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kuttler k analysis functions complex many var 2022

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Textbook in PDF format This book is on multi-variable real analysis with an introduction to complex analysis. It is for advanced undergraduate students and beginning graduate students. It is NOT SUITABLE as a first course in calculus. I assume the reader has had a course in calculus which does an honest job of presenting the Riemann integral of a function of one variable and knows the usual things about completeness of R and its algebraic properties although these things are reviewed. Also,it is expected that the reader knows the usual elementary things found in an undergraduate linear algebra course, such as row operations and linear transformations, linear independence and vector spaces. I have included the Lebesgue integral instead of the more usual Riemann integral because it seems to me that the Riemann integral is too technically difficult. This book has a modern approach to real analysis and an introduction to complex analysis. It also has an introduction to functional analysis if there is time, but I have tried to write it in a manner which would allow the omission of these topics if it were desired to only include real and complex analysis. The main direction in the complex analysis part is toward classical nineteenth century analysis although it does include an introduction to methods of complex analysis in spectral theory of operators on a Banach space. I am presenting some very interesting theorems more than once. I think it is good to see different ways of proving them. Often these theorems appear for the first time in the exercises. I hope that by doing so, better understanding of both subjects will be acquired. For example, the introduction to the zeta function is heavily dependent on real Lebesgue theory of integration. I have also tried to include all of the hard topology which is usually omitted but nevertheless used in analysis books. This includes things like the Brouwer fixed point theorem and the Jordan curve theorem. Preliminary Topics Basic Notions. Basic Topology and Algebra. Stone Weierstrass Approximation Theorem. Real Analysis The Derivative, a Linear Transformation. Line Integrals and Curves. Measures and Measurable Functions. The Abstract Lebesgue Integral. Positive Linear Functionals. Basic Function Spaces. Change of Variables. Some Fundamental Functions and Transforms. Abstract Analysis Banach Spaces. Representation Theorems. Complex Analysis Fundamentals. Isolated Singularities and Analytic Functions. Mapping Theorems. Spectral Theory of Linear Maps. A Green’s Theorem for a Jordan Curve

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Kuttler K. Analysis of Functions of Complex and Many Var. 2022.pdf 33.4 MB

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kuttler k analysis functions complex many var 2022

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