R. Michael Range - Böcker

This is an introductory text in several complex variables, using methods of integral representations. It begins with elementary local results, discusses basic new concepts of the multi-dimensional theory such as pseudoconvexity and holomorphic convexity, and leads up to complete proofs of fundamental global results, both classical and new. The use of integral representation techniques makes it possible to treat the subject with a minimum of prerequisites, and it has the further advantage that it uses the multivariable forms of theorem with which the students are already acquainted. This book also provides a systematic introduction to integral representation methods and their applications, including a simplified proof of C. Fefferman's famous mapping theorem.

This is an introductory text in several complex variables, using methods of integral representations. It begins with elementary local results, discusses basic new concepts of the multi-dimensional theory such as pseudoconvexity and holomorphic convexity, and leads up to complete proofs of fundamental global results, both classical and new. The use of integral representation techniques makes it possible to treat the subject with a minimum of prerequisites, and it has the further advantage that it uses the multivariable forms of theorem with which the students are already acquainted. This book also provides a systematic introduction to integral representation methods and their applications, including a simplified proof of C. Fefferman's famous mapping theorem.

This unique book provides a new and well-motivated introduction to calculus and analysis, historically significant fundamental areas of mathematics that are widely used in many disciplines. It begins with familiar elementary high school geometry and algebra, and develops important concepts such as tangents and derivatives without using any advanced tools based on limits and infinite processes that dominate the traditional introductions to the subject. This simple algebraic method is a modern version of an idea that goes back to René Descartes and that has been largely forgotten. Moving beyond algebra, the need for new analytic concepts based on completeness, continuity, and limits becomes clearly visible to the reader while investigating exponential functions.The author carefully develops the necessary foundations while minimizing the use of technical language. He expertly guides the reader to deep fundamental analysis results, including completeness, key differential equations, definite integrals, Taylor series for standard functions, and the Euler identity. This pioneering book takes the sophisticated reader from simple familiar algebra to the heart of analysis. Furthermore, it should be of interest as a source of new ideas and as supplementary reading for high school teachers, and for students and instructors of calculus and analysis.

This unique book provides a new and well-motivated introduction to calculus and analysis, historically significant fundamental areas of mathematics that are widely used in many disciplines. It begins with familiar elementary high school geometry and algebra, and develops important concepts such as tangents and derivatives without using any advanced tools based on limits and infinite processes that dominate the traditional introductions to the subject. This simple algebraic method is a modern version of an idea that goes back to René Descartes and that has been largely forgotten. Moving beyond algebra, the need for new analytic concepts based on completeness, continuity, and limits becomes clearly visible to the reader while investigating exponential functions.The author carefully develops the necessary foundations while minimizing the use of technical language. He expertly guides the reader to deep fundamental analysis results, including completeness, key differential equations, definite integrals, Taylor series for standard functions, and the Euler identity. This pioneering book takes the sophisticated reader from simple familiar algebra to the heart of analysis. Furthermore, it should be of interest as a source of new ideas and as supplementary reading for high school teachers, and for students and instructors of calculus and analysis.

Unlock the mysteries of Calculus with a fresh approach rooted in simplicity and historical insight. This book reintroduces a nearly forgotten idea from René Descartes (1596-1650), showing how the fundamental concepts of Calculus can be understood using just basic algebra. Starting with rational functions — the core of early Calculus — this method allows the reader to grasp the rules for derivatives without the intimidating concepts of limits or real numbers, making the subject more accessible than ever.But the journey doesn't stop there. While attempting to apply this algebraic approach to exponential functions, the reader will encounter the limitations of simple methods, revealing the necessity for more advanced mathematical tools. This natural progression leads to the discovery of continuity, the approximation process, and ultimately, the introduction of real numbers and limits. These deeper concepts pave the way for understanding differentiable functions, seamlessly bridging the gap between elementary algebra and the profound ideas that underpin Calculus.Whether you're a student, educator, or math enthusiast, this book offers a unique pathway to mastering Calculus. By connecting historical context with modern mathematical practice, it provides a richer, more motivating learning experience. For those looking to dive even deeper, the author's 2015 book, What is Calculus? From Simple Algebra to Deep Analysis, is the perfect next step.