DeLyra J. Solutions for Complex Calculus...Vol 1S. 2019

Textbook in PDF format

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 of the text 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. This is Volume 1S, and is the companion volume to Volume 1, which is dedicated to the complex calculus. It includes all the 117 problems proposed in the text, with complete solutions, which are detailed and commented. The solutions are organized according to the 16 chapters of the corresponding volume of the text.
Number, the Language of Science
The Simplicity of Complex Numbers
Elementary Functions, but Not Quite
Even Less Elementary Functions
Geometrical Aspects of the Functions
Border Effects in Capacitors
Complex Calculus I: Differentiation
Complex Calculus II: Integration
Complex Derivatives and Integrals
Complex Inequalities and Series
Series, Limits and Convergence
Representation of Functions by Series
Convergence Criteria and Proofs
Laurent Series and Residues
Calculation of Integrals by Residues
Residues on Riemann Surfaces