Mathematical analysis is the branch of mathematics that studies functions, spaces, and operators through quantitative 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 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 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. 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 433 BCE.
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
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