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Mathematics

# Mathematics - Mathematical analysis which mathematicians...

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Mathematical analysis, which mathematicians refer to simply as analysis , has its beginnings in the rigorous formulation of infinitesimal calculus . It is a branch of pure mathematics that includes the theories of differentiation , integration and measure , limits, infinite series , [1] and analytic functions . These theories are often studied in the context of real numbers , complex numbers , and real and complex functions . However, they can also be defined and studied in any space of mathematical objects that has a definition of nearness (a topological space ) or, more specifically, distance (a metric space ). Contents [ hide ] 1 History 2 Subdivisions 3 Topological spaces, metric spaces 4 Calculus of finite differences, discrete calculus or discrete analysis 5 See also 6 Notes 7 References 8 External links [ edit ] History 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 . [2] 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. [3] In India , the 12th century mathematician Bhāskara II gave examples of the derivative and used what is now known as Rolle's theorem . In the 14th century, Madhava of Sangamagrama developed infinite series expansions, like the power series and the Taylor series , of functions such as sine , cosine , tangent and arctangent . Alongside his development of the Taylor series of the trigonometric functions , he also estimated the magnitude of the error terms created by truncating these series and gave a rational approximation of an infinite series. His followers at the Kerala school of astronomy and mathematics further expanded his works, up to the 16th century. In Europe, during the later half of the 17th century, Newton and Leibniz independently developed infinitesimal calculus , which grew, with the stimulus of applied work that continued

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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 mathematical function . [4]
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Mathematics - Mathematical analysis which mathematicians...

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