{{short description|Branch of mathematics}} {{use dmy dates|date=May 2021|cs1-dates=y}} [[File:Attracteur étrange de Lorenz.png|thumb|upright=1.2|A [[strange attractor]] arising from a [[differential equation]]. Differential equations are an important area of mathematical analysis with many applications in science and engineering.]] '''Mathematical analysis''' is the branch of [[mathematics]] that studies [[function (mathematics)|functions]], spaces, and operators through quantitative methods of approximation and convergence. It grew out of [[calculus]], especially the use of [[derivative]]s and [[integral]]s to study variable quantities, and in the 19th century its foundations were reformulated with greater rigor. Basic objects of study in mathematical analysis include the [[real number]]s, [[function (mathematics)|functions]], [[sequence (mathematics)|sequences]], [[series (mathematics)|series]], and [[limit (mathematics)|limits]]. Analysis has remained closely connected with applications in the sciences, where it is used to study equations, approximate one object by another, and estimate the accuracy of such approximations. Modern analysis studies these questions in many settings, including [[Euclidean space]]s, [[metric space]]s, [[topological space]]s, [[measure space]]s, and [[function space]]s. Its major areas include [[complex analysis]], [[functional analysis]], [[measure theory]], [[harmonic analysis]], and the theory of [[ordinary differential equation|ordinary]] and [[partial differential equation]]s. == History == [[Image:Archimedes pi.svg|thumb|right|300px|[[Archimedes]] used the [[method of exhaustion]] to compute the [[area]] inside a circle by finding the area of [[regular polygon]]s with more and more sides. This was an early but informal example of a [[limit (mathematics)|limit]], one of the most basic concepts in mathematical analysis.]] ===Ancient=== Mathematical analysis formally developed in the 17th century during the [[Scientific Revolution]],{{cite book|last=Jahnke|first=Hans Niels|title=A History of Analysis|series=History of Mathematics |url=https://books.google.com/books?id=CVRZEXFVsZkC&pg=PR7|date=2003|volume=24 |publisher=[[American Mathematical Society]]|isbn=978-0821826232|page=7|access-date=2015-11-15|archive-date=2016-05-17|archive-url=https://web.archive.org/web/20160517180439/https://books.google.com/books?id=CVRZEXFVsZkC&pg=PR7|url-status=live|doi=10.1090/hmath/024}} but many of its ideas can be traced back to earlier mathematicians. Early results in analysis were implicitly present in the early days of [[Greek mathematics|ancient Greek mathematics]]. For instance, an [[geometric series|infinite geometric sum]] is implicit in [[Zeno of Elea|Zeno's]] [[Zeno's paradoxes#Dichotomy paradox|paradox of the dichotomy]]. (Strictly speaking, the point of the paradox is to deny that the infinite sum exists.) Later, [[Greek mathematics|Greek mathematicians]] such as [[Eudoxus of Cnidus|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. The explicit use of [[infinitesimals]] appears in Archimedes' ''[[The Method of Mechanical Theorems]]'', a work rediscovered in the 20th century.{{cite book|last=Pinto|first=J. Sousa|title=Infinitesimal Methods of Mathematical Analysis|url=https://books.google.com/books?id=bLbfhYrhyJUC&pg=PA7|date=2004|publisher=Horwood Publishing|isbn=978-1898563990|page=8|access-date=2015-11-15|archive-date=2016-06-11|archive-url=https://web.archive.org/web/20160611045431/https://books.google.com/books?id=bLbfhYrhyJUC&pg=PA7|url-status=live}} In Asia, the [[Chinese mathematics|Chinese mathematician]] [[Liu Hui]] used the method of exhaustion in the 3rd century CE to find the area of a circle.{{cite book|series=Chinese studies in the history and philosophy of science and technology|volume=130|title=A comparison of Archimedes' and Liu Hui's studies of circles|first1=Liu|last1=Dun|first2=Dainian|last2=Fan|first3=Robert Sonné|last3=Cohen|publisher=Springer|date=1966|isbn=978-0-7923-3463-7|page=279|url=https://books.google.com/books?id=jaQH6_8Ju-MC|access-date=2015-11-15|archive-date=2016-06-17|archive-url=https://web.archive.org/web/20160617055211/https://books.google.com/books?id=jaQH6_8Ju-MC|url-status=live}}, [https://books.google.com/books?id=jaQH6_8Ju-MC&pg=PA279 Chapter, p. 279] {{Webarchive|url=https://web.archive.org/web/20160526221958/https://books.google.com/books?id=jaQH6_8Ju-MC&pg=PA279 |date=2016-05-26 }} From Jain literature, it appears that Hindus were in possession of the formulae for the sum of the [[arithmetic series|arithmetic]] and [[geometric series|geometric]] series as early as the 4th century BCE.{{cite journal | title = On the Use of Series in Hindu Mathematics | author = Singh, A. N. | journal = Osiris | volume = 1 |date = 1936 | pages = 606–628 | doi = 10.1086/368443 | jstor = 301627 | s2cid = 144760421 | url = https://www.jstor.org/stable/301627}} [[Bhadrabahu|Ācārya Bhadrabāhu]] uses the sum of a geometric series in his Kalpasūtra in {{BCE|433}}.{{cite journal | title = Summation of Convergent Geometric Series and the concept of approachable Sunya | author = K. B. Basant, Satyananda Panda | journal = Indian Journal of History of Science | volume = 48 |date = 2013 | pages = 291–313 | url = https://insa.nic.in/writereaddata/UpLoadedFiles/IJHS/Vol48_2_7_KBBasant.pdf}} ===Medieval=== During the medieval period, mathematicians in several traditions developed methods that anticipated later topics in analysis, including quadrature, infinite series, infinitesimal reasoning, approximation, and the mathematical study of motion. In the Islamic world, [[Ibn al-Haytham]] worked on sums of powers and area problems, and [[Ibrahim ibn Sinan]] generalized Archimedean methods in the quadrature of the parabola.{{cite journal |last=Katz |first=Victor J. |title=Ideas of Calculus in Islam and India |journal=Mathematics Magazine |volume=68 |issue=3 |year=1995 |pages=163–174 |doi=10.2307/2691411 }}{{cite web |last=O'Connor |first=John J. |author-link=John J. O'Connor (mathematician) |last2=Robertson |first2=Edmund F. |author2-link=Edmund F. Robertson |title=Ibrahim ibn Sinan |website=MacTutor History of Mathematics Archive |publisher=University of St Andrews |url=https://mathshistory.st-andrews.ac.uk/Biographies/Ibrahim/ |access-date=19 June 2026 }} In medieval Europe, the [[Oxford Calculators]] studied uniformly accelerated motion, and [[Nicole Oresme]] gave a graphical proof of the [[mean speed theorem]], representing displacement by the area under a velocity-time graph. Oresme also used infinite series and proved the divergence of the [[harmonic series (mathematics)|harmonic series]].{{cite encyclopedia |last=Stillwell |first=John Colin |title=Analysis: Models of motion in medieval Europe |encyclopedia=Encyclopaedia Britannica |publisher=Encyclopaedia Britannica, Inc. |url=https://www.britannica.com/science/analysis-mathematics/Models-of-motion-in-medieval-Europe |access-date=19 June 2026 }} In India, [[Bhāskara II]] used infinitesimal reasoning and stated results related to what is now called [[Rolle's theorem]], and the [[Kerala school of astronomy and mathematics]], especially [[Madhava of Sangamagrama|Madhava]], developed infinite series for trigonometric functions.{{citation |last=Seal |first=Brajendranath |author-link=Brajendra Nath Seal |title=The positive sciences of the ancient Hindus |journal=Nature |volume=97 |issue=2426 |page=177 |date=1916 |doi=10.1038/097177a0 |bibcode=1916Natur..97..177. |hdl=2027/mdp.39015004845684 |s2cid=3958488 |hdl-access=free }} These developments were not mathematical analysis in the modern sense, but were important precursors to calculus and analysis. Questions of the nature of the [[continuum (set theory)|continuum]] were important to medieval European philosophers. In particular, whether the continuum could be infinitely divided, whether it consisted of points. The works of Aristotle, which only became more widely available in Europe in the earth 13th century, held that the continuum was not made of points: if it were, the points would have to be side-by-side which would be incompatible with the inseparability of the continuum. In the 14th century, [[Thomas Bradwardine]] described the continuum as being made of an infinite collection of [[infinitesimal]]s in ''Tractatus de continuo'', but not of points. [[William of Occam]] held, contrarily, that the continuum is made of actual points. Aristotle had also written on the nature of infinity, classifying two different kinds of infinity: [[potential infinity]] and [[actual infinity]]. For medieval European philosophers, the nature of infinity posed problems for the ideas of infinite divisibility and infinitesimals, because if a subdivision of a continuum was infinitesimal, its ratio to the whole continuum would need to be infinite, and various paradoxes would result. [[Richard Swineshead]] in ''Liber calculationum'' wrote that such ratios should simply be left undefined, and that the arguments about infinity do not proceed as arguments about finite quantities. These are some of the earliest seeds in the European tradition in which Leibniz's views on the continuum would emerge some three centuries years later, with Leibniz explicitly crediting Swineshead.{{cite book |last=Boyer |first=Carl B. |title=The History of the Calculus and Its Conceptual Development |publisher=Dover Publications |location=New York |year=1959 |orig-year=1939 |pages=66–70 }} ===Renaissance=== {{see also|History of calculus}} In the early seventeenth century, methods that anticipated the calculus became increasingly systematic. [[Johannes Kepler]] used infinitesimal and summation arguments in problems of area and volume, [[Galileo Galilei]] connected mathematics with the study of motion, [[Bonaventura Cavalieri]] developed the method of indivisibles, and [[Evangelista Torricelli]] extended such methods in geometry and mechanics. These works did not yet constitute mathematical analysis in the modern sense, but they helped shift the subject from classical geometric constructions toward general methods for treating continuous quantities, tangents, areas, volumes, and motion. The subsequent development of differential and integral calculus by Newton and Leibniz became the starting point for much of later analysis.{{cite book |last=Boyer |first=Carl B. |title=The History of the Calculus and Its Conceptual Development |publisher=Dover Publications |location=New York |year=1959 |orig-year=1939 |pages=101–150 }} ===Modern=== ====Foundations==== The modern foundations of mathematical analysis were established in 17th century Europe. 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.{{cite web | last = Pellegrino | first = Dana | title = Pierre de Fermat | url = http://www.math.rutgers.edu/~cherlin/History/Papers2000/pellegrino.html | access-date = 2008-02-24 | archive-date = 2008-10-12 | archive-url = https://web.archive.org/web/20081012024028/http://www.math.rutgers.edu/~cherlin/History/Papers2000/pellegrino.html | url-status = live }} 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 [[Isaac Newton|Newton]] and [[Gottfried Leibniz|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 differential equation|ordinary]] and [[partial differential equation]]s, [[Fourier analysis]], and [[generating function]]s. During this period, calculus techniques were applied to approximate [[discrete mathematics|discrete problems]] by continuous ones. ====Modernization==== In the 18th century, [[Leonhard Euler|Euler]] introduced the notion of a [[function (mathematics)|mathematical function]].{{cite book| last = Dunham| first = William| title = Euler: The Master of Us All| url = https://archive.org/details/eulermasterofusa0000dunh| url-access = registration| date = 1999| publisher =The Mathematical Association of America | page= [https://archive.org/details/eulermasterofusa0000dunh/page/17 17]}} Real analysis began to emerge as an independent subject when [[Bernard Bolzano]] introduced the modern definition of continuity in 1816,*{{cite book |first=Roger |last=Cooke |author-link=Roger Cooke (mathematician) |title=The History of Mathematics: A Brief Course |publisher=Wiley-Interscience |date=1997 |isbn=978-0471180821 |page=[https://archive.org/details/historyofmathema0000cook/page/379 379] |chapter=Beyond the Calculus |quote=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) |chapter-url=https://archive.org/details/historyofmathema0000cook/page/379 }} but Bolzano's work did not become widely known until the 1870s. In 1821, [[Augustin Louis Cauchy|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 [[infinitesimal]]s. 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]]. [[Siméon Denis Poisson|Poisson]], [[Joseph Liouville|Liouville]], [[Joseph Fourier|Fourier]] and others studied partial differential equations and [[harmonic analysis]]. The contributions of these mathematicians and others, such as [[Karl Weierstrass|Weierstrass]], developed the [[(ε, δ)-definition of limit]] approach, thus founding the modern field of mathematical analysis. Around the same time, [[Bernhard Riemann|Riemann]] introduced his theory of [[integral|integration]], and made significant advances in complex analysis. Towards the end of the 19th century, mathematicians started worrying that they were assuming the existence of a [[Continuum (set theory)|continuum]] of [[real number]]s without proof. [[Richard Dedekind|Dedekind]] then constructed the real numbers by [[Dedekind cut]]s, in which irrational numbers are formally defined, which serve to fill the "gaps" between rational numbers, thereby creating a [[complete metric space|complete]] set: the continuum of real numbers, which had already been developed by [[Simon Stevin]] in terms of [[decimal expansion]]s. Around that time, the attempts to refine the [[theorem]]s of [[Riemann integral|Riemann integration]] led to the study of the "size" of the set of [[Classification of discontinuities|discontinuities]] of real functions. Also, various [[pathological (mathematics)|pathological objects]], (such as [[nowhere continuous function]]s, continuous but [[Weierstrass function|nowhere differentiable functions]], and [[space-filling curve]]s), commonly known as "monsters", began to be investigated. In this context, [[Camille Jordan|Jordan]] developed his theory of [[Jordan measure|measure]], [[Georg Cantor|Cantor]] developed what is now called [[naive set theory]], and [[René-Louis Baire|Baire]] proved the [[Baire category theorem]]. In the early 20th century, calculus was formalized using an axiomatic [[set theory]]. [[Henri Lebesgue|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. [[David Hilbert|Hilbert]] introduced [[Hilbert space]]s to solve [[integral equation]]s. The idea of [[normed vector space]] was in the air, and in the 1920s [[Stefan Banach|Banach]] created [[functional analysis]]. == Important concepts == ===Real numbers=== The [[real numbers]] provide the standard setting for much of classical analysis. Their completeness, often expressed by the [[least-upper-bound property]], underlies basic results about [[limit (mathematics)|limits]], [[continuous function|continuity]], [[derivative|differentiation]], and [[integral|integration]]. ===Approximation and convergence=== Approximation plays a fundamental role in many areas of mathematics. An example is the [[limit of a sequence]] of real numbers. A sequence of real numbers is a family a_{1}, a_{2}\ldots of real numbers, each indexed by a natural number. A sequence is said to ''converge'' to a limit L if almost all members of the sequence are arbitrarily close to L. More precisely, this means that for any error tolerance \epsilon, all of the members of the sequence are within \epsilon of L except possibly for finitely many members of the sequence. Formally, for any error \epsilon > 0 there is an integer N such that |a_n-L|<\epsilon whenever n>N. This example shows the use of approximation: the elements a_n ''approximate'' the number L, and the error tolerance is \epsilon. However, convergence alone does not provide information on how good the approximation is, and many results in analysis concern the quality of approximation. Another example comes from differential calculus. A real function f is differentiable at a point a if there is a linear function L(x) = f(a) + f'(a)(x-a) that approximates the function well near the point x=a. But "well" here only means that the approximation error |f(x) - L(x)| = o(x-a) where o(x-a) [[little-o notation|is a function]] that tends to zero faster than x-a as x\to a. Better approximations provide more uniform and quantitative estimates on the size of the error term. [[Taylor's theorem]], for example, states that for a twice continuously-differentiable function on a closed interval containing a |f(x) - L(x)| \le M (x-a)^2 where M can be estimated explicitly using the second derivative. This allows the error in the linear approximation to determined much more precisely than differentiability at the point. The [[inequality]] |f(x)-L(x)| \le M(x-a)^2 is an example of what is called an ''estimate'' in analysis. Estimates are inqualities that are used to quantify the error in an approximation, as well as more generally to express that a certain operation is controlled (bounded) by some other operation. === Continuity === Continuity also plays a key role in analysis. In elementary analysis, the idea of a [[continuous function]] is introduced using an [[epsilon-delta definition]]. Roughly, a function is continuous at a point if sufficiently small changes in the input produce small changes in the output. This rules out "jumps" in the graph of the function, or other kinds of pathological oscillatory behavior. Continuity is important in analysis because it allows local control of a function to imply global conclusions when combined with additional hypotheses such as connectedness or compactness. For example, continuous real-valued functions on intervals have the [[intermediate value property]], and continuous real-valued functions on compact sets attain [[extreme value theorem|maximum and minimum values]]. Continuous functions on compact metric spaces are also [[uniformly continuous]], meaning that the function oscillates on a comparable scale throughout the domain. These results are basic tools in calculus, optimization, differential equations, and approximation theory. Continuous functions generalize readily to metric spaces and other [[topological space]]s. Spaces of continuous functions are among the most studied in [[functional analysis]], where they provide useful structural probes of a space. In differential equations, continuity is also a threshold regularity condition: in more advanced analysis, regularity theorems often show that objects first defined only weakly, [[almost everywhere]], or [[distribution (mathematics)|distributionally]] have continuous representatives under additional hypotheses, and can thus be treated as honest functions rather than more general objects. === Metric spaces === {{Main|Metric space}} A metric space is a [[Set (mathematics)|set]] where a notion of [[distance]] (called a [[metric (mathematics)|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 space]]s, and the [[integer]]s. Much of [[functional analysis]] is concerned with [[function space|spaces of functions]], which can be given the structure of a metric space, such as [[Banach space]]s and [[Hilbert space]]s. In many of these examples, the metric comes from a [[normed space|norm]]. For example, the space of continuous real-valued functions C([0,1]) on the [[unit interval]] is a Banach space under the [[supremum norm]]. The spaces of primary importance in [[measure theory]] and [[harmonic analysis]] are the [[Lp spaces|L''p'' spaces]], which are metric spaces whose metrics again come from a norm, are [[complete metric space|complete]]; i.e., they are Banach spaces. Metric spaces are extremely convenient in analysis because many of the strong approximation results that hold in Euclidean space carry over to metric spaces with minimal changes. For example, compactness in metric spaces is equivalent to sequential compactness, so limiting arguments on metric spaces can be comparatively straightforward in metric spaces as opposed to in more general spaces in analysis. [[Compact metric space]]s and [[complete metric space]]s are especially important in analysis, where many arguments require the existence of limits. === Complex variables === {{main|Complex analysis}} [[Complex number]]s provide another important tool for many analytic equations. A complex-valued function of a complex variable is [[holomorphic function|holomorphic]] if its derivative exists as a complex limit at each point of its domain. Holomorphic functions are much more rigid than differentiable functions of a real variable: they are [[analytic function]]s, represented locally by convergent power series. An important tool in complex variables is [[contour integration]], in which functions are integrated along curves in the complex plane. The [[Cauchy integral theorem]], [[Cauchy integral formula]], and [[residue theorem]] relate the values of a holomorphic or meromorphic function to its behavior on curves and near singularities. These results allow one to shift contours when integrating a holomorphic function, provided the deformation of the contour does not cross a singularity. This is useful in evaluating many real integrals, and in the study of functions through their singularities. In [[operator theory]] and [[spectral theory]], the [[resolvent formalism|resolvent]] of an operator encodes information about its [[spectrum of an operator|spectrum]] and often allows functions of operators to be defined by complex integration. Complex variables thus appear in connection with differential and integral equations, eigenvalue problems, and the theory of linear operators. === Measures, averaging, and probability === [[Measure theory]] gives a systematic way of assigning sizes to subsets of a space, in a way that generalizes length, area, and volume in Euclidean space. Abstractly, measure theory begins by specifying a class of sets which are [[measurable set|measurable]], that is, sets that have a measure. The measurable sets form a [[sigma algebra]], meaning that one can take countable unions and intersections, as well as set complements. Measures are then defined in such a way as to be compatible with the set operations on the measurable sets. In measure theory, pointwise convergence of functions can be replaced with the notion of convergence [[almost everywhere]], that is, convergence at every point except a set whose measure is zero. Convergence almost everywhere is much more convenient in measure theory: often various types of mean convergence cannot control a precise set on which pointwise convergence fails, but can ensure that the bad set has measure zero. Functions are often identified if they agree almost everywhere, that is, off a set of measure zero. Thus measure zero sets are often in practice simply ignored. The Lebesgue integral extends the [[Riemann integral]], and is better adapted to the limiting processes of analysis. The idea of the Lebesgue integral, for a non-negative function f, is that it possible to form a [[symmetric decreasing rearrangement|rearrangement]] of its values to form a decreasing function, using the measure of its [[superlevel set]]s to define the rearrangement. Integration of decreasing functions have better limit properties under integration, and that good behavior can be transferred to the Lebesgue integral. One has theorems like [[monotone convergence]], [[Fatou's lemma]], and the [[dominated convergence theorem]] that simplify many limit arguments. Measure theory also supplies the underlying mathematics of [[probability theory]]. A [[probability space]] is a measure space whose total measure is one, and [[expected value]]s are integrals with respect to this probability measure. Many function spaces in analysis are defined using measures. The [[Lp space|L''p'' spaces]] consist of functions whose powers are integrable, with functions identified when they agree almost everywhere. These spaces are important throughout analysis, such as in [[harmonic analysis]] and [[partial differential equation]]s. === Decomposing functions into simpler pieces === Another theme in analysis is that of decomposing functions into simpler pieces, determined by symmetry, scale, or oscillation. The prototypical example is that of [[Fourier series]], which decomposes a [[periodic function]] into basic [[sinusoid]]s. The Fourier series takes the form of a [[trigonometric series]] \sum_{n=-\infty}^\infty c_n e^{in\theta} where c_n are complex numbers and \theta is the independent variable. The Fourier series is one instance of an [[eigenfunction expansion]], with the exponentials e^{in\theta} being the eigenfunctions of the [[rotation group]] acting on the circle. It has the property of being an orthogonal expansion: any two of the eigenfunctions are orthogonal in the [[Hilbert space]] of [[square integrable function]]s on the circle. Eigenfunction expansions appear in many areas in mathematical analysis, and particularly in its applications to the sciences where symmetry is often important. An example of a different sort of eigenfunction expansion is the decomposition of a function on the sphere into [[spherical harmonics]]. Again, this is an orthogonal expansion, and orthogonality of the expansion leads to a tractable isolation of each space. Expansions can be used to study convergence and approximation, smoothness and oscillation, decay, and solutions of differential equations. [[Sobolev space]]s, for example, relate the smoothness of functions to the decay of their Fourier coefficients. These representations can exhibit structural information that may be difficult to see from the original form of a function. [[Fourier analysis]] is the study of such decompositions, chiefly focused on the [[Fourier transform]] and some of its generalizations. === Dynamics and evolution === {{Main|Dynamical system|Differential equation|Ergodic theory|C0 semigroup}} Many problems in analysis concern how quantities change over time or under repeated application of a rule. This leads to [[ordinary differential equation]]s and [[partial differential equation]]s, where one studies functions whose derivatives satisfy prescribed relations. For repeated applications of a rule, one studies iterates of a function or transformation, giving rise to discrete [[dynamical system]]s. Analytic questions in dynamics include existence and uniqueness of solutions, [[stability theory|stability]], approximation of trajectories, long-time behavior, and dependence on [[initial conditions]]. For example, a differential equation may determine a [[flow (mathematics)|flow]] on a space, while iteration of a map produces an orbit. Analytic tools are used to determine whether such orbits converge, remain bounded, become periodic, or exhibit more complicated behavior. Dynamics is an important application of analytic methods, and leads to ideas and techniques within analysis itself. In [[ergodic theory]], the key objects are transformations that preserve a measure, and one asks how time averages along orbits relate to space averages over the whole system. The basic theory concerns whether time averages of functions become equal to their spatial averages under the orbit of a system, and how rapidly that approximation takes place. In functional analysis, evolution problems are often studied using one-parameter families of operators, such as [[semigroup theory|operator semigroups]], which generalize the exponential function from numbers or matrices to infinite-dimensional spaces. ===Operators and spectral theory=== Many areas of analysis study operators, like [[differential operator]]s, [[integral operator]]s, or [[linear transformation]]s on a [[function space]] or other [[topological vector space]]. Operators can encode information such as the evolution of a system, a differential or [[integral equation]], or a [[quantum state]] or [[observable]]. The [[spectral theory]] of operators allows operators to be broken into pieces and represented, generalizing aspects of the [[eigenvalue]] decomposition from linear algebra to infinite dimensions. As in linear algebra, it is often possible to understand an operator more deeply through its spectral decomposition. == Main branches == === Calculus === {{Main|Calculus}} Calculus is a branch of analysis that deals primarily with calculation and applications.{{cite encyclopedia |last=Stillwell |first=John Colin |title=Analysis |encyclopedia=Encyclopaedia Britannica |publisher=Encyclopaedia Britannica, Inc. |url=https://www.britannica.com/science/analysis-mathematics |access-date=18 June 2026 }}{{cite book |editor-last1=Aleksandrov |editor-first1=A. D. |editor-last2=Kolmogorov |editor-first2=A. N. |editor-last3=Lavrent'ev |editor-first3=M. A. |translator-last1=Gould |translator-first1=S. H. |translator-last2=Bartha |translator-first2=T. |title=Mathematics: Its Content, Methods, and Meaning |volume=1 |publisher=MIT Press |location=Cambridge, Massachusetts |year=1963 }}, Chapter 1 The two branches of calculus are: [[integral calculus]], which deals with the study of averages, accumulation, and area; and [[differential calculus]], which studies rates of change, [[linear approximation]], and the [[derivative]]. Examples of applications of calculus include how to work with approximations such as [[Taylor approximation]]s and [[Taylor series]], integration in elementary terms, numerical approximations to integration, and basic [[optimization]]. Vector analysis, also called vector calculus, is part of calculus that deals with [[vector-valued function]]s.{{cite book |last1=Borisenko |first1=A. I. |last2=Tarapov |first2=I. E. |title=Vector and Tensor Analysis with Applications (Dover Books on Mathematics) |date=1979 |publisher=Dover Books on Mathematics}} === Real analysis === {{Main|Real analysis}} Real analysis (traditionally, the "theory of functions of a real variable") is a branch of mathematical analysis dealing with the [[real number]]s and real-valued functions of a real variable.{{cite book |last=Rudin |first=Walter |author-link=Walter Rudin |title=Principles of Mathematical Analysis |url=https://archive.org/details/principlesofmath00rudi |url-access=registration |series=Walter Rudin Student Series in Advanced Mathematics |date=1976 |edition=3rd |publisher=McGraw–Hill |isbn=978-0070542358}}{{cite book |last=Abbott |first=Stephen |title=Understanding Analysis |series=Undergraduate Texts in Mathematics |isbn=978-0387950600 |date=2001 |location=New York |publisher=Springer-Verlag}} In particular, it deals with the properties of real [[function (mathematics)|functions]] and [[sequence]]s, including [[Limit of a sequence|convergence]] and [[limit of a function|limits]] of [[sequence]]s of real numbers, providing rigorous foundations for calculus, and [[continuous function|continuity]], [[smoothness]] and related properties of real-valued functions. === Complex analysis === {{Main|Complex analysis}} Complex analysis (traditionally known as the "theory of functions of a complex variable") is the branch of mathematical analysis that investigates [[Function (mathematics)|functions]] of [[complex numbers]].{{cite encyclopedia |last=Stillwell |first=John Colin |title=Analysis |encyclopedia=Encyclopaedia Britannica |publisher=Encyclopaedia Britannica, Inc. |url=https://www.britannica.com/science/analysis-mathematics |access-date=18 June 2026 }}{{cite book |editor-last1=Aleksandrov |editor-first1=A. D. |editor-last2=Kolmogorov |editor-first2=A. N. |editor-last3=Lavrent'ev |editor-first3=M. A. |translator-last1=Gould |translator-first1=S. H. |translator-last2=Bartha |translator-first2=T. |title=Mathematics: Its Content, Methods, and Meaning |volume=1 |publisher=MIT Press |location=Cambridge, Massachusetts |year=1963 }}, Chapter 1 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 function]]s of complex variables (or, more generally, [[meromorphic function]]s). Because the separate [[real number|real]] and [[imaginary number|imaginary]] parts of any analytic function must satisfy [[Laplace's equation]], complex analysis is widely applicable to two-dimensional problems in [[physics]]. === Functional analysis === {{Main|Functional analysis}} Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of [[vector space]]s endowed with some kind of limit-related structure (e.g. [[Inner product space#Definition|inner product]], [[Norm (mathematics)#Definition|norm]], [[Topological space#Definitions|topology]], etc.) and the [[linear transformation|linear operators]] acting upon these spaces and respecting these structures in a suitable sense.{{cite encyclopedia |last=Stillwell |first=John Colin |title=Analysis |encyclopedia=Encyclopaedia Britannica |publisher=Encyclopaedia Britannica, Inc. |url=https://www.britannica.com/science/analysis-mathematics |access-date=18 June 2026 }}{{cite book |editor-last1=Aleksandrov |editor-first1=A. D. |editor-last2=Kolmogorov |editor-first2=A. N. |editor-last3=Lavrent'ev |editor-first3=M. A. |translator-last1=Gould |translator-first1=S. H. |translator-last2=Bartha |translator-first2=T. |title=Mathematics: Its Content, Methods, and Meaning |volume=1 |publisher=MIT Press |location=Cambridge, Massachusetts |year=1963 }}, Chapter 1 The historical roots of functional analysis lie in the study of [[function space|spaces of functions]] and the formulation of properties of transformations of functions such as the [[Fourier transform]] as transformations defining [[continuous function|continuous]], [[unitary operator|unitary]] etc. operators between function spaces. This point of view turned out to be particularly useful for the study of [[differential equations|differential]] and [[integral equations]]. === Fourier analysis === {{Main|Fourier analysis}} Fourier analysis is a branch of mathematical analysis concerned with the representation of [[function (mathematics)|function]]s and [[signal]]s as the superposition of basic [[wave]]s.{{cite encyclopedia |last=Stillwell |first=John Colin |title=Analysis |encyclopedia=Encyclopaedia Britannica |publisher=Encyclopaedia Britannica, Inc. |url=https://www.britannica.com/science/analysis-mathematics |access-date=18 June 2026 }}{{cite book |last1=Stein |first1=Elias M. |last2=Shakarchi |first2=Rami |title=Real Analysis: Measure Theory, Integration, and Hilbert Spaces |series=Princeton Lectures in Analysis |volume=3 |publisher=Princeton University Press |location=Princeton, New Jersey |year=2005 |isbn=978-0-691-11386-9 |page=vii }} This includes the study of the notions of [[Fourier series]] and [[Fourier transform]]s, and of their generalizations. Fourier analysis also includes the study of [[linear differential operator]]s with constant coefficients and their generalizations to [[pseudodifferential operators]]. [[Microlocal analysis]] is a subfield of Fourier analysis concerned with how to localize the singularities of functions, and the how they propagate under differential and pseudodifferetial operators. === Harmonic analysis === {{Main|Harmonic analysis}} Harmonic analysis is a branch of mathematics having origins in the study of [[harmonic function]]s, and especially their boundary values. Harmonic function theory leads naturally to the study of [[function space]]s like [[Hardy space]]s, and in particular criteria for membership in such spaces and estimates of operators. It includes many of the methods of Fourier analysis, but also other decomposition methods such as [[Calderón–Zygmund lemma|Calderón–Zygmund decompositions]] which break functions into parts that can be handled by Fourier methods and parts that can be handled by local methods.{{cite book |last1=Stein |first1=Elias M. |last2=Shakarchi |first2=Rami |title=Functional Analysis: Introduction to Further Topics in Analysis |series=Princeton Lectures in Analysis |volume=4 |publisher=Princeton University Press |location=Princeton, New Jersey |year=2011 |isbn=978-0-691-11387-6 }} [[Abstract harmonic analysis]] is a related tradition, which generalizes Fourier methods to groups other than those to which classical Fourier methods apply. === Differential equations === {{Main|Differential equation}} A differential equation is a [[mathematics|mathematical]] [[equation]] for an unknown [[function (mathematics)|function]] of one or several [[Variable (mathematics)|variables]] that relates the values of the function itself and its [[derivative]]s of various [[Derivative#Higher derivatives|orders]].{{cite book|first = Edward L.|last = Ince|title =Ordinary Differential Equations|publisher = Dover Publications|date = 1956|isbn=978-0486603490|url = https://books.google.com/books?id=mbyqAAAAQBAJ}}[[Witold Hurewicz]], ''Lectures on Ordinary Differential Equations'', Dover Publications, {{isbn|0486495108}}{{cite encyclopedia |last=Stillwell |first=John Colin |title=Analysis |encyclopedia=Encyclopaedia Britannica |publisher=Encyclopaedia Britannica, Inc. |url=https://www.britannica.com/science/analysis-mathematics |access-date=18 June 2026 }} 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 system (mathematics)|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 of motion|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 [[equations of motion|equation of motion]]) may be solved explicitly. === Measure theory === {{Main|Measure (mathematics)}} A measure on a [[set (mathematics)|set]] is a systematic way to assign a number to each suitable [[subset]] of that set, intuitively interpreted as its size.{{cite book|author-link = Terence Tao|first = Terence|last = Tao|date = 2011|title = An Introduction to Measure Theory| series=Graduate Studies in Mathematics | volume=126 |publisher = American Mathematical Society|isbn = 978-0821869192|url = https://books.google.com/books?id=HoGDAwAAQBAJ|access-date = 2018-10-26|archive-date = 2019-12-27|archive-url = https://web.archive.org/web/20191227145317/https://books.google.com/books?id=HoGDAwAAQBAJ|url-status = live|doi=10.1090/gsm/126}}{{cite encyclopedia |last=Stillwell |first=John Colin |title=Analysis |encyclopedia=Encyclopaedia Britannica |publisher=Encyclopaedia Britannica, Inc. |url=https://www.britannica.com/science/analysis-mathematics |access-date=18 June 2026 }} 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-dimensional Euclidean space \mathbb{R}^n. For instance, the Lebesgue measure of the [[Interval (mathematics)|interval]] \left[0, 1\right] in the [[real line|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 [[Extended real number line|+∞]] to (certain) subsets of a set 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 [[Sigma-algebra|\sigma-algebra]]. This means that the empty set, countable [[union (set theory)|unions]], countable [[intersection (set theory)|intersections]] and [[complement (set theory)|complements]] of measurable subsets are measurable. [[Non-measurable set]]s 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 === {{Main|Numerical analysis}} Numerical analysis is the study of [[algorithm]]s that use numerical [[approximation]] (as opposed to general [[symbolic computation|symbolic manipulations]]) for the problems of mathematical analysis (as distinguished from [[discrete mathematics]]).{{cite book |last=Hildebrand |first=Francis B. | author-link=Francis B. Hildebrand | title=Introduction to Numerical Analysis | edition=2nd |date=1974 |publisher=McGraw-Hill |isbn= 978-0070287617}} 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 equation]]s appear in [[celestial mechanics]] (planets, stars and galaxies); [[numerical linear algebra]] is important for [[data analysis]]; [[stochastic differential equation]]s and [[Markov chain]]s are essential in simulating living cells for medicine and biology. === Geometric analysis and global analysis === {{Main|Geometric analysis|Global analysis}} Geometric analysis is the branch of analysis concerned with the study of [[manifold]]s, often [[Riemannian manifold]]s. Global questions of Riemannian geometry are often studied.{{cite encyclopedia |last=Stillwell |first=John Colin |title=Analysis |encyclopedia=Encyclopaedia Britannica |publisher=Encyclopaedia Britannica, Inc. |url=https://www.britannica.com/science/analysis-mathematics |access-date=18 June 2026 }} One example is the [[spectral geometry]] of the [[Laplace–Beltrami operator]], which generalizes the problem of [[hearing the shape of a drum]], for instance. Other global questions in geometric analysis include partial differential equations on Riemannian manfiolds, such as the [[Yamabe equation]] and the [[Ricci flow]]. The latter led to the proof of the [[Poincaré conjecture]]. === Convex analysis === {{Main|Convex analysis}} Convex analysis is the branch of analysis concerned with [[convex function]]s, [[convex set]]s, and applications to optimization and [[linear programming]]. Convex functions generally have more powerful methods available for ensuring the existence and uniqueness of minima, particularly when combined with [[lower semicontinuity]]. There are many dualities in convex optimization, often expressed in terms of the [[convex conjugate]], which allow an optimization problem to be paired with a [[dual problem]], which gives a useful criterion for checking optimality of a putative solution or determining how much slack there is in a numerical solution. === Calculus of variations === {{Main|Calculus of variations}} Calculus of variations{{cite encyclopedia |last=Stillwell |first=John Colin |title=Analysis |encyclopedia=Encyclopaedia Britannica |publisher=Encyclopaedia Britannica, Inc. |url=https://www.britannica.com/science/analysis-mathematics |access-date=18 June 2026 }} is the study of finding optimal solutions to minimization problems in infinite dimensional spaces. A basic example is to find a [[geodesic]] on a [[surface (mathematics)|surface]]: the endpoints are given, and one must find a path among the infinite-dimensional space of all possible paths, that minimizes the arc length. Many problems in [[partial differential equations]] admit a more natural variational characterization, and this can lead to notions of [[weak solution]]s, which are often more suited to analytic methods and numerical work. [[Direct methods in the calculus of variations]] formulate variational problems as convex optimization problems. [[Optimal transport]] is the sub-branch of the calculus of variations concerned with the formulation and solution of the problem of transporting stuff from one place to another subject to cost constraints. This sub-branch also shares many ideas with convex analysis, including duality theorems such as [[Transportation_theory_(mathematics)#Monge_and_Kantorovich_formulations|Kantorovich duality]], which is a form of [[Legendre transform|Legendre duality]]. The ideas of optimal transportation have wide and unexpected applications, such as the [[Wasserstein metric]] and its applications to [[machine learning]]. ===Probability theory and stochastic analysis=== {{main|Probability theory}} Probability theory is closely connected with mathematical analysis through [[measure theory]].{{cite book |last1=Stein |first1=Elias M. |last2=Shakarchi |first2=Rami |title=Functional Analysis: Introduction to Further Topics in Analysis |series=Princeton Lectures in Analysis |volume=4 |publisher=Princeton University Press |location=Princeton, New Jersey |year=2011 |isbn=978-0-691-11387-6 }} In the modern axiomatic formulation, a probability space is a measure space of total measure one, [[random variable]]s are measurable functions, and [[expected value]]s are integrals with respect to a [[probability measure]]. Analytic methods are central in the study of convergence of random variables, [[martingale (probability)|martingale]]s, [[stochastic process]]es, [[Brownian motion]], [[stochastic differential equation]]s, and [[ergodic]] theory. Probability theory is an area of mathematics in its own right, but many of its modern foundations and methods belong to analysis. Stochastic analysis studies analytic questions involving random processes, including [[stochastic integration]], stochastic differential equations, and connections with partial differential equations and functional analysis. == Other topics == * [[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|''p''-adic analysis]], the study of analysis within the context of [[p-adic number|''p''-adic numbers]], which differs in some interesting and surprising ways from its real and complex counterparts. * [[Non-standard analysis]], which investigates the [[hyperreal number]]s and their functions and gives a [[rigour#Mathematical rigour|rigorous]] treatment of [[infinitesimal]]s and infinitely large numbers. * [[Computable analysis]], the study of which parts of analysis can be carried out in a [[computability theory|computable]] manner. * [[Set-valued analysis]] – applies ideas from analysis and topology to set-valued functions. * [[Idempotent analysis]] – analysis in the context of an [[idempotent semiring]], where the lack of an additive inverse is compensated somewhat by the idempotent rule A + A = A. ** [[Tropical analysis]] – analysis of the idempotent semiring called the [[tropical semiring]] (or [[max-plus algebra]]/[[min-plus algebra]]). * [[Constructive analysis]], which is built upon a foundation of [[constructive logic|constructive]], rather than classical, logic and set theory. * [[Intuitionistic analysis]], which is developed from constructive logic like constructive analysis but also incorporates [[choice sequence]]s. * [[Paraconsistent analysis]], which is built upon a foundation of [[paraconsistent logic|paraconsistent]], rather than classical, logic and set theory. * [[Smooth infinitesimal analysis]], which is developed in a smooth topos. * [[Dynamical systems]], which is the study of the behavior of systems under time-evolution. == Applications == Techniques from analysis are also found in other areas such as: === Physical sciences === The vast majority of [[classical mechanics]], [[Theory of relativity|relativity]], and [[quantum mechanics]] is based on applied analysis, and [[differential equation]]s in particular. Examples of important differential equations include [[Newton's second law]], the [[Schrödinger equation]], and the [[Einstein field equations]]. [[Functional analysis]] is also a major factor in [[quantum mechanics]]. === Signal processing === When processing signals, such as [[Sound|audio]], [[radio wave]]s, 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.{{cite book |title=Theory and Application of Digital Signal Processing |last1=Rabiner |first1=L. R. |last2=Gold |first2=B. |location=Englewood Cliffs, New Jersey |publisher=[[Prentice-Hall]] |date=1975 |isbn=978-0139141010 |url=https://archive.org/details/theoryapplicatio00rabi |url-access=registration}} === 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 game]]s * [[Differential geometry]], the application of calculus to specific mathematical spaces known as [[manifold]]s that possess a complicated internal structure but behave in a simple manner locally. * [[Differentiable manifolds]] * [[Differential topology]] * [[Partial differential equations]] == Notable textbooks == * [[Leonhard Euler]] (1748) ''[[Introductio in analysin infinitorum]]'' * [[A. L. Cauchy]] (1821) ''[[Cours d'analyse]]'' * [[Camille Jordan]] (1882) ''[[Cours d%27analyse de l%27%C3%89cole polytechnique]]'' * [[G. H. Hardy]] (1908) ''[[A Course of Pure Mathematics]]'' * [[E. T. Whittaker]] & [[G. N. Watson]] (1915) [[A Course of Modern Analysis]] * [[George Pólya]] & [[Gábor Szegő]] (1925) ''[[Problems and Theorems in Analysis]]'' (two volumes){{cite book |title=Problems and Theorems in Analysis I: Series. Integral Calculus. Theory of Functions | id={{ASIN|3540636404|country=ca}} }}{{cite book |title=Problems and Theorems in Analysis II: Theory of Functions. Zeros. Polynomials. Determinants. Number Theory. Geometry | id={{ASIN|3540636862|country=ca}} }} * [[Walter Rudin]] (1953) ''[[Principles of Mathematical Analysis]]''{{cite book |title=Principles of Mathematical Analysis | id={{ASIN|0070856133|country=ca}} }} * [[Elias M. Stein]] & Rami Shakarchi (2003, 2011) ''[[Princeton Lectures in Analysis]]'' (four volumes) == See also == {{Portal|Mathematics}} * [[Arithmetization of analysis]] * [[Constructive analysis]] * [[History of calculus]] * [[Hypercomplex analysis]] * [[Multiple rule-based problems]] * [[Multivariable calculus]] * [[Paraconsistent logic]] * [[Smooth infinitesimal analysis]] * [[Timeline of calculus and mathematical analysis]] == References == {{Duplicated citations|reason=DuplicateReferences script detected: * https://www.britannica.com/science/analysis-mathematics (refs: 17, 23, 27, 29, 35, 37, 39, 40) |date=August 2026}} {{cite encyclopedia |title=analysis {{!}} mathematics |url=https://www.britannica.com/topic/analysis-mathematics |access-date=2015-07-31 |encyclopedia=Encyclopædia Britannica |author-first=John Colin |author-last=Stillwell |author-link=John Colin Stillwell |date= |archive-date=2015-07-26 |archive-url=https://web.archive.org/web/20150726223522/https://www.britannica.com/topic/analysis-mathematics |url-status=live}} {{cite book |author-first=John Colin |author-last=Stillwell |author-link=John Colin Stillwell |title=Mathematics and its History |edition=2nd |publisher=[[Springer Science+Business Media Inc.]] |isbn=978-0387953366 |date=2004 |chapter=Infinite Series |page=170 |quote=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 series 1⁄2 + 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}} {{cite book |author-last=Smith |author-first=David Eugene |author-link=David Eugene Smith |date=1958 |title=History of Mathematics |url=https://archive.org/details/historyofmathema0002smit |url-access=registration |publisher=[[Dover Publications]] |isbn=978-0486204307}} {{cite book |author-link=Lawrence Craig Evans |author-first=Lawrence Craig |author-last=Evans |title=Partial Differential Equations |publisher=[[American Mathematical Society]] |location=Providence |date=1998 |isbn=978-0821807729}} {{cite book |author-last=Rudin |author-first=Walter |author-link=Walter Rudin |title=Functional Analysis |publisher=[[McGraw-Hill Science]] |date=1991 |isbn=978-0070542365 |url=https://archive.org/details/functionalanalys0000rudi |url-access=registration}} {{cite book |author-last=Conway |author-first=John Bligh |author-link=John Bligh Conway |title=A Course in Functional Analysis |edition=2nd |publisher=[[Springer-Verlag]] |date=1994 |isbn=978-0387972459 |url=https://books.google.com/books?id=ix4P1e6AkeIC |access-date=2016-02-11 |archive-date=2020-09-09 |archive-url=https://web.archive.org/web/20200909165657/https://books.google.com/books?id=ix4P1e6AkeIC |url-status=live}} {{cite book |author-last=Ahlfors |author-first=Lars Valerian |author-link=Lars Valerian Ahlfors |title=Complex Analysis |location=New York |publisher=[[McGraw-Hill]] |edition=3rd |date=1979 |isbn=978-0070006577 |url=https://books.google.com/books?id=2MRuus-5GGoC }} == Further reading == * {{anchor|Mathematics: Its Content, Methods, and Meaning}}{{cite book |editor-last1=Aleksandrov |editor-first1=A. D. |editor-link1=Aleksandr Danilovich Aleksandrov |editor-last3=Lavrent'ev |editor-first3=M. A. |editor-link3=Mikhail Alekseevich Lavrentyev |editor-last2=Kolmogorov |editor-first2=A. N. |editor-link2=Andrey Nikolaevich Kolmogorov |translator-first1=S. H. |translator-last1=Gould |volume=1–3 |date=March 1969 |title=Mathematics: Its Content, Methods, and Meaning |edition=2nd |publisher=[[The M.I.T. Press]] / [[American Mathematical Society]] |publication-place=Cambridge, Massachusetts }} * {{cite book |author-last=Apostol |author-first=Tom M. |author-link=Tom M. Apostol |date=1974 |title=Mathematical Analysis |edition=2nd |publisher=[[Addison–Wesley]] |isbn=978-0201002881}} * {{cite book |author-last=Binmore |author-first=Kenneth George |author-link=Kenneth George Binmore |date=1981 |orig-date=1981 |title=The foundations of analysis: a straightforward introduction |url=https://archive.org/details/foundationsofana0000binm |url-access=registration |publisher=[[Cambridge University Press]]}} * {{cite book |author-last1=Johnsonbaugh |author-first1=Richard |author-link1=Richard Johnsonbaugh |author-first2=William Elmer |author-last2=Pfaffenberger |date=1981 |title=Foundations of mathematical analysis |location=New York |publisher=[[M. Dekker]]}} * {{cite encyclopedia |author-first=Sergey Mikhailovich [Серге́й Миха́йлович] |author-last=Nikol'skiĭ [Нико́льский] |author-link=Sergey Mikhailovich Nikolsky |date=2002 |url=https://encyclopediaofmath.org/wiki/Mathematical_analysis |title=Mathematical analysis |encyclopedia=[[Encyclopaedia of Mathematics]] |editor-link=Michiel Hazewinkel |editor-first=Michiel |editor-last=Hazewinkel |publisher=[[Springer-Verlag]] |isbn=978-1402006098}} * {{cite book |author-first1=Nicola |author-last1=Fusco |author-link1=Nicola Fusco |author-first2=Paolo |author-last2=Marcellini |author-link2=Paolo Marcellini |author-first3=Carlo |author-last3=Sbordone |date=1996 |title=Analisi Matematica Due |language=it |publisher={{ill|Liguori Editore|it}} |isbn=978-8820726751}} * {{cite book |author-last=Rombaldi |author-first=Jean-Étienne |date=2004 |title=Éléments d'analyse réelle : CAPES et agrégation interne de mathématiques |language=fr |publisher=[[EDP Sciences]] |isbn=978-2868836816}} * {{cite book |title=Principles of Mathematical Analysis |author-last=Rudin |author-first=Walter |author-link=Walter Rudin |publisher=[[McGraw-Hill]] |date=1976 |isbn=978-0070542358 |edition=3rd |location=New York}} * {{cite book |title=Real and Complex Analysis |author-last=Rudin |author-first=Walter |author-link=Walter Rudin |publisher=[[McGraw-Hill]] |date=1987 |isbn=978-0070542341 |edition=3rd |location=New York}} * {{cite book |title=A Course Of Modern Analysis: An Introduction to the General Theory of Infinite Processes and of Analytic Functions; with an Account of the Principal Transcendental Functions |title-link=Whittaker and Watson |author-last1=Whittaker |author-first1=Edmund Taylor |author-link1=Edmund Taylor Whittaker |author-last2=Watson |author-first2=George Neville |author-link2=George Neville Watson |date=1927-01-02 |edition=4th |publisher=[[at the University Press]] |publication-place=Cambridge |isbn=0521067944 }} (vi+608 pages) (reprinted: 1935, 1940, 1946, 1950, 1952, 1958, 1962, 1963, 1992) * {{cite web |url=http://www.math.harvard.edu/~ctm/home/text/class/harvard/114/07/html/home/course/course.pdf |archive-url=https://web.archive.org/web/20070419024458/http://www.math.harvard.edu/~ctm/home/text/class/harvard/114/07/html/home/course/course.pdf |archive-date=2007-04-19 |url-status=live |title=Real Analysis – Course Notes}} ==External links== {{Wikiquote}} {{Commons category}} * [http://www.economics.soton.ac.uk/staff/aldrich/Calculus%20and%20Analysis%20Earliest%20Uses.htm Earliest Known Uses of Some of the Words of Mathematics: Calculus & Analysis] * [http://www.jirka.org/ra/ Basic Analysis: Introduction to Real Analysis] by Jiri Lebl ([[Creative Commons|Creative Commons BY-NC-SA]]) * [https://www.britannica.com/topic/analysis-mathematics Mathematical Analysis – Encyclopædia Britannica] * [http://mathworld.wolfram.com/topics/CalculusandAnalysis.html Calculus and Analysis] {{Analysis-footer}} {{Areas of mathematics}} {{Industrial and applied mathematics}} {{Authority control}} [[Category:Mathematical analysis| ]]