1910 as a generalization of the uniformization theorem that every such surface is conformally equivalent to either the Riemann sphere or the complex plane with slits parallel to the real axis removed.

- Ruel V. Churchill (1960)
*Complex variables and applications*, 2nd edition, §1.9 Regions in the complex plane, pp. 16 to 18, McGraw-Hill - Constantin Carathéodory (1954)
*Theory of Functions of a Complex Variable*, v. I, p. 97, Chelsea Publishing. - Howard Eves (1966)
*Functions of a Complex Variable*, p. 105, Prindle, Weber & Schmidt.

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