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Hermitian Analysis: From Fourier Series to Cauchy-Riemann Geometry provides a coherent, integrated look at various topics from analysis. It begins with Fourier series, continues with Hilbert spaces, discusses the Fourier transform on the real line, and then turns to the heart of the book: geometric considerations in several complex variables. The final chapter includes complex differential forms, geometric inequalities from one and several complex variables, finite unitary groups, proper mappings, and naturally leads to the Cauchy-Riemann geometry of the unit sphere. The book thus takes the reader from the unit circle to the unit sphere. This textbook will be a useful resource for upper-undergraduate students who intend to continue with mathematics, graduate students interested in analysis, and researchers interested in some basic aspects of CR Geometry. It will also be useful for students in physics and engineering, as it includes topics in harmonic analysis arising in these subjects. The inclusion of an appendix and more than 270 exercises makes this book suitable for a capstone undergraduate Honors class.

Fourier series --- Harmonic analysis --- Civil & Environmental Engineering --- Engineering & Applied Sciences --- Operations Research --- Analysis (Mathematics) --- Functions, Potential --- Potential functions --- Fourier integrals --- Series, Fourier --- Series, Trigonometric --- Trigonometric series --- Mathematics. --- Fourier analysis. --- Differential equations. --- Differential geometry. --- Fourier Analysis. --- Differential Geometry. --- Ordinary Differential Equations. --- Global differential geometry. --- Differential Equations. --- 517.91 Differential equations --- Differential equations --- Geometry, Differential --- Analysis, Fourier --- Mathematical analysis --- Fourier series. --- Harmonic analysis. --- Banach algebras --- Calculus --- Mathematics --- Bessel functions --- Harmonic functions --- Time-series analysis --- Fourier analysis --- Differential geometry

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Spherical functions. --- Euclidean algorithm. --- Algorithm of Euclid --- Continued division --- Division, Continued --- Euclid algorithm --- Euclidian algorithm --- Euclid's algorithm --- Algorithms --- Number theory --- Functions, Spherical --- Spherical harmonics --- Transcendental functions --- Spheroidal functions --- Funcions esferoïdals --- Algorismes --- Algorisme d'Euclides --- Algoritmes --- Àlgebra --- Algorismes computacionals --- Algorismes genètics --- Anàlisi numèrica --- Funcions recursives --- Programació (Matemàtica) --- Programació (Ordinadors) --- Teoria de màquines --- Traducció automàtica --- Funcions harmòniques

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Functions of complex variables. --- Inequalities (Mathematics). --- Mathematical analysis.

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This monograph systematically explores the theory of rational maps between spheres in complex Euclidean spaces and its connections to other areas of mathematics. Synthesizing research from the last forty years, the author aims for accessibility by balancing abstract concepts with concrete examples. Numerous computations are worked out in detail, and more than 100 optional exercises are provided throughout for readers wishing to better understand challenging material. The text begins by presenting core concepts in complex analysis and a wide variety of results about rational sphere maps. The susbequent chapters discuss combinatorial and optimization results about monomial sphere maps, groups associated with rational sphere maps, relevant complex and CR geometry, and some geometric properties of rational sphere maps. Fifteen open problems appear in the final chapter, with references provided to appropriate parts of the text. These problems will encourage readers to apply the material to future research. Rational Sphere Maps will be of interest to researchers and graduate students studying several complex variables and CR geometry. Mathematicians from other areas, such as number theory, optimization, and combinatorics, will also find the material appealing.

Algebraic geometry --- Differential geometry. Global analysis --- Analytical spaces --- Mathematical analysis --- Mathematics --- analyse (wiskunde) --- topologie (wiskunde) --- complexe veranderlijken --- differentiaal geometrie --- statistiek --- wiskunde --- geometrie

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Mathematics. --- Problem solving.