List of real analysis topics

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This is a list of articles that are considered real analysis topics.

Contents

See also: glossary of real and complex analysis.

General topics

Limits

Sequences and series

(see also list of mathematical series )

Summation methods

More advanced topics

  • Convolution
  • Farey sequence – the sequence of completely reduced fractions between 0 and 1
  • Oscillation – is the behaviour of a sequence of real numbers or a real-valued function, which does not converge, but also does not diverge to +∞ or −∞; and is also a quantitative measure for that.
  • Indeterminate forms – algebraic expressions gained in the context of limits. The indeterminate forms include 00, 0/0, 1, ∞  ∞, ∞/∞, 0 × ∞, and ∞0.

Convergence

Convergence tests

Functions

Continuity

Distributions

Variation

Derivatives

Differentiation rules

Differentiation in geometry and topology

see also List of differential geometry topics

Integrals

(see also Lists of integrals)

Integration and measure theory

see also List of integration and measure theory topics

Fundamental theorems

Foundational topics

Numbers

Real numbers

Specific numbers

Sets

Maps

Applied mathematical tools

Infinite expressions

Inequalities

See list of inequalities

Means

Orthogonal polynomials

Spaces

Measures

Field of sets

Historical figures

See also

Related Research Articles

<span class="mw-page-title-main">Complex analysis</span> Branch of mathematics studying functions of a complex variable

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 helpful in many branches of mathematics, including 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.

<span class="mw-page-title-main">Holomorphic function</span> Complex-differentiable (mathematical) function

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 . 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. Holomorphic functions are the central objects of study in complex analysis.

In mathematics, a series is, roughly speaking, an addition of infinitely many quantities, one after the other. The study of series is a major part of calculus and its generalization, mathematical analysis. Series are used in most areas of mathematics, even for studying finite structures through generating functions. The mathematical properties of infinite series make them widely applicable in other quantitative disciplines such as physics, computer science, statistics and finance.

<span class="mw-page-title-main">Mean value theorem</span> On the existence of a tangent to an arc parallel to the line through its endpoints

In mathematics, the mean value or Lagrange theorem from the branch of mathematical analysis states, roughly, that for a "smooth" arc between two endpoints on a plane, there is at least one point on the arc whose tangent line is parallel to the line connecting its endpoints.

In mathematics, the branch of real analysis 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.

<span class="mw-page-title-main">Dirac delta function</span> Generalized function whose value is zero everywhere except at zero

In mathematical analysis, the Dirac delta function, also known as the unit impulse, is a generalized function on the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire real line is equal to one. Since there is no function having this property, modelling the delta "function" rigorously involves the use of limits or, as is common in mathematics, measure theory and the theory of distributions.

<span class="mw-page-title-main">Mathematical analysis</span> Branch of mathematics

Analysis is the branch of mathematics dealing with continuous functions, limits, and related theories, such as differentiation, integration, measure, infinite sequences, series, and analytic functions.

<span class="mw-page-title-main">Uniform convergence</span> Mode of convergence of a function sequence

In the mathematical field of analysis, uniform convergence is a mode of convergence of functions stronger than pointwise convergence. A sequence of functions converges uniformly to a limiting function on a set as the function domain if, given any arbitrarily small positive number , a number can be found such that each of the functions differs from by no more than at every pointin. Described in an informal way, if converges to uniformly, then how quickly the functions approach is "uniform" throughout in the following sense: in order to guarantee that differs from by less than a chosen distance , we only need to make sure that is larger than or equal to a certain , which we can find without knowing the value of in advance. In other words, there exists a number that could depend on but is independent of , such that choosing will ensure that for all . In contrast, pointwise convergence of to merely guarantees that for any given in advance, we can find such that, for that particular, falls within of whenever .

<span class="mw-page-title-main">Taylor's theorem</span> Approximation of a function by a truncated power series

In calculus, Taylor's theorem gives an approximation of a -times differentiable function around a given point by a polynomial of degree , called the -th-order Taylor polynomial. For a smooth function, the Taylor polynomial is the truncation at the order of the Taylor series of the function. The first-order Taylor polynomial is the linear approximation of the function, and the second-order Taylor polynomial is often referred to as the quadratic approximation. There are several versions of Taylor's theorem, some giving explicit estimates of the approximation error of the function by its Taylor polynomial.

<span class="mw-page-title-main">Lipschitz continuity</span> Strong form of uniform continuity

In mathematical analysis, Lipschitz continuity, named after German mathematician Rudolf Lipschitz, is a strong form of uniform continuity for functions. Intuitively, a Lipschitz continuous function is limited in how fast it can change: there exists a real number such that, for every pair of points on the graph of this function, the absolute value of the slope of the line connecting them is not greater than this real number; the smallest such bound is called the Lipschitz constant of the function. For instance, every function that is defined on an interval and has a bounded first derivative is Lipschitz continuous.

<span class="mw-page-title-main">Harmonic function</span> Functions in mathematics

In mathematics, mathematical physics and the theory of stochastic processes, a harmonic function is a twice continuously differentiable function where U is an open subset of that satisfies Laplace's equation, that is, everywhere on U. This is usually written as or

<span class="mw-page-title-main">Cauchy's integral formula</span> Provides integral formulas for all derivatives of a holomorphic function

In mathematics, Cauchy's integral formula, named after Augustin-Louis Cauchy, is a central statement in complex analysis. It expresses the fact that a holomorphic function defined on a disk is completely determined by its values on the boundary of the disk, and it provides integral formulas for all derivatives of a holomorphic function. Cauchy's formula shows that, in complex analysis, "differentiation is equivalent to integration": complex differentiation, like integration, behaves well under uniform limits – a result that does not hold in real analysis.

In mathematical analysis, Cesàro summation assigns values to some infinite sums that are not necessarily convergent in the usual sense. The Cesàro sum is defined as the limit, as n tends to infinity, of the sequence of arithmetic means of the first n partial sums of the series.

In mathematics, a divergent series is an infinite series that is not convergent, meaning that the infinite sequence of the partial sums of the series does not have a finite limit.

In mathematics, the Riesz–Fischer theorem in real analysis is any of a number of closely related results concerning the properties of the space L2 of square integrable functions. The theorem was proven independently in 1907 by Frigyes Riesz and Ernst Sigismund Fischer.

In mathematics, the study of interchange of limiting operations is one of the major concerns of mathematical analysis, in that two given limiting operations, say L and M, cannot be assumed to give the same result when applied in either order. One of the historical sources for this theory is the study of trigonometric series.

<span class="mw-page-title-main">Hilbert space</span> Type of topological vector space

In mathematics, Hilbert spaces allow the methods of linear algebra and calculus to be generalized from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise naturally and frequently in mathematics and physics, typically as function spaces. Formally, a Hilbert space is a vector space equipped with an inner product that induces a distance function for which the space is a complete metric space. A Hilbert space is a special case of a Banach space.

Most of the terms listed in Wikipedia glossaries are already defined and explained within Wikipedia itself. However, glossaries like this one are useful for looking up, comparing and reviewing large numbers of terms together. You can help enhance this page by adding new terms or writing definitions for existing ones.

This is a glossary of concepts and results in real analysis and complex analysis in mathematics.