Integration with Complex Numbers

Abstract

Abstract This introductory text on complex analysis focuses on how to evaluate challenging improper (real) integrals, or their Cauchy principal values if need be, by associating them with (complex) contour integrals. On the way to this goal it explains in detail the basic arithmetic, algebra and analysis of complex numbers and functions: particularly the Cauchy–Riemann equations, Cauchy’s theorem, Cauchy’s integral formula, Taylor’s theorem, Laurent’s theorem and Cauchy’s residue theorem. Recognising that many non-specialist cohorts need to acquire skill and confidence in these techniques, great care is taken to allow time for consolidation of fundamental ideas before proceeding to more sophisticated ones, and stress is laid on worked examples to explain ideas and applications, informal diagrams to build insight, roughwork initial explorations to help seek out solution strategies and—above all—suites of exercises in which the learner can develop and reinforce competence: learning through doing being the hallmark of the working textbook. Substantial revision sections on real analysis and calculus are built into the text for learners who may require additional preparation. An appended final chapter addresses some more advanced topics, such as uniform convergence, that are relevant to why certain key theorems work. Specimen solutions for many exercises will be made available to instructors upon application to the publishers.