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Analysis is the branch of mathematics dealing with continuous functions, limits, and related theories, such as differentiation, integration, measure, infinite sequences, series, and analytic functions.^[1][2]

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).

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 CE to find the area of a circle.^[7] From Jain literature, it appears that Hindus were in possession of the formulae for the sum of the arithmetic and geometric series as early as the 4th century BCE.^[8] Ācārya Bhadrabāhu uses the sum of a geometric series in his Kalpasūtra in 433 BCE.^[9]

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 used infinitesimal 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.

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.

In the 18th century, Euler introduced the notion of a 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 19th 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.

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 limits, continuity, differentiation, and integration.

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 list of real numbers indexed by a natural number. A sequence is said to converge to a limit if almost all members of the sequence are arbitrarily close to . More precisely, this means that for any error tolerance , all of the members of the sequence are within of except possibly for finitely many members of the sequence. Formally, for any error there is an integer such that whenever .

This example shows the use of approximation: the elements approximate the number , and the error tolerance is . 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 is differentiable at a point if there is a linear function that approximates the function well near the point . But “well” here only means that the approximation error where is a function that tends to zero faster than as . 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 where 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 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 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 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 spaces. 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 distributionally have continuous representatives under additional hypotheses, and can thus be treated as honest functions rather than more general objects.

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.

Much of functional analysis is concerned with spaces of functions, which can be given the structure of a metric space, such as Banach spaces and Hilbert spaces. In many of these examples, the metric comes from a norm. For example, the space of continuous real-valued functions 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 L^p spaces, which are metric spaces whose metrics again come from a norm, are 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 spaces and complete metric spaces are especially important in analysis, where many arguments require the existence of limits.

Complex numbers provide another important tool for many analytic equations. A complex-valued function of a complex variable is 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 functions, 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 of an operator encodes information about its 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.

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, 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 , is that it possible to form a rearrangement of its values to form a decreasing function, using the measure of its superlevel sets 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 values are integrals with respect to this probability measure.

Many function spaces in analysis are defined using measures. The 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 equations.

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 sinusoids. The Fourier series takes the form of a trigonometric series where are complex numbers and is the independent variable.

The Fourier series is one instance of an eigenfunction expansion, with the exponentials 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 functions 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 spaces, 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.

Many problems in analysis concern how quantities change over time or under repeated application of a rule. This leads to ordinary differential equations and partial differential equations, 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 systems.

Analytic questions in dynamics include existence and uniqueness of solutions, stability, approximation of trajectories, long-time behavior, and dependence on initial conditions. For example, a differential equation may determine a 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 operator semigroups, which generalize the exponential function from numbers or matrices to infinite-dimensional spaces.

Many areas of analysis study operators, like differential operators, integral operators, or linear transformations 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.

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 (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 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.

Free or noncommutative analysis is a sub-branch of functional analysis that deals with spaces that are, in some ways, noncommutative. It is related to noncommutative function theory, which is a noncommutative generalization of complex analysis, as well as free probability theory and noncommutative geometry. In free analysis, emphasis is placed on working with noncommutative variables and functions, especially over spaces or algebras that are noncommutative.

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.

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.

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 -dimensional Euclidean space . For instance, the Lebesgue measure of the interval 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 . 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 -algebra. This means that the empty set, 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 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 analysis, also called vector calculus, is a branch of mathematical analysis dealing with vector-valued functions.^[26]

Scalar analysis is a branch of mathematical analysis dealing with values related to scale as opposed to direction. Values such as temperature are scalar because they describe the magnitude of a value without regard to direction, force, or displacement that value may or may not have.

• 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 Dirac-like operators, termed in general as monogenic or Clifford analytic functions.
• p-adic analysis, the study of analysis within the context of p-adic numbers, which differs in some interesting and surprising ways from its real and complex counterparts.
• Non-standard analysis, which investigates the hyperreal numbers and their functions and gives a rigorous treatment of infinitesimals and infinitely large numbers.
• Computable analysis, the study of which parts of analysis can be carried out in a computable manner.
• Stochastic calculus – analytical notions developed for stochastic processes.
• Set-valued analysis – applies ideas from analysis and topology to set-valued functions.
• Convex analysis, the study of convex sets and 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, rather than classical, logic and set theory.
• Intuitionistic analysis, which is developed from constructive logic like constructive analysis but also incorporates choice sequences.
• Paraconsistent analysis, which is built upon a foundation of paraconsistent, rather than classical, logic and set theory.
• Smooth infinitesimal analysis, which is developed in a smooth topos.

Techniques from analysis are also found in other areas such as:

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 Newton’s second law, the Schrödinger equation, and the Einstein field equations.

Functional analysis is also a major factor in quantum mechanics.

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 Fourier-transforming a signal, manipulating the Fourier-transformed data in a simple way, and reversing the transformation.^[27]

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

• Leonhard Euler (1748) Introductio in analysin infinitorum
• A. L. Cauchy (1821) Cours d’analyse
• Camille Jordan (1882) Cours_d’analyse de l’École 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)^[28][29]
• Walter Rudin (1953) Principles of Mathematical Analysis^[30]
• Elias M. Stein & Rami Shakarchi (2003, 2011) Princeton Lectures in Analysis (four volumes)

• 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

• Aleksandrov, A. D.; Kolmogorov, A. N.; Lavrent’ev, M. A., eds. (March 1969). Mathematics: Its Content, Methods, and Meaning. Vol. 1–3. Translated by Gould, S. H. (2nd ed.). Cambridge, Massachusetts: The M.I.T. Press / American Mathematical Society.
• Apostol, Tom M. (1974). Mathematical Analysis (2nd ed.). Addison–Wesley. ISBN 978-0201002881.
• Binmore, Kenneth George (1981) [1981]. The foundations of analysis: a straightforward introduction. Cambridge University Press.
• Johnsonbaugh, Richard; Pfaffenberger, William Elmer (1981). Foundations of mathematical analysis. New York: M. Dekker.
• Nikol’skiĭ [Нико́льский], Sergey Mikhailovich [Серге́й Миха́йлович] (2002). “Mathematical analysis”. In Hazewinkel, Michiel (ed.). Encyclopaedia of Mathematics. Springer-Verlag. ISBN 978-1402006098.
• Fusco, Nicola; Marcellini, Paolo; Sbordone, Carlo (1996). Analisi Matematica Due (in Italian). Liguori Editore [it]. ISBN 978-8820726751.
• Rombaldi, Jean-Étienne (2004). Éléments d’analyse réelle : CAPES et agrégation interne de mathématiques (in French). EDP Sciences. ISBN 978-2868836816.
• Rudin, Walter (1976). Principles of Mathematical Analysis (3rd ed.). New York: McGraw-Hill. ISBN 978-0070542358.
• Rudin, Walter (1987). Real and Complex Analysis (3rd ed.). New York: McGraw-Hill. ISBN 978-0070542341.
• Whittaker, Edmund Taylor; Watson, George Neville (1927-01-02). 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 (4th ed.). Cambridge: at the University Press. ISBN 0521067944. {{cite book}}: ISBN / Date incompatibility (help) (vi+608 pages) (reprinted: 1935, 1940, 1946, 1950, 1952, 1958, 1962, 1963, 1992)
• “Real Analysis – Course Notes” (PDF). Archived (PDF) from the original on 2007-04-19.

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