mathematical analysis a strange attractor arising from a differential equation differential equations are an important area of mathematical analysis with many applications to science and engineering mathematical analysis is the branch of mathematics dealing with limits and related theories such as differentiation integration measure infinite series and analytic functions these theories are usually studied in the context of real and complex numbers and functions analysis evolved from calculus which involves the elementary concepts and techniques of analysis analysis may be distinguished from geometry however it can be applied to any space of mathematical objects that has a definition of nearness a topological space or specific distances between objects a metric space history 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 mathematical analysis formally developed in the 17th century during the scientific revolution 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 the modern foundations of mathematical analysis were established in 17th century europe descartes and fermat independently developed analytic geometry and a few decades later newton and leibniz independently developed infinitesimal calculus which grew with the stimulus of applied work that continued through the 18thcentury 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 but bolzanos 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 definition of limit approach thus founding the modern field of mathematical analysis in the middle of the 19th century riemann introduced his theory of integration the last third of the century saw the arithmetization of analysis by weierstrass who thought that geometric reasoning was inherently misleading and introduced the epsilondelta 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 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 monsters nowhere continuous functions continuous but nowhere differentiable functions spacefilling curves 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 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 important concepts metric spaces 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 md where m displaystyle m is a set and d displaystyle d is a metric on m displaystyle m ie a function d m m r displaystyle dcolon mtimes mrightarrow mathbb r such that for any x y z m displaystyle xyzin m the following holds d x y 0 displaystyle dxy0 if and only if x y displaystyle xy identity of indiscernibles d x y d y x displaystyle dxydyx symmetry and d x z d x y d y z displaystyle dxzleq dxydyz triangle inequality by taking the third property and letting z x displaystyle zx it can be shown that d x y 0 displaystyle dxygeq 0 nonnegative sequences and limits 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 singlyinfinite sequence has a limit if it approaches some point x called the limit as n becomes very large that is for an abstract sequence an with n running from 1 to infinity understood the distance between an and x approaches 0 as n denoted lim n a n x displaystyle lim nto infty anx main branches real analysis 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 realvalued functions of a real variable 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 realvalued functions complex analysis 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 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 laplaces equation complex analysis is widely applicable to twodimensional problems in physics functional analysis 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 limitrelated structure eg inner product norm topology etc and the linear operators acting upon these spaces and respecting these structures in a suitable sense 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 differential equations 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 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 newtons 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 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 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 in the real numbers is its length in the everyday sense of the wordspecifically 1 technically a measure is a function that assigns a nonnegative 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 subcollection of all subsets the socalled measurable subsets which are required to form a displaystyle sigma algebra this means that countable unions countable intersections and complements of measurable subsets are measurable nonmeasurable 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 nontrivial consequence of the axiom of choice numerical analysis 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 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 21stcentury 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 other topics calculus of variations deals with extremizing functionals as opposed to ordinary calculus which deals with functions harmonic analysis deals with the representation of functions or signals as the superposition of basic waves 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 clifford analysis the study of clifford valued functions that are annihilated by dirac or diraclike operators termed in general as monogenic or clifford analytic functions padic analysis the study of analysis within the context of padic numbers which differs in some interesting and surprising ways from its real and complex counterparts nonstandard analysis which investigates the hyperreal numbers and their functions and gives a rigorous treatment of infinitesimals and infinitely large numbers 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 setvalued analysis applies ideas from analysis and topology to setvalued functions convex analysis the study of convex sets and functions tropical analysis or idempotent analysis analysis in the context of the semiring of the maxplus algebra where the lack of an additive inverse is compensated somewhat by the idempotent rule aaa when transferred to the tropical setting many nonlinear problems become linear nonnewtonian calculus alternatives to the classical calculus of newton and leibniz applications techniques from analysis are also found in other areas such as physical sciences the vast majority of classical mechanics relativity and quantum mechanics is based on applied analysis and differential equations in particular examples of important differential equations include newtons second law the schrdinger equation and the einstein field equations functional analysis is also a major factor in quantum mechanics signal processing when processing signals such as audio 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 fouriertransforming a signal manipulating the fouriertransformed data in a simple way and reversing the transformation other areas of mathematics techniques from analysis are used in many areas of mathematics including 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 differentiable manifolds differential topology partial differential equations analysis portal constructive analysis history of calculus nonclassical analysis paraconsistent logic smooth infinitesimal analysis timeline of calculus and mathematical analysis notes aleksandrov a d kolmogorov a n lavrentev m a eds 1984 mathematics its content methods and meaning translated by gould s h hirsch k a bartha t translation edited by s h gould 2nd ed mit press published in cooperation with the american mathematical society apostol tom m 1974 mathematical analysis 2nd ed addisonwesley isbn9780201002881 binmore k g 19801981 the foundations of analysis a straightforward introduction cambridge university press johnsonbaugh richard pfaffenberger w e 1981 foundations of mathematical analysis new york m dekker nikolskii s m 2002 mathematical analysis in hazewinkel michiel encyclopaedia of mathematics springerverlag isbn1402006098 archived from the original on 9 april 2006 nicola fusco paolo marcellini carlo sbordone 1996 analisi matematica due in italian liguori editore isbn9788820726751cs1 maint multiple names authors list link rombaldi jeantienne 2004 lments danalyse relle capes et agrgation interne de mathmatiques in french edp sciences isbn286883681x rudin walter 1976 principles of mathematical analysis 3rd ed new york mcgrawhill isbn007054235x rudin walter 1987 real and complex analysis 3rd ed new york mcgrawhill isbn0070542341 smith david e 1958 history of mathematics dover publications isbn0486204308 whittaker e t watson g n 1927 a course of modern analysis 4th ed cambridge university press isbn0521588073 real analysis course notes pdf wikiquote has quotations related to mathematical analysis wikimedia commons has media related to mathematical analysis earliest known uses of some of the words of mathematics calculus analysis basic analysis introduction to real analysis by jiri lebl creative commons byncsa mathematical analysisencyclopdia britannica calculus and analysis vteareas of mathematics outline topic lists branchesalgebra linear multilinear abstract elementary arithmetic number theory calculus analysis differential equations dynamical systems numerical analysis optimization functional analysis geometry discrete algebraic analytic differential finite topology trigonometry foundations philosophy of mathematics mathematical logic set theory category theory applied mathematical physics probability mathematical statistics statistics game theory information theory computer science computation control theory others history of mathematics recreational mathematics mathematics and art mathematics education order theory graph theory divisions pure applied discrete computational category portal commons wikiproject vteindustrial and applied mathematicscomputational computational logic algorithms design analysis information theory coding theory cryptography discrete computer algebra computational number theory combinatorics graph theory discrete geometry mathematicalphysicsalgebraic operator algebra representation theory renormalization group feynman integral mtheory analytic differential geometry fourier analysis harmonic analysis functional analysis operator theory analysis approximation theory numerical analysis differential equations dynamical systems control theory variational calculus probability theory distributionsrandom variables stochastic processes analysis path integral stochastic variational calculus decision sciences statistics operations research optimization game theory mathematical economics mathematical finance other applications biology chemistry psychology sociology the unreasonable effectiveness of mathematics in the natural sciences category mathematics portal outline topics list authority control ndl 00564620 