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Download E-books Analysis II: Differential and Integral Calculus, Fourier Series, Holomorphic Functions (Universitext) PDF

Functions in R and C, together with the speculation of Fourier sequence, Fourier integrals and a part of that of holomorphic features, shape the focal subject of those volumes. in response to a path given by way of the writer to massive audiences at Paris VII collage for a few years, the exposition proceeds slightly nonlinearly, mixing rigorous arithmetic skilfully with didactical and ancient issues. It units out to demonstrate the range of attainable techniques to the most effects, as a way to start up the reader to tools, the underlying reasoning, and basic principles. it really is appropriate for either educating and self-study. In his general, own kind, the writer emphasizes rules over calculations and, heading off the condensed kind often present in textbooks, explains those principles with out parsimony of phrases. The French variation in 4 volumes, released from 1998, has met with resounding good fortune: the 1st volumes are actually to be had in English.

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2 – ideas of calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . three – Truncated expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . four – Truncated growth of a quotient . . . . . . . . . . . . . . . . . . . . . five – Gauss’ convergence criterion . . . . . . . . . . . . . . . . . . . . . . . . . . 6 – The hypergeometric sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 – Asymptotic examine of the equation xex = t . . . . . . . . . . . . . . eight – Asymptotics of the roots of sin x. log x = 1 . . . . . . . . . . . . . nine – Kepler’s equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 – Asymptotics of the Bessel capabilities . . . . . . . . . . . . . . . . . . § 2. Summation formulae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . eleven – Cavalieri and the sums 1k + 2k + . . . + nk . . . . . . . . . . . . . 12 – Jakob Bernoulli . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . thirteen – the facility sequence for cot z . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 – Euler and the facility sequence for arctan x . . . . . . . . . . . . . . . . 15 – Euler, Maclaurin and their summation formulation . . . . . . . . . sixteen – The Euler-Maclaurin formulation with the rest . . . . . . . . . . 17 – Calculating an quintessential by way of the trapezoidal rule . . . . . . . . . 18 – The sum 1 + half + . . . + 1/n, the infinite product for the Γ functionality, and Stirling’s formulation . . . . . . . . . . . . . . . . . . 19 – Analytic continuation of the zeta functionality . . . . . . . . . . . . . 195 195 195 197 198 two hundred 202 204 206 208 210 213 224 224 226 231 234 238 239 241 242 247 Contents VII – Harmonic research and Holomorphic services . . . . . . . . . 1 – Cauchy’s necessary formulation for a circle . . . . . . . . . . . . . . . . . . § 1. research at the unit circle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 – services and measures at the unit circle . . . . . . . . . . . . . . three – Fourier coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . four – Convolution product on T . . . . . . . . . . . . . . . . . . . . . . . . . . . . five – Dirac sequences in T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . § 2. simple theorems on Fourier sequence . . . . . . . . . . . . . . . . . . . . . 6 – completely convergent Fourier sequence . . . . . . . . . . . . . . . . . . . 7 – Hilbertian calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . eight – The Parseval-Bessel equality . . . . . . . . . . . . . . . . . . . . . . . . . . nine – Fourier sequence of differentiable capabilities . . . . . . . . . . . . . . . . 10 – Distributions on T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . § three. Dirichlet’s procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . eleven – Dirichlet’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 – Fej´er’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . thirteen – Uniformly convergent Fourier sequence . . . . . . . . . . . . . . . . . . . § four. Analytic and holomorphic capabilities . . . . . . . . . . . . . . . . . . . . . . . 14 – Analyticity of the holomorphic services . . . . . . . . . . . . . . 15 – the utmost precept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . sixteen – services analytic in an annulus. Singular issues. Meromorphic services . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 – Periodic holomorphic features . . . . . . . . . . . . . . . . . . . . . . 18 – The theorems of Liouville and of d’Alembert-Gauss . . . . . 19 – Limits of holomorphic services . . . . . . . . . . . . . . . . . . . . . . 20 – Infinite items of holomorphic features . . . . . . . . . . . . . § five. Harmonic features and Fourier sequence . . . . . . . . . . . . . . . . . . . . . 21 – Analytic features defined by means of a Cauchy necessary . . . . . . . . 22 – Poisson’s functionality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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