Domain of a function in the context of "Function of several real variables"

In number theory, an arithmetic, arithmetical, or number-theoretic function is generally any function whose domain is the set of positive integers and whose range is a subset of the complex numbers. Hardy & Wright include in their definition the requirement that an arithmetical function "expresses some arithmetical property of n". There is a larger class of number-theoretic functions that do not fit this definition, for example, the prime-counting functions. This article provides links to functions of both classes.

An example of an arithmetic function is the divisor function whose value at a positive integer n is equal to the number of divisors of n.

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.

In mathematical analysis, a domain or region is a non-empty, connected, and open set in a topological space. In particular, it is any non-empty connected open subset of the real coordinate space R or the complex coordinate space C. A connected open subset of coordinate space is frequently used for the domain of a function.

The basic idea of a connected subset of a space dates from the 19th century, but precise definitions vary slightly from generation to generation, author to author, and edition to edition, as concepts developed and terms were translated between German, French, and English works. In English, some authors use the term domain, some use the term region, some use both terms interchangeably, and some define the two terms slightly differently; some avoid ambiguity by sticking with a phrase such as non-empty connected open subset.

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 for every ${\displaystyle x_{0}}$ in its domain, its Taylor series about ${\displaystyle x_{0}}$ converges to the function in some neighborhood of ${\displaystyle x_{0}}$. This is stronger than merely being infinitely differentiable at ${\displaystyle x_{0}}$, and therefore having a well-defined Taylor series; the Fabius function provides an example of a function that is infinitely differentiable but not analytic.

In mathematics, a function from a set X to a set Y assigns to each element of X exactly one element of Y. The set X is called the domain of the function and the set Y is called the codomain of the function.

Functions were originally the idealization of how a varying quantity depends on another quantity. For example, the position of a planet is a function of time. Historically, the concept was elaborated with the infinitesimal calculus at the end of the 17th century, and, until the 19th century, the functions that were considered were differentiable (that is, they had a high degree of regularity). The concept of a function was formalized at the end of the 19th century in terms of set theory, and this greatly increased the possible applications of the concept.

In mathematics, the graph of a function ${\displaystyle f}$ is the set of ordered pairs ${\displaystyle (x,y)}$, where ${\displaystyle f(x)=y.}$ In the common case where ${\displaystyle x}$ and ${\displaystyle f(x)}$ are real numbers, these pairs are Cartesian coordinates of points in a plane and often form a curve.The graphical representation of the graph of a function is also known as a plot.

In the case of functions of two variables – that is, functions whose domain consists of pairs ${\displaystyle (x,y)}$ –, the graph usually refers to the set of ordered triples ${\displaystyle (x,y,z)}$ where ${\displaystyle f(x,y)=z}$. This is a subset of three-dimensional space; for a continuous real-valued function of two real variables, its graph forms a surface, which can be visualized as a surface plot.

In mathematical analysis, the maximum and minimum of a function are, respectively, the greatest and least value taken by the function. Known generically as extrema, they may be defined either within a given range (the local or relative extrema) or on the entire domain (the global or absolute extrema) of a function. Pierre de Fermat was one of the first mathematicians to propose a general technique, adequality, for finding the maxima and minima of functions.

As defined in set theory, the maximum and minimum of a set are the greatest and least elements in the set, respectively. Unbounded infinite sets, such as the set of real numbers, have no minimum or maximum.

Vertical position or vertical location is a position along a vertical direction (the plumb line direction) above or below a given vertical datum (a reference level surface, such as mean sea level).Vertical distance or vertical separation is the distance between two vertical positions.Many vertical coordinates exist for expressing vertical position: depth, height, altitude, elevation, etc.Points lying on an equigeopotential surface are said to be on the same vertical level, as in a water level.A function with domain along the vertical line is called a vertical distribution or vertical profile.

In mathematics (specifically multivariable calculus), a multiple integral is a definite integral of a function of several real variables, for instance, f(x, y) or f(x, y, z).

Integrals of a function of two variables over a region in ${\displaystyle \mathbb {R} ^{2}}$ (the real-number plane) are called double integrals, and integrals of a function of three variables over a region in ${\displaystyle \mathbb {R} ^{3}}$ (real-number 3D space) are called triple integrals. For repeated antidifferentiation of a single-variable function, see the Cauchy formula for repeated integration.