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A Guide to Advanced Real Analysis is an outline of the core material in the standard graduate-level real analysis course. It is intended as a resource for students in such a course as well as others who wish to learn or review the subject. On the abstract level, it covers the theory of measure and integration and the basics of point set topology, functional analysis, and the most important types of function spaces. On the more concrete level, it also deals with the applications of these general theories to analysis on Euclidean space: the Lebesgue integral, Hausdorff measure, convolutions, Fourier series and transforms, and distributions. The relevant definitions and major theorems are stated in detail. Proofs, however, are generally presented only as sketches, in such a way that the key ideas are explained but the technical details are omitted. In this way a large amount of material is presented in a concise and readable form.

Mathematical analysis. --- Functions of real variables. --- Real variables --- Functions of complex variables --- 517.1 Mathematical analysis --- Mathematical analysis --- Functions of real variables

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This book studies the geometric theory of polynomials and rational functions in the plane. Any theory in the plane should make full use of the complex numbers and thus the early chapters build the foundations of complex variable theory, melding together ideas from algebra, topology and analysis. In fact, throughout the book, the author introduces a variety of ideas and constructs theories around them, incorporating much of the classical theory of polynomials as he proceeds. These ideas are used to study a number of unsolved problems, bearing in mind that such problems indicate the current limitations of our knowledge and present challenges for the future. However, theories also lead to solutions of some problems and several such solutions are given including a comprehensive account of the geometric convolution theory. This is an ideal reference for graduate students and researchers working in this area.

Polynomials. --- Functions of complex variables. --- Complex variables --- Elliptic functions --- Functions of real variables --- Algebra

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Convex functions. --- Subdifferentials. --- Calculus, Subdifferential --- Subdifferential calculus --- Convex functions --- Functions, Convex --- Functions of real variables

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Convex Cones

Functional analysis --- Cone. --- Cône. --- Analyse fonctionnelle --- Functions of real variables. --- Convex bodies. --- Functions of real variables --- Convex bodies --- Cone --- Geometry, Descriptive --- Geometry, Solid --- Convex domains --- Real variables --- Functions of complex variables --- Géometrie convexe

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Geometric Function Theory is that part of Complex Analysis which covers the theory of conformal and quasiconformal mappings. Beginning with the classical Riemann mapping theorem, there is a lot of existence theorems for canonical conformal mappings. On the other side there is an extensive theory of qualitative properties of conformal and quasiconformal mappings, concerning mainly a prior estimates, so called distortion theorems (including the Bieberbach conjecture with the proof of the Branges). Here a starting point was the classical Scharz lemma, and then Koebe's distortion theorem.

Geometric function theory. --- Functions of complex variables. --- Complex variables --- Elliptic functions --- Functions of real variables --- Function theory, Geometric --- Functions of complex variables