Taylor series in the context of "Power series"

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 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.

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 holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space ⁠${\displaystyle \mathbb {C} ^{n}}$⁠. The existence of a complex derivative in a neighbourhood is a very strong condition: It implies that a holomorphic function is infinitely differentiable and locally equal to its own Taylor series (is analytic). Holomorphic functions are the central objects of study in complex analysis.

Though the term analytic function is often used interchangeably with "holomorphic function", the word "analytic" is defined in a broader sense to denote any function (real, complex, or of more general type) that can be written as a convergent power series in a neighbourhood of each point in its domain. That all holomorphic functions are complex analytic functions, and vice versa, is a major theorem in complex analysis.

In mathematics, a recurrence relation is an equation according to which the ${\displaystyle n}$th term of a sequence of numbers is equal to some combination of the previous terms. Often, only ${\displaystyle k}$ previous terms of the sequence appear in the equation, for a parameter ${\displaystyle k}$ that is independent of ${\displaystyle n}$; this number ${\displaystyle k}$ is called the order of the relation. If the values of the first ${\displaystyle k}$ numbers in the sequence have been given, the rest of the sequence can be calculated by repeatedly applying the equation.

In linear recurrences, the nth term is equated to a linear function of the ${\displaystyle k}$ previous terms. A famous example is the recurrence for the Fibonacci numbers,$${\displaystyle F_{n}=F_{n-1}+F_{n-2}}$$where the order ${\displaystyle k}$ is two and the linear function merely adds the two previous terms. This example is a linear recurrence with constant coefficients, because the coefficients of the linear function (1 and 1) are constants that do not depend on ${\displaystyle n.}$ For these recurrences, one can express the general term of the sequence as a closed-form expression of ${\displaystyle n}$. As well, linear recurrences with polynomial coefficients depending on ${\displaystyle n}$ are also important, because many common elementary functions and special functions have a Taylor series whose coefficients satisfy such a recurrence relation (see holonomic function).

Brook Taylor FRS (18 August 1685 – 29 December 1731) was an English mathematician and barrister best known for several results in mathematical analysis. Taylor's most famous developments are Taylor's theorem and the Taylor series, essential in the infinitesimal approach of functions in specific points.