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Mathematical analysis

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Mathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of infinitesimal calculus. It is a branch of pure mathematics that includes the theories of differentiation, integration and measure, limits, infinite series, and analytic functions. These theories are often studied in the context of real numbers, complex numbers, and real and complex functions. However, they can also be defined and studied in any space of mathematical objects that has a definition of nearness (a topological space) or, more specifically, distance (a metric space).


Early results in analysis were implicitly present in the early days of ancient Greek mathematics. For instance, an infinite geometric sum is implicit in Zeno's paradox of the dichotomy. Later, Greek mathematicians such as Eudoxus and Archimedes made more explicit, but informal, use of the concepts of limits and convergence when they used the method of exhaustion to compute the area and volume of regions and solids. In India, the 12th century mathematician Bhāskara II gave examples of the derivative and used what is now known as Rolle's theorem.

In the 14th century, Madhava of Sangamagrama developed infinite series expansions, like the power series and the Taylor series, of functions such as sine, cosine, tangent and arctangent. Alongside his development of the Taylor series of the trigonometric functions, he also estimated the magnitude of the error terms created by truncating these series and gave a rational approximation of an infinite series. His followers at the Kerala school of astronomy and mathematics further expanded his works, up to the 16th century.

In Europe, during the later half of the 17th century, Newton and Leibniz independently developed infinitesimal calculus, which grew, with the stimulus of applied work that continued through the 18th century, into analysis topics such as the calculus of variations, ordinary and partial differential equations, Fourier analysis, and generating functions. During this period, calculus techniques were applied to approximate discrete problems by continuous ones.

In the 18th century, Euler introduced the notion of mathematical function. Real analysis began to emerge as an independent subject when Bernard Bolzano introduced the modern definition of continuity in 1816. but Bolzano's work did not become widely known until the 1870s. In 1821, Cauchy began to put calculus on a firm logical foundation by rejecting the principle of the generality of algebra widely used in earlier work, particularly by Euler. Instead, Cauchy formulated calculus in terms of geometric ideas and infinitesimals. Thus, his definition of continuity required an infinitesimal change in x to correspond to an infinitesimal change in y. He also introduced the concept of the Cauchy sequence, and started the formal theory of complex analysis. Poisson, Liouville, Fourier and others studied partial differential equations and harmonic analysis. The contributions of these mathematicians and others, such as Weierstrass, developed the epsilontic approach, thus founding the modern field of mathematical analysis.

In the middle of the century Riemann introduced his theory of integration. The last third of the 19th century saw the arithmetization of analysis by Weierstrass, who thought that geometric reasoning was inherently misleading, and introduced the "epsilon-delta" definition of limit. Then, mathematicians started worrying that they were assuming the existence of a continuum of real numbers without proof. Dedekind then constructed the real numbers by Dedekind cuts, in which irrational numbers are formally defined, which serve to fill the "gaps" between rational numbers, thereby creating a complete set: the continuum of real numbers. Around that time, the attempts to refine the theorems of Riemann integration led to the study of the "size" of the set of discontinuities of real functions.

Also, " monsters" ( nowhere continuous functions, continuous but nowhere differentiable functions, space-filling curves) began to be created. In this context, Jordan developed his theory of measure, Cantor developed what is now called naive set theory, and Baire proved the Baire category theorem. In the early 20th century, calculus was formalized using an axiomatic set theory. Lebesgue solved the problem of measure, and Hilbert introduced Hilbert spaces to solve integral equations. The idea of normed vector space was in the air, and in the 1920s Banach created functional analysis.


Mathematical analysis includes the following subfields.

  • Differential equations
  • Real analysis, the rigorous study of derivatives and integrals of functions of real variables. This includes the study of sequences and their limits, series.
    • Multivariable calculus
    • Real analysis on time scales - a unification of real analysis with calculus of finite differences
  • Measure theory - given a set, the study of how to assign to each suitable subset a number, intuitively interpreted as the size of the subset.
  • Vector calculus
  • Functional analysis studies spaces of functions and introduces concepts such as Banach spaces and Hilbert spaces.
  • Calculus of variations deals with extremizing functionals, as opposed to ordinary calculus which deals with functions.
  • Harmonic analysis deals with Fourier series and their abstractions.
  • Geometric analysis involves the use of geometrical methods in the study of partial differential equations and the application of the theory of partial differential equations to geometry.
  • Complex analysis, the study of functions from the complex plane to itself which are complex differentiable (that is, holomorphic).
    • Several complex variables
  • Hypercomplex analysis or Clifford analysis
  • p-adic analysis, the study of analysis within the context of p-adic numbers, which differs in some interesting and surprising ways from its real and complex counterparts.
  • Non-standard analysis, which investigates the hyperreal numbers and their functions and gives a rigorous treatment of infinitesimals and infinitely large numbers. It is normally classed as model theory.
  • Numerical analysis, the study of algorithms for approximating the problems of continuous mathematics.
  • Computable analysis, the study of which parts of analysis can be carried out in a computable manner.
  • Stochastic calculus - analytical notions developed for stochastic processes.
  • Set-valued analysis - applies ideas from analysis and topology to set-valued functions.
  • Tropical analysis (or idempotent analysis) - analysis in the context of the semiring of the max-plus algebra where the lack of an additive inverse is compensated somewhat by the idempotent rule A+A=A. When transferred to the tropical setting, many nonlinear problems become linear.

Classical analysis would normally be understood as any work not using functional analysis techniques, and is sometimes also called hard analysis; it also naturally refers to the more traditional topics. The study of differential equations is now shared with other fields such as dynamical systems, though the overlap with conventional analysis is large.

Analysis in other areas:

  • Analytic number theory
  • Analytic combinatorics
  • Continuous probability
  • Differential entropy in information theory
  • Differential games
  • Differential geometry, the application of calculus to specific mathematical spaces known as manifolds that possess a complicated internal structure but behave in a simple manner locally.
  • Differential topology

Topological spaces, metric spaces

The motivation for studying mathematical analysis in the wider context of topological or metric spaces is three-fold:

  1. The same basic techniques have proved applicable to a wider class of problems (e.g., the study of function spaces).
  2. A greater understanding of analysis in more abstract spaces frequently proves to be directly applicable to classical problems. For example, in Fourier analysis, functions are expressed in terms of a certain infinite sum of trigonometric functions. Thus Fourier analysis might be used to decompose a sound into a unique combination of pure tones of various pitches. The "weights", or coefficients, of the terms in the Fourier expansion of a function can be thought of as components of a vector in an infinite dimensional space known as a Hilbert space. Study of functions defined in this more general setting thus provides a convenient method of deriving results about the way functions vary in space as well as time or, in more mathematical terms, partial differential equations, where this technique is known as separation of variables.
  3. The conditions needed to prove the particular result are stated more explicitly. The analyst then becomes more aware exactly what aspect of the assumption is needed to prove the theorem.

Calculus of finite differences, discrete calculus or discrete analysis

As the above section on topological spaces makes clear, analysis isn't just about continuity in the traditional sense of real numbers. Analysis is fundamentally about functions, the spaces that the functions act on and the function spaces that the functions themselves are members of. A discrete function f(n) is usually called a sequence a(n). A sequence could be a finite sequence from some data source or an infinite sequence from a discrete dynamical system. A discrete function could be defined explicitly by a list, or by a formula for f(n) or it could be given implicitly by a recurrence relation or difference equation. A difference equation is the discrete equivalent of a differential equation and can be used to approximate the latter or studied in its own right. Every question and method about differential equations has a discrete equivalent for difference equations. For instance where there are integral transforms in harmonic analysis for studing continuous functions or analog signals, there are discrete transforms for discrete functions or digital signals. As well as the discrete metric there are more general discrete or finite metric spaces and finite topological spaces.

Web pages

  • Earliest Known Uses of Some of the Words of Mathematics: Calculus & Analysis
  • Basic Analysis: Introduction to Real Analysis by Jiri Lebl (Creative Commons BY-NC-SA)
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