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finition of nearness a topological space or specific distances between objects a metric space contents 1 history 1 1 ancient 1 2 medieval 1 3 modern 1 3 1 foundations 1 3 2 modernization 2 important concepts 2 1 metric spaces 2 2 sequences and limits 3 main branches 3 1 real analysis 3 2 complex analysis 3 3 functional analysis 3 4 harmonic analysis 3 5 differential equations 3 6 measure theory 3 7 numerical analysis 3 8 vector analysis 3 9 scalar analysis 3 10 tensor analysis 4 other topics 5 applications 5 1 physical sciences 5 2 signal processing 5 3 other areas of mathematics 6 famous textbooks 7 see also 8 references 9 further reading 10 external links history edit archimedes used the method of exhaustion to compute the area inside a circle by finding the area of regular polygons with more and more sides this was an early but informal example of a limit one of the most basic concepts in mathematical analysis ancient edit mathematical analysis formally developed in the 17th century during the scientific 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 of ancient greek mathematics for instance an infinite geometric sum is implicit in zeno s paradox of the dichotomy 4 strictly speaking the point of the paradox is to deny that the infinite sum exists 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 5 the explicit use of infinitesimals appears in archimedes the method of mechanical theorems a work rediscovered in the 20th century 6 in asia the chinese mathematician liu hui used the method of exhaustion in the 3rd century ad 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 the arithmetic and geometric series as early as the 4th century b c 8 ācārya bhadrabāhu uses the sum of a geometric series in his kalpasūtra in 433 b c 9 in indian mathematics particular instances of arithmetic series have been found to implicitly occur in vedic literature as early as 2000 b c medieval edit zu chongzhi established a method that would later be called cavalieri s principle to find the volume of a sphere in the 5th century 10 in the 12th century the indian mathematician bhāskara ii gave examples of derivatives and used what is now known as rolle s theorem 11 in the 14th century madhava of sangamagrama developed infinite series expansions now called taylor series of functions such as sine cosine tangent and arctangent 12 alongside his development of taylor series of trigonometric functions he also estimated the magnitude of the error terms resulting of truncating these series and gave a rational approximation of some infinite series his followers at the kerala school of astronomy and mathematics further expanded his works up to the 16th century modern edit foundations edit the modern foundations of mathematical analysis were established in 17th century europe 3 this began when fermat and descartes developed analytic geometry which is the precursor to modern calculus fermat s method of adequality allowed him to determine the maxima and minima of functions and the tangents of curves 13 descartes s publication of la géométrie in 1637 which introduced the cartesian coordinate system is considered to be the establishment of mathematical analysis it would be a few decades later that 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 modernization edit in the 18th century euler introduced the notion of mathematical function 14 real analysis began to emerge as an independent subject when bernard 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 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 ε δ definition of limit approach thus founding the modern field of mathematical analysis around the same time riemann introduced his theory of integration and made significant advances in complex analysis towards the end of the 19 th century 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 which had already been developed by simon stevin in terms of decimal expansions 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 various pathological objects such as nowhere continuous functions continuous but nowhere differentiable functions and space filling curves commonly known as monsters began to be investigated 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 greatly improved measure theory and introduced his own theory of integration now known as lebesgue integration which proved to be a big improvement over riemann s 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 important concepts edit metric spaces edit main article metric space in mathematics a metric space is a set where a notion of distance called a metric between elements of the set is defined much of analysis happens in some metric space the most commonly used are the real line the complex plane euclidean space other vector spaces and the integers examples of analysis without a metric include measure theory which describes size rather than distance and functional analysis which studies topological vector spaces that need not have any sense of distance formally a metric space is an ordered pair m d displaystyle m d where m displaystyle m is a set and d displaystyle d is a metric on m displaystyle m i e a function d m m r displaystyle d colon m times m rightarrow mathbb r such that for any x y z m displaystyle x y z in m the following holds d x y 0 displaystyle d x y geq 0 with equality if and only if x y displaystyle x y identity of indiscernibles d x y d y x displaystyle d x y d y x symmetry and d x z d x y d y z displaystyle d x z leq d x y d y z triangle inequality by taking the third property and letting z x displaystyle z x it can be shown that d x y 0 displaystyle d x y geq 0 non negative sequences and limits edit main article sequence see also limit of a sequence a sequence is an ordered list like a set it contains members also called elements or terms 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 a function whose domain is a countable totally ordered set such as the natural numbers one of the most important properties of a sequence is convergence informally a sequence converges if it has a limit continuing informally a singly infinite sequence has a limit if it approaches some point x called the limit as n becomes very large that is for an abstract sequence a n with n running from 1 to infinity understood the distance between a n and x approaches 0 as n denoted lim n a n x displaystyle lim _ n to infty a_ n x main branches edit real analysis edit main article real analysis real analysis traditionally the theory of functions of a real variable is a branch of mathematical analysis dealing with the real numbers and real valued functions of a real variable 16 17 in particular it deals with the analytic properties of real functions and sequences including convergence and limits of sequences of real numbers the calculus of the real numbers and continuity smoothness and related properties of real valued functions complex analysis edit main article complex analysis 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 18 it is useful in many branches of mathematics including algebraic geometry number theory applied mathematics as well as in physics including hydrodynamics thermodynamics mechanical engineering electrical engineering and particularly quantum field theory complex analysis is particularly concerned with the analytic functions of complex variables or more generally meromorphic functions because the separate real and imaginary parts of any analytic function must satisfy laplace s equation complex analysis is widely applicable to two dimensional problems in physics functional analysis edit main article functional analysis functional analysis is a branch of mathematical analysis the core of which is formed by the study of vector spaces endowed with some kind of limit related structure e g inner product norm topology etc and the linear 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 of spaces of functions and the formulation of properties of transformations of functions such as the fourier transform as transformations defining continuous unitary etc operators between function spaces this point of view turned out to be particularly useful for the study of differential and integral equations harmonic analysis edit main article harmonic analysis harmonic analysis is a branch of mathematical analysis concerned with the representation of functions and signals as the superposition of basic waves this includes the study of the notions of fourier series and fourier transforms fourier analysis and of their generalizations harmonic analysis has applications in areas as diverse as music theory number theory representation theory signal processing quantum mechanics tidal analysis and neuroscience differential equations edit main article differential equations a differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders 21 22 23 differential equations play a prominent role in engineering physics economics biology and other disciplines differential equations arise in many areas of science and technology specifically whenever a deterministic 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 in classical 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 an equation of motion may be solved explicitly measure theory edit main article measure mathematics a measure on a set is a systematic way to assign a number to each suitable subset 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 the lebesgue measure on a euclidean space which assigns the conventional length area and volume of euclidean geometry to suitable subsets of the n displaystyle n dimensional euclidean space r n displaystyle mathbb r n for instance the lebesgue measure of the interval 0 1 displaystyle left 0 1 right in the real 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 x displaystyle x it must assign 0 to the empty 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 a consistent size to each subset of a given set while satisfying the other axioms of a measure one only finds trivial examples like the counting measure this problem was resolved by defining measure only on a sub collection of all subsets the so called measurable subsets which are required to form a σ displaystyle sigma algebra this means that countable unions countable intersections and complements 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 the axiom of choice numerical analysis edit main article numerical analysis numerical analysis is the study of algorithms that use numerical approximation as opposed to general symbolic manipulations for the problems of mathematical analysis as distinguished from discrete mathematics 25 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 numerical analysis naturally finds applications in all fields of engineering and the physical sciences but in the 21st century the life sciences and even the arts have adopted elements of scientific computations ordinary differential equations appear in celestial mechanics planets stars and galaxies numerical linear algebra is important for data analysis stochastic differential equations and markov chains are essential in simulating living cells for medicine and biology vector anal...