Complex Analysis (Undergraduate Texts in Mathematics) Joseph Bak (Author), Donald J. Newman (Author)


Product Details

• Hardcover: 331 pages
• Publisher: Springer; 3rd ed. edition (August 6, 2010)
• Language: English
• ISBN-10: 1441972870
• ISBN-13: 978-1441972873
• Product Dimensions: 9.2 x 6.1 x 0.9 inches
  

Contents
Preface to the Third Edition ......................................... v
Preface to the Second Edition ........................................ vii
1 The Complex Numbers ......................................... 1
Introduction .................................................... 1
1.1TheField of Complex Numbers .............................. 1
1.2 The Complex Plane ......................................... 4
1.3TheSolution of the Cubic Equation ........................... 9
1.4 Topological Aspects of the Complex Plane ..................... 12
1.5 StereographicProjection; The Point at Infinity .................. 16
Exercises ...................................................... 18
2 Functions of the Complex Variable z ............................. 21
Introduction .................................................... 21
2.1 Analytic Polynomials ....................................... 21
2.2PowerSeries .............................................. 25
2.3Differentiability and Uniqueness of Power Series ................ 28
Exercises ...................................................... 32
3 Analytic Functions ............................................. 35
3.1Analyticity and the Cauchy-Riemann Equations ................. 35
3.2 The Functions ez
,sin z,cos z ................................. 40
Exercises ...................................................... 41
4 Line Integrals and Entire Functions .............................. 45
Introduction .................................................... 45
4.1 Properties of the Line Integral ................................ 45
4.2 The Closed Curve Theorem for Entire Functions ................ 52
Exercises ...................................................... 56
5 Properties of Entire Functions ................................... 59
5.1 The Cauchy Integral Formula and Taylor Expansion
for Entire Functions ........................................ 59
5.2Liouville Theorems and the Fundamental Theorem of Algebra; The
Gauss-Lucas Theorem ...................................... 65
5.3Newton’s Method and Its Application to Polynomial Equations .... 68
Exercises ...................................................... 74
6 Properties of Analytic Functions ................................. 77
Introduction .................................................... 77
6.1 The Power Series Representation for Functions Analytic inaDisc .. 77
6.2Analytic inanArbitrary Open Set ............................. 81
6.3TheUniqueness, Mean-Value, and Maximum-Modulus Theorems;
Critical Points and Saddle Points .............................. 82
Exercises ...................................................... 90
7 Further Properties of Analytic Functions ......................... 93
7.1 The Open Mapping Theorem; Schwarz’ Lemma ................. 93
7.2 The Converse of Cauchy’s Theorem: Morera’s Theorem; The
Schwarz Reflection Principle and AnalyticArcs ................. 98
Exercises ...................................................... 104
8 Simply Connected Domains ..................................... 107
8.1 The General Cauchy Closed Curve Theorem .................... 107
8.2 The Analytic Function log z .................................. 113
Exercises ...................................................... 116
9 Isolated Singularities of an Analytic Function ..................... 117
9.1 Classification of Isolated Singularities; Riemann’sPrinciple and the
Casorati-Weierstrass Theorem ................................ 117
9.2 Laurent Expansions ......................................... 120
Exercises ...................................................... 126
10 The Residue Theorem ............ ..... ......................... 129
10.1Winding Numbers and the Cauchy Residue Theorem............. 129
10.2 Applications of the Residue Theorem .......................... 135
Exercises ...................................................... 141
11 Applications of the Residue Theorem to the Evaluation of Integrals
and Sums ..................................................... 143
Introduction .................................................... 143
11.1Evaluation of Definite Integrals by Contour Integral Techniques ... 143
11.2 Application of Contour Integral Methods to Evaluation
and Estimation of Sums ..................................... 151
Exercises ...................................................... 158
12 Further Contour Integral Techniques ............................ 161
12.1Shifting the Contour of Integration ............................ 161
12.2AnEntire Function Bounded inEveryDirection ................. 164
Exercises ...................................................... 167
13 Introduction to ConformalMapping ............................. 169
13.1 Conformal Equivalence ..................................... 169
13.2Special Mappings .......................................... 175
13.3 Schwarz-Christoffel Transformations .......................... 187
Exercises ...................................................... 192
14 The Riemann Mapping Theorem ................................ 195
14.1 ConformalMapping and Hydrodynamics ....................... 195
14.2TheRiemann Mapping Theorem .............................. 200
14.3 Mapping Properties of Analytic Functions on
Closed Domains ... ........................................ 204
Exercises ...................................................... 213
15 Maximum-Modulus Theorems
for Unbounded Domains .......... ..... ......................... 215
15.1 A General Maximum-Modulus Theorem ....................... 215
15.2 The Phragmén-Lindelöf Theorem ............................. 218
Exercises ...................................................... 223
16 Harmonic Functions ............................................ 225
16.1Poisson Formulae and the Dirichlet Problem .................... 225
16.2Liouville Theorems for Re f ; Zeroes of Entire Functions
of Finite Order ............................................. 233
Exercises ...................................................... 238
17 Different Forms of Analytic Functions ............................ 241
Introduction .................................................... 241
17.1Infinite Products ........................................... 241
17.2Analytic Functions Defined by Definite Integrals ................ 249
17.3Analytic Functions Defined by Dirichlet Series .................. 251
Exercises ...................................................... 255
18 Analytic Continuation; The Gamma
and Zeta Functions ............................................. 257
Introduction .................................................... 257
18.1PowerSeries .............................................. 257
18.2Analytic Continuation of Dirichlet Series ....................... 263
18.3 The Gamma and Zeta Functions .............................. 265
Exercises ...................................................... 271

19 Applications to Other Areas of Mathematics ...................... 273
Introduction .................................................... 273
19.1AVariation Problem ........................................ 273
19.2 The Fourier Uniqueness Theorem ............................. 275
19.3AnInfinite System of Equations .............................. 277
19.4 Applications to Number Theory .............................. 278
19.5AnAnalyticProofofThePrime Number Theorem............... 285
Exercises ...................................................... 290
Answers ........................................................... 291
References ......................................................... 319
Appendices ........................................................ 321
Index ............................................................. 325