Mathematical analysis is the branch of mathematics that studies functions, spaces, and operators through methods of approximation and convergence. It grew out of calculus, especially the use of derivatives and integrals 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 numbers, functions, sequences, series, and limits. Analysis is closely connected with applications in the sciences, providing methods for formulating and analyzing mathematical models, constructing approximations, and estimating their errors.
Typical analytical questions concern how functions, sequences, and operators behave under limiting processes and perturbations: convergence and continuity, regularity and stability, quantitative bounds, and the accuracy of approximations. Modern analysis studies these questions in many settings, including Euclidean spaces, metric spaces, topological spaces, measure spaces, and function spaces. Its major areas include complex analysis, functional analysis, measure theory, harmonic analysis, and the theory of ordinary and partial differential equations.
History
Ancient Mathematical analysis formally developed in the 17th century during the Scientific Revolution, 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. (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. The explicit use of infinitesimals appears in Archimedes' The Method of Mechanical Theorems, a work rediscovered in the 20th century. In Asia, the Chinese mathematician Liu Hui used the method of exhaustion in the 3rd century CE to find the area of a circle., Chapter, p. 279 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. Ācārya Bhadrabāhu uses the sum of a geometric series in his Kalpasūtra in .
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. 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. 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, developed infinite series for trigonometric functions. 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 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 infinitesimals 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.
Renaissance
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.
Modern Origins In 17th century Europe, new methods for geometry and mathematical physics laid the groundwork for mathematical analysis. Descartes's La Géométrie (1637), together with Fermat's related work, introduced algebraic methods into the study of curves and established the foundations of analytic geometry. Fermat's method of adequality allowed him to determine tangents of curves and their the maxima and minima. Shortly thereafter, Wallis published Arithmetica Infinitorum, which extended ideas of Cavalieri on quadrature and the method of indivisibles. Newton published some of his earliest mathematical works amid this milieu, and was focused on power series. At the same time, motivation from physics, exemplified by the discoveries of Kepler and Galileo, called for more general constructions than the classical constructions of particular figures.
The culminating work of Newton and Leibniz in the later 17th century gave these strands a new and more general form. The differential and integral calculi made change, tangent problems, quadrature, and inverse tangent problems parts of a common method. Their independent work is commonly treated as the emergence of infinitesimal analysis as a distinct mathematical discipline. Physics was profoundly influenced by the new mathematics. For example, Newton's second law of motion could assume a quantitative form: that the force was related to a second derivative.
During the eighteenth century, mathematicians extended these methods and increasingly reconceived analysis in terms of algebraic relations and functions rather than solely geometric quantities. Euler gave the concept of function a central role, while Lagrange and others developed methods involving power series, differential equations, and the calculus of variations.
Modernization In the 18th century, Euler introduced the notion of a mathematical function. Real analysis began to emerge as an independent subject when Bernard Bolzano introduced the modern definition of continuity in 1816,* 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.
Important concepts Analysis studies functions and their generalizations by methods involving limits. As the subject developed, these methods were extended to more complicated objects, including functionals and operators.
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 limits, continuity, differentiation, and 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 of real numbers, each 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