Mathematical analysis

Analysis is the generic name given to any branch of mathematics which depends upon the concepts of limits and convergence, and studies closely related topics such as continuity, integration, differentiability and transcendental functions. These topics are often studied in the context of real numbers, complex numbers, and their functions. However, they can also be defined and studied in any space of mathematical objects that is equipped with a definition of "nearness" (a topological space) or "distance" (a metric space). Mathematical analysis has its beginnings in the rigorous formulation of calculus.

History

Greek mathematicians such as Eudoxus and Archimedes made 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. In the 12th century the Indian mathematician Bhaskara gave an example of what would now be called a "differential coefficient" and the basic idea behind what is now known as Rolle's theorem. The 14th century Indian mathematician Madhava of Sangamagrama expressed various trigonometric functions as infinite series, and estimated the magnitude of the error terms created by truncating these series.

Related Topics:
Greek - Eudoxus - Archimedes - Method of exhaustion - 12th century - Indian mathematician - Bhaskara - Rolle's theorem - 14th century - Madhava of Sangamagrama - Trigonometric - Infinite series

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In Europe, analysis originated in the 17th century, with the independent invention of calculus by Newton and Leibniz. In the 17th and 18th centuries, analysis topics such as the calculus of variations, ordinary and partial differential equations, Fourier analysis and generating functions were developed mostly in applied work. Calculus techniques were applied successfully to approximate discrete problems by continuous ones.

Related Topics:
17th century - Newton - Leibniz - 18th centuries - Calculus of variations - Ordinary - Partial differential equation - Fourier analysis - Generating function - Discrete problems

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All through the 18th century the definition of the concept of function was a subject of debate among mathematicians. In the 19th century, Cauchy was the first to put calculus on a firm logical foundation by introducing the concept of the Cauchy sequence. He also started the formal theory of complex analysis. Poisson, Liouville, Fourier and others studied partial differential equations and harmonic analysis.

Related Topics:
Function - 19th century - Cauchy - Cauchy sequence - Complex analysis - Poisson - Liouville - Fourier - Harmonic analysis

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In the middle of the century Riemann introduced his theory of integration. The last third of the 19th century saw the arithmetization of analysis by Weierstrass, who thought that geometric reasoning was inherently misleading, and introduced the ε-δ definition of limit.

Related Topics:
Riemann - Integration - Weierstrass - Limit

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Then, mathematicians started worrying that they were assuming the existence of a continuum of real numbers without proof. Dedekind then constructed the real numbers by Dedekind cuts. 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.

Related Topics:
Continuum - Real number - Dedekind - Dedekind cut - Theorem - Riemann integration - Discontinuities

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Also, "monsters" (nowhere continuous functions, continuous but nowhere differentiable functions, space-filling curves) began to be created. 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 axiomatic set theory. Lebesgue solved the problem of measure, and Hilbert introduced Hilbert spaces to solve integral equations. The idea of normed vector space was in the air, and in the 1920s Banach created functional analysis.

Related Topics:
Monsters - Nowhere continuous - Space-filling curve - Jordan - Measure - Cantor - Naive set theory - Baire - Baire category theorem - 20th century - Axiomatic set theory - Lebesgue - Hilbert - Hilbert space - Integral equation - Normed vector space - 1920s - Banach - Functional analysis

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