Astrange attractor arising from adifferential equation. Differential equations are an important area of mathematical analysis with many applications in science and engineering.
These theories are usually studied in the context ofreal andcomplex numbers andfunctions. Analysis evolved fromcalculus, which involves the elementary concepts and techniques of analysis.Analysis may be distinguished fromgeometry; however, it can be applied to anyspace ofmathematical objects that has a definition of nearness (atopological space) or specific distances between objects (ametric space).
Archimedes used themethod of exhaustion to compute thearea inside a circle by finding the area ofregular polygons with more and more sides. This was an early but informal example of alimit, one of the most basic concepts in mathematical analysis.
Mathematical analysis formally developed in the 17th century during theScientific Revolution,[3] but many of its ideas can be traced back to earlier mathematicians. Early results in analysis were implicitly present in the early days ofancient Greek mathematics. For instance, aninfinite geometric sum is implicit inZeno'sparadox of the dichotomy.[4] (Strictly speaking, the point of the paradox is to deny that the infinite sum exists.) Later,Greek mathematicians such asEudoxus andArchimedes made more explicit, but informal, use of the concepts of limits and convergence when they used themethod of exhaustion to compute the area and volume of regions and solids.[5] The explicit use ofinfinitesimals appears in Archimedes'The Method of Mechanical Theorems, a work rediscovered in the 20th century.[6] In Asia, theChinese mathematicianLiu Hui used the method of exhaustion in the 3rd century CE to find the area of a circle.[7] From Jain literature, it appears that Hindus were in possession of the formulae for the sum of thearithmetic andgeometric series as early as the 4th century BCE.[8]Ācārya Bhadrabāhu uses the sum of a geometric series in his Kalpasūtra in 433 BCE.[9]
In the 18th century,Euler introduced the notion of amathematical function.[14] Real analysis began to emerge as an independent subject whenBernard Bolzano introduced the modern definition of continuity in 1816,[15] 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 thegenerality of algebra widely used in earlier work, particularly by Euler. Instead, Cauchy formulated calculus in terms of geometric ideas andinfinitesimals. Thus, his definition of continuity required an infinitesimal change inx to correspond to an infinitesimal change iny. He also introduced the concept of theCauchy sequence, and started the formal theory ofcomplex analysis.Poisson,Liouville,Fourier and others studied partial differential equations andharmonic analysis. The contributions of these mathematicians and others, such asWeierstrass, developed the(ε, δ)-definition of limit approach, thus founding the modern field of mathematical analysis. Around the same time,Riemann introduced his theory ofintegration, and made significant advances in complex analysis.
Towards the end of the 19th century, mathematicians started worrying that they were assuming the existence of acontinuum ofreal numbers without proof.Dedekind then constructed the real numbers byDedekind cuts, in which irrational numbers are formally defined, which serve to fill the "gaps" between rational numbers, thereby creating acomplete set: the continuum of real numbers, which had already been developed bySimon Stevin in terms ofdecimal expansions. Around that time, the attempts to refine thetheorems ofRiemann integration led to the study of the "size" of the set ofdiscontinuities of real functions.
A sequence is an ordered list. Like aset, it containsmembers (also calledelements, orterms). Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence. Most precisely, a sequence can be defined as afunction whose domain is acountabletotally ordered set, such as thenatural numbers.
One of the most important properties of a sequence isconvergence. Informally, a sequence converges if it has alimit. Continuing informally, a (singly-infinite) sequence has a limit if it approaches some pointx, called the limit, asn becomes very large. That is, for an abstract sequence (an) (withn running from 1 to infinity understood) the distance betweenan andx approaches 0 asn → ∞, denoted
Real analysis (traditionally, the "theory of functions of a real variable") is a branch of mathematical analysis dealing with thereal numbers and real-valued functions of a real variable.[16][17] In particular, it deals with the analytic properties of realfunctions andsequences, includingconvergence andlimits ofsequences of real numbers, thecalculus of the real numbers, andcontinuity,smoothness and related properties of real-valued functions.
Complex analysis is particularly concerned with theanalytic functions of complex variables (or, more generally,meromorphic functions). Because the separatereal andimaginary parts of any analytic function must satisfyLaplace's equation, complex analysis is widely applicable to two-dimensional problems inphysics.
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study ofvector spaces endowed with some kind of limit-related structure (e.g.inner product,norm,topology, etc.) and thelinear operators acting upon these spaces and respecting these structures in a suitable sense.[19][20] The historical roots of functional analysis lie in the study ofspaces of functions and the formulation of properties of transformations of functions such as theFourier transform as transformations definingcontinuous,unitary etc. operators between function spaces. This point of view turned out to be particularly useful for the study ofdifferential andintegral equations.
Differential equations arise in many areas of science and technology, specifically whenever adeterministic relation involving some continuously varying quantities (modeled by functions) and their rates of change in space or time (expressed as derivatives) is known or postulated. This is illustrated inclassical mechanics, where the motion of a body is described by its position and velocity as the time value varies.Newton's laws allow one (given the position, velocity, acceleration and various forces acting on the body) to express these variables dynamically as a differential equation for the unknown position of the body as a function of time. In some cases, this differential equation (called anequation of motion) may be solved explicitly.
A measure on aset is a systematic way to assign a number to each suitablesubset of that set, intuitively interpreted as its size.[24] In this sense, a measure is a generalization of the concepts of length, area, and volume. A particularly important example is theLebesgue measure on aEuclidean space, which assigns the conventionallength,area, andvolume ofEuclidean geometry to suitable subsets of the-dimensional Euclidean space. For instance, the Lebesgue measure of theinterval in thereal numbers is its length in the everyday sense of the word – specifically, 1.
Technically, a measure is a function that assigns a non-negative real number or+∞ to (certain) subsets of a set. It must assign 0 to theempty set and be (countably) additive: the measure of a 'large' subset that can be decomposed into a finite (or countable) number of 'smaller' disjoint subsets, is the sum of the measures of the "smaller" subsets. In general, if one wants to associate aconsistent size toeach subset of a given set while satisfying the other axioms of a measure, one only finds trivial examples like thecounting measure. This problem was resolved by defining measure only on a sub-collection of all subsets; the so-calledmeasurable subsets, which are required to form a-algebra. This means that the empty set, countableunions, countableintersections andcomplements of measurable subsets are measurable.Non-measurable sets in a Euclidean space, on which the Lebesgue measure cannot be defined consistently, are necessarily complicated in the sense of being badly mixed up with their complement. Indeed, their existence is a non-trivial consequence of theaxiom of choice.
Modern numerical analysis does not seek exact answers, because exact answers are often impossible to obtain in practice. Instead, much of numerical analysis is concerned with obtaining approximate solutions while maintaining reasonable bounds on errors.
Scalar analysis is a branch of mathematical analysis dealing with values related to scale as opposed to direction. Values such as temperature are scalar because they describe the magnitude of a value without regard to direction, force, or displacement that value may or may not have.
Clifford analysis, the study of Clifford valued functions that are annihilated by Dirac or Dirac-like operators, termed in general as monogenic or Clifford analytic functions.
p-adic analysis, the study of analysis within the context ofp-adic numbers, which differs in some interesting and surprising ways from its real and complex counterparts.
Idempotent analysis – analysis in the context of anidempotent semiring, where the lack of an additive inverse is compensated somewhat by the idempotent rule A + A = A.
When processing signals, such asaudio,radio waves, light waves,seismic waves, and even images, Fourier analysis can isolate individual components of a compound waveform, concentrating them for easier detection or removal. A large family of signal processing techniques consist of Fourier-transforming a signal, manipulating the Fourier-transformed data in a simple way, and reversing the transformation.[27]
Differential geometry, the application of calculus to specific mathematical spaces known asmanifolds that possess a complicated internal structure but behave in a simple manner locally.
^Stillwell, John Colin (2004). "Infinite Series".Mathematics and its History (2nd ed.).Springer Science+Business Media Inc. p. 170.ISBN978-0387953366.Infinite series were present in Greek mathematics, [...] There is no question that Zeno's paradox of the dichotomy (Section 4.1), for example, concerns the decomposition of the number 1 into the infinite series1⁄2 +1⁄22 +1⁄23 +1⁄24 + ... and that Archimedes found the area of the parabolic segment (Section 4.4) essentially by summing the infinite series 1 +1⁄4 +1⁄42 +1⁄43 + ... =4⁄3. Both these examples are special cases of the result we express as summation of a geometric series
^Rajagopal, C. T.; Rangachari, M. S. (June 1978). "On an untapped source of medieval Keralese Mathematics".Archive for History of Exact Sciences.18 (2):89–102.doi:10.1007/BF00348142.S2CID51861422.
^*Cooke, Roger (1997)."Beyond the Calculus".The History of Mathematics: A Brief Course. Wiley-Interscience. p. 379.ISBN978-0471180821.Real analysis began its growth as an independent subject with the introduction of the modern definition of continuity in 1816 by the Czech mathematician Bernard Bolzano (1781–1848)
^Real Analysis: Measure Theory, Integration, and Hilbert Spaces.ASIN0691113866.
^Ahlfors, Lars (1979-01-01).Complex Analysis: An Introduction to the Theory of Analytic Functions of One Complex Variable. McGraw-Hill Education.ISBN978-0070006577.
