Complex Calculus: Mathematical Methods for Physics and Engineering - Volume 1

Description

There is a longstanding conflict between extension and depth in the teaching of mathematics to physics students. This text intends to present an approach that tries to track what could be called the ``middle way'' in this conflict. It is the result of several years of experience of the author teaching the mathematical physics courses at the Physics Institute of the University of São Paulo. The text is organized in the form of relatively short chapters, each appropriate for exposition in one lecture. Each chapter includes a list of proposed problems, which have varied levels of difficulty, including practice problems, problems that complete and extend the material presented in the text, and some longer and more difficult problems, which are presented as challenges to the students. There are complete solutions available, detailed and commented, to all the problems proposed, which are presented in separate volumes. This volume is dedicated to the complex calculus. This is a more practical and less abstract version of complex analysis and of the study of analytic functions. This does not mean that there are no proofs in the text, since all the fundamental theorems are proved with a good level of rigor. The text starts from the very beginning, with the definition of complex numbers, and proceeds up to the study of integrals on the complex plane and on Riemann surfaces. The facts and theorems established here will be used routinely in all the subsequent volumes of this series of books. The development is based on an analogy with vector fields and with electrostatics, emphasizing interpretations and proofs that have a geometrical character. The approach is algorithmic and emphasizes the representation of functions by series, with detailed discussion of the convergence issues. 1 Number, the Language of Science 2 The Simplicity of Complex Numbers 3 Elementary Functions, but Not Quite 4 Even Less Elementary Functions 5 Geometrical Aspects of the Functions 6 Border Effects in Capacitors 7 Complex Calculus I: Differentiation 8 Complex Calculus II: Integration 9 Complex Derivatives and Integrals 10 Complex Inequalities and Series 11 Series, Limits and Convergence 12 Representation of Functions by Series 13 Convergence Criteria and Proofs 14 Laurent Series and Residues 15 Calculation of Integrals by Residues 16 Residues on Riemann Surfaces A Continued Fractions B Series for the Square Roots of Integers C The Stirling Approximation D Some Facts from Real Analysis E Complete Singularity Classification F The Complex Gamma Function