Introduction to Complex Analysis

by Priestley, H. A.

• ISBN13: 9780198525615
• ISBN10: 0198525613
• Edition: 2nd
• Format: Hardcover
• Copyright: 2003-10-30
• Publisher: Oxford University Press

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