9780198525615

Introduction to Complex Analysis

by
  • ISBN13:

    9780198525615

  • ISBN10:

    0198525613

  • Edition: 2nd
  • Format: Hardcover
  • Copyright: 2003-10-30
  • Publisher: Oxford University Press

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Summary

This title includes the following features: Best-selling text in itsfield; Substantially expanded introductory chapters; Contains carefully gradedexercies and worked examples; Based on over 17 years of teachingexperience

Table of Contents

Notation and terminology xiii
The complex plane
1(11)
Complex numbers
1(2)
Algebra in the complex plane
3(4)
Conjugation, modulus, and inequalities
7(2)
Exercises
9(3)
Geometry in the complex plane
12(18)
Lines and circles
12(5)
The extended complex plane and the Riemann sphere
17(5)
Mobius transformations
22(4)
Exercises
26(4)
Topology and analysis in the complex plane
30(17)
Open sets and closed sets in the complex plane
30(5)
Convexity and connectedness
35(4)
Limits and continuity
39(4)
Exercises
43(4)
Paths
47(9)
Introducing curves and paths
47(4)
Properties of paths and contours
51(3)
Exercises
54(2)
Holomorphic functions
56(11)
Differentiation and the Cauchy--Riemann equations
56(3)
Holomorphic functions
59(5)
Exercises
64(3)
Complex series and power series
67(11)
Complex series
68(3)
Power series
71(3)
A proof of the Differentiation theorem for power series
74(2)
Exercises
76(2)
A cornucopia of holomorphic functions
78(13)
The exponential function
78(2)
Complex trigonometric and hyperbolic functions
80(3)
Zeros and periodicity
83(1)
Argument, logarithms, and powers
84(2)
Holomorphic branches of some simple multifunctions
86(2)
Exercises
88(3)
Conformal mapping
91(16)
Conformal mapping
91(4)
Some standard conformal mappings
95(2)
Mappings of regions by standard mappings
97(5)
Building conformal mappings
102(2)
Exercises
104(3)
Multifunctions
107(12)
Branch points and multibranches
107(5)
Cuts and holomorphic branches
112(6)
Exercises
118(1)
Integration in the complex plane
119(9)
Integration along paths
119(5)
The Fundamental theorem of calculus
124(2)
Exercises
126(2)
Cauchy's theorem: basic track
128(14)
Cauchy's theorem
129(5)
Deformation
134(3)
Logarithms again
137(3)
Exercises
140(2)
Cauchy's theorem: advanced track
142(9)
Deformation and homotopy
142(3)
Holomorphic functions in simply connected regions
145(1)
Argument and index
146(3)
Cauchy's theorem revisited
149(1)
Exercises
150(1)
Cauchy's formulae
151(10)
Cauchy's integral formula
151(3)
Higher-order derivatives
154(5)
Exercises
159(2)
Power series representation
161(15)
Integration of series in general and power series in particular
161(2)
Taylor's theorem
163(4)
Multiplication of power series
167(1)
A primer on uniform convergence
168(6)
Exercises
174(2)
Zeros of holomorphic functions
176(12)
Characterizing zeros
176(2)
The Identity theorem and the Uniqueness theorem
178(5)
Counting zeros
183(2)
Exercises
185(3)
Holomorphic functions: further theory
188(6)
The Maximum modulus theorem
188(1)
Holomorphic mappings
189(3)
Exercises
192(2)
Singularities
194(17)
Laurent's theorem
194(6)
Singularities
200(5)
Meromorphic functions
205(2)
Exercises
207(4)
Cauchy's residue theorem
211(10)
Residues and Cauchy's residue theorem
211(2)
Calculation of residues
213(6)
Exercises
219(2)
A technical toolkit for contour integration
221(13)
Evaluating real integrals by contour integration
221(2)
Inequalities and limits
223(2)
Estimation techniques
225(4)
Improper and principal-value integrals
229(3)
Exercises
232(2)
Applications of contour integration
234(22)
Integrals of rational functions
234(3)
Integrals of other functions with a finite number of poles
237(4)
Integrals involving functions with infinitely many poles
241(2)
Integrals involving multifunctions
243(2)
Evaluation of definite integrals: overview (basic track)
245(2)
Summation of series
247(1)
Further techniques
248(3)
Exercises
251(5)
The Laplace transform
256(22)
Basic properties and evaluation of Laplace transforms
256(3)
Inversion of Laplace transforms
259(8)
Applications
267(7)
Exercises
274(4)
The Fourier transform
278(11)
Introducing the Fourier transform
278(2)
Evaluation and inversion
280(2)
Applications
282(5)
Exercises
287(2)
Harmonic functions and conformal mapping
289(20)
Harmonic functions
289(7)
The Dirichlet problem and its solution by conformal mapping
296(3)
Further examples of conformal mapping
299(7)
Exercises
306(3)
Appendix: new perspectives
309(10)
The Prime number theorem
309(4)
The Bieberbach conjecture
313(1)
Julia sets and the Mandelbrot set
314(5)
Bibliography 319(2)
Notation index 321(2)
Index 323

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