Titlebook: Essential Real Analysis; Michael Field Textbook 2017 Springer International Publishing AG 2017 use of euler maclaurin formula.Euler-Maclau

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Basic Properties of Real Numbers, Sequences and Continuous Functions,niversity level courses on calculus. Also included are short, but complete, review sections on complex numbers, the existence and uniqueness of the integral of a continuous function using an elementary method not dependent on uniform continuity, and the construction and properties of exponential and logarithmic functions.

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Functions,is shown how to construct smooth (infinitely differentiable) functions with specific properties. Bernstein’s construction is used to give a constructive proof of the Weierstrass approximation theorem—uniform approximation of continuous functions by polynomials. The definition of a real analytic func

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Metric Spaces, the Bolzano-Weierstrass theorem. The emphasis is on sequential properties (such as sequential continuity and sequential compactness) rather than more abstract topological notions (though all the foundational results are covered). The Arzelá-Ascoli theorem is proved, giving a generalization of the B

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Differential Calculus on ,,heorems proved are the mean value theorem, the inverse and implicit function theorems, the rank theorem, symmetry of higher order derivatives, Taylor’s theorem and the strong form of local existence and uniqueness theorem for ODEs using methods based on smooth uniform approximation. Statements and p

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1615-2085 ial Real Analysis .will appeal to students in pure and applied mathematics, as well as scientists looking to acquire a firm footing in mathematical analysis. Numerous exercises of varying difficulty, including 978-3-319-67545-9978-3-319-67546-6Series ISSN 1615-2085 Series E-ISSN 2197-4144

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Textbook 2017tric spaces includes applications to ordinary differential equations andfractals.. .Essential Real Analysis .will appeal to students in pure and applied mathematics, as well as scientists looking to acquire a firm footing in mathematical analysis. Numerous exercises of varying difficulty, including

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Future Challenges for Autonomous Systems,f. However, with the rapid growth of populations since the industrial revolution, and the evolution of a world food and energy market, famines now affect more people and more nations than ever before. Thus, the vicissitudes of climate have begun to loom large in world affairs.