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E-raamat: Integration with Complex Numbers: A Primer on Complex Analysis

(Honorary Senior Lecturer, Queen's University Belfast), (Professor in Mathematics, National University of Ireland, Galway)
  • Formaat: 256 pages
  • Ilmumisaeg: 18-Apr-2022
  • Kirjastus: Oxford University Press
  • Keel: eng
  • ISBN-13: 9780192661579
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Integration with Complex Numbers: A Primer on Complex AnalysisSuurem pilt
  • Formaat: 256 pages
  • Ilmumisaeg: 18-Apr-2022
  • Kirjastus: Oxford University Press
  • Keel: eng
  • ISBN-13: 9780192661579
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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 iintroductory gradualness.

Arvustused

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. * zb Math Open *

1 Background part A
1(10)
1.1 Introduction
1(1)
1.2 Revision 1: sets
2(2)
1.3 Revision 2: sequences
4(2)
1.4 Revision 3: series
6(5)
2 What are complex numbers?
11(30)
2.1 How do we handle them?
14(6)
2.2 Navigating around the complex plane
20(9)
2.3 Sequences and series of complex numbers
29(4)
2.4 Powers and roots: de Moivre's theorem
33(4)
2.5 Exercises
37(4)
3 Background part B
41(22)
3.1 Real functions
41(3)
3.2 Limits of real functions
44(5)
3.3 Continuity of real functions
49(3)
3.4 Differentiation of real functions
52(5)
3.5 A very brief look at partial differentiation
57(4)
3.6 Exercises
61(2)
4 Complex functions
63(22)
4.1 Introduction
63(1)
4.2 Limits, continuity, differentiation (again)
64(9)
4.3 Cauchy--Riemann
73(5)
4.4 Surprises!
78(3)
4.5 Exercises
81(4)
5 Background part C
85(24)
5.1 Introduction
85(2)
5.2 Integration by inspection
87(1)
5.3 Integration by parts
88(2)
5.4 Integration by substitution, or change of variable
90(2)
5.5 A look at improper integrals
92(4)
5.6 Cauchy principal values-a (slightly) more advanced topic
96(9)
5.7 Exercises
105(4)
6 Paths in the complex plane
109(30)
6.1 Introduction
109(2)
6.2 Functions from R to C
111(3)
6.3 Paths and contours
114(6)
6.4 Combining paths
120(2)
6.5 Connected sets and domains
122(3)
6.6 Integrating along a contour
125(10)
6.7 Exercises
135(4)
7 Cauchy's theorem(s)
139(22)
7.1 Introduction
139(1)
7.2 Baby Cauchy
139(3)
7.3 The triangular contour case
142(3)
7.4 The star domain case
145(5)
7.5 The general case
150(3)
7.6 Cauchy's integral formula
153(4)
7.7 Exercises
157(4)
8 Taylor's theorem
161(18)
8.1 Introduction
161(2)
8.2 Taylor series
163(3)
8.3 Examples
166(3)
8.4 Zeros
169(5)
8.5 Exercises
174(5)
9 Residues
179(20)
9.1 Laurent's theorem
179(11)
9.2 The residue theorem
190(3)
9.3 Residue calculation tools
193(2)
9.4 Exercises
195(4)
10 Reality from complexity
199(54)
10.1 Integrating `around the unit circle'
199(12)
10.2 Integrating `around an infinite semicircle'
211(12)
10.3 Spiking it with trig
223(5)
10.4 Some special case techniques
228(18)
10.5 The Gaussian integral-complex analysis showing off
246(3)
10.6 Exercises
249(4)
11 The repair shop for broken promises
253(14)
11.1 The field axioms
253(1)
11.2 L'Hopital's rule for complex functions
254(1)
11.3 Swopping summation and integration
255(9)
11.4 Smoothness: analytical and geometrical
264(3)
Suggestions for further or supplementary reading 267(2)
Index 269
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.
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