Integration with Complex Numbers: A Primer on Complex Analysis

ISBN : 9780192846075

Brian McMaster; Aisling McCluskey
256 ページ
156 x 234 mm

Complex analysis, more than almost any other undergraduate topic in mathematics, runs the full pure/applied gamut from the most subtle, difficult, and ingenious proofs to the most direct, hands-on, engineering-based applications. This creates challenges for the instructor as much as for the very wide range of students whose various programmes require a secure grasp of complex analysis. Its techniques are indispensable to many, but skill in the use of a mathematical tool is hazardous and fallible without a sound understanding of why and when that tool is the right one to pick up. This kind of understanding develops only by combining careful exploration of ideas, analysis of proofs, and practice across a range of exercises. Integration with Complex Numbers: A Primer on Complex Analysis offers a reader-friendly contemporary balance between idea, proof, and practice, informed by several decades of classroom experience and a seasoned understanding of the backgrounds, motivation, and competing time pressures of today's student cohorts. To achieve its aim of supporting and sustaining such cohorts through those aspects of complex analysis that they encounter in first and second-year study, it also balances competing needs to be self-contained, comprehensive, accessible, and engaging - all in sufficient but not in excessive measures. In particular, it begins where most students are likely to be, and invests the time and effort that are required in order to deliver accessibility and introductory gradualness.


1 Background Part A: set, sequence, series
2 What are complex numbers?
3 Background Part B: real functions and their limits
4 Complex functions
5 Background Part C: real integration
6 Paths in the complex plane
7 Cauchy's theorem(s)
8 Taylor's theorem
9 Residues
10 Reality from complexity
11 The repair shop for broken promises


Brian McMaster studied at Queen's University Belfast (QUB), UK and has served his alma mater department in various capacities including those of Adviser of Studies, Head of Research, and Associate Director of Education. He has written for over sixty refereed journal articles, mostly in the area of analytic topology but incorporating a smattering of applications in disciplines as diverse as probabilistic metric spaces and decision support theory. In 2018 he received the accolade for Special Recognition for Inspirational Teaching awarded by QUB Students Union. His teaching interests focus around analysis (real and complex) and set theory and their development into various fields, especially that of analytic topology. ; Since her appointment at the National University of Ireland Galway in 1992, Aisling McCluskey has established a meaningful and rewarding academic career, maintaining an active research profile whilst holding the teaching and learning of mathematics central to her academic endeavour. She has received several institutional and national awards for excellence in teaching, most recently in 2020 and 2021. She is currently a Governor at NUI Galway and Head of School.