Real and Complex Analysis

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ISBN: 9781584888062 | 1584888067
Cover: Hardcover
Copyright: 12/8/2009

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Unlike other undergraduate-level texts, Real and Complex Analysis develops both the real and complex theory together. It takes a unified, elegant approach to the theory that is consistent with the recommendations of the MAA's 2004 Curriculum Guide. By presenting real and complex analysis together, the authors illustrate the connections and differences between these two branches of analysis right from the beginning. This combined development also allows for a more streamlined approach to real and complex function theory. More than 1,000 exercises enhance the text and ancillary materials are available on the book's website.

Preface	p. xv
Acknowledgments	p. xvii
The Authors	p. xix
The Spaces R, R^k, and C	p. 1
The Real Numbers R	p. 1
Properties of the Real Numbers R	p. 2
The Absolute Value	p. 7
Intervals in R	p. 10
The Real Spaces R^k	p. 10
Properties of the Real Spaces R^k	p. 11
Inner Products and Norms on R^k	p. 14
Intervals in R^k	p. 18
The Complex Numbers C	p. 19
An Extension of R²	p. 19
Properties of Complex Numbers	p. 21
A Norm on C and the Complex Conjugate of z	p. 24
Polar Notation and the Arguments of z	p. 26
Circles, Disks, Powers, and Roots	p. 30
Matrix Representation of Complex Numbers	p. 34
Supplementary Exercises	p. 35
Point-Set Topology	p. 41
Bounded Sets	p. 42
Bounded Sets in X	p. 42
Bounded Sets in R	p. 44
Spheres, Balls, and Neighborhoods	p. 47
Classification Of Points	p. 50
Interior, Exterior, and Boundary Points	p. 50
Limit Points and Isolated Points	p. 53
Open And Closed Sets	p. 55
Open Sets	p. 55
Closed Sets	p. 58
Relatively Open and Closed Sets	p. 61
Density	p. 62
Nested Intervals And The Bolzano-Weierstrass Theorem	p. 63
Nested Intervals	p. 63
The Bolzano-Weierstrass Theorem	p. 66
Compactness And Connectedness	p. 69
Compact Sets	p. 69
The Heine-Borel Theorem	p. 71
Connected Sets	p. 72
Supplementary Exrcises	p. 75
Limits and Convergence	p. 83
Definitions And First Properties	p. 84
Definitions and Examples	p. 84
First Properties of Sequences	p. 89
Convergence Results For Seqences	p. 90
General Results for Sequences in X	p. 90
Special Results for Sequences in R and C	p. 92
Topological Results For Sequences	p. 97
Subsequences in X	p. 97
The Limit Superior and Limit Inferior	p. 100
Cauchy Sequences and Completeness	p. 104
Properties Of Infinite Series	p. 108
Definition and Examples of Series in X	p. 108
Basic Results for Series in X	p. 110
Special Series	p. 115
Testing for Absolute Convergence in X	p. 120
Manipulation Of Series In R	p. 123
Rearrangements of Series	p. 123
Multiplication of Series	p. 125
Definition of e^x for x ¿ R	p. 128
Supplementary Exercises	p. 128
Functions: Definitions and Limits	p. 135
Definitions	p. 135
Notation and Definitions	p. 136
Complex Functions	p. 137
Functions As Mappings	p. 139
Images and Preimages	p. 139
Bounded Functions	p. 141
Combining Functions	p. 142
One-to-One Functions and Onto Functions	p. 144
Inverse Functions	p. 147
Some Elementary Complex Functions	p. 148
Complex Polynomials and Rational Functions	p. 148
The Complex Square Root Function	p. 149
The Complex Exponential Function	p. 150
The Complex Logarithm	p. 151
Complex Trigonometric Functions	p. 154
Limits Of Functions	p. 156
Definition and Examples	p. 156
Properties of Limits of Functions	p. 160
Algebraic Results for Limits of Functions	p. 163
Supplementry Exercises	p. 171
Functions: Continuity and Convergence	p. 177
Continuity	p. 177
Definitions	p. 177
Examples of Continuity	p. 179
Algebraic Properties of Continuous Functions	p. 184
Topological Properties and Characterizations	p. 187
Real Continuous Functions	p. 191
Uniform Continuity	p. 198
Definition and Examples	p. 198
Topological Properties and Consequences	p. 201
Continuous Extensions	p. 203
Seqences And Series Of Functions	p. 208
Definitions and Examples	p. 208
Uniform Convergence	p. 210
Series of Functions	p. 216
The Tietze Extension Theorem	p. 219
Supplementary Exercises	p. 222
The Derivative	p. 233
The Derivative For â : D¹R	p. 234
Three Definitions Are Better Than One	p. 234
First Properties and Examples	p. 238
Local Extrema Results and the Mean Value Theorem	p. 247
Taylor Polynomials	p. 250
Differentiation of Sequences and Series of Functions	p. 255
The Derivative For â : D^kR	p. 257
Definition	p. 258
Partial Derivatives	p. 260
The Gradient and Directional Derivatives	p. 262
Higher-Order Partial Derivatives	p. 266
Geometric Interpretation of Partial Derivatives	p. 268
Some Useful Results	p. 269
The Derivative For â : D^kR^P	p. 273
Definition	p. 273
Some Useful Results	p. 283
Differentiability Classes	p. 289
The Derivative For â: DC	p. 291
Three Derivative Definitions Again	p. 292
Some Useful Results	p. 295
The Cauchy-Riemann Equations	p. 297
The z and z Derivatives	p. 305
The Inverse And Implict Function Theorems	p. 309
Some Technical Necessities	p. 310
The Inverse Function Theorem	p. 313
The Implicit Function Theorem	p. 318
Supplementary Exeprcises	p. 321
Real Integration	p. 335
The Integral Of â : [a, b]R	p. 335
Definition of the Riemann Integral	p. 335
Upper and Lower Sums and Integrals	p. 339
Relating Upper and Lower Integrals to Integrals	p. 346
Properties Of The Riemann Integral	p. 349
Classes of Bounded Integrable Functions	p. 349
Elementary Properties of Integrals	p. 354
The Fundamental Theorem of Calculus	p. 360
Further Development Of Integration Theory	p. 363
Improper Integrals of Bounded Functions	p. 363
Recognizing a Sequence as a Riemann Sum	p. 366
Change of Variables Theorem	p. 366
Uniform Convergence and Integration	p. 367
Vector-Valued And Line Integrals	p. 369
The Integral of â : [a, b] R^p	p. 369
Curves and Contours	p. 372
Line Integrals	p. 377
Supplementary Exrcises	p. 381
Complex Integration	p. 387
Introduction To Complex Intergrals	p. 387
Integration over an Interval	p. 387
Curves and Contours	p. 390
Complex Line Integrals	p. 393
Further Development Of Complex Line Integrals	p. 400
The Triangle Lemma	p. 400
Winding Numbers	p. 404
Antiderivatives and Path-Independence	p. 408
Integration in Star-Shaped Sets	p. 410
Cauchy's Integral Theorem And Its Consequnces	p. 415
Auxiliary Results	p. 416
Cauchy's Integral Theorem	p. 420
Deformation of Contours	p. 423
Cauchy's Integral Formula	p. 428
The Various Forms of Cauchy's Integral Formula	p. 428
The Maximum Modulus Theorem	p. 433
Cauchy's Integral Formula for Higher-Order Derivatives	p. 435
Further Properties Of Complex Differentiable Functions	p. 438
Harmonic Functions	p. 438
A Limit Result	p. 439
Morera's Theorem	p. 440
Liouville's Theorem	p. 441
The Fundamental Theorem of Algebra	p. 442
Appendices: Winding Numbers Revisited	p. 443
A Geometric Interpretation	p. 443
Winding Numbers of Simple Closed Contours	p. 447
Supplementary Exercises	p. 450
Taylor Series, Laurent Series, and the Residue Calculus	p. 455
Power Series	p. 456
Definition, Properties, and Examples	p. 456
Manipulations of Power Series	p. 464
Taylor Series	p. 473
Analytic Functions	p. 481
Definition and Basic Properties	p. 481
Complex Analytic Functions	p. 483
Laurent's Theorem For Complex Functions	p. 457
Singularities	p. 493
Definitions	p. 493
Properties of Functions Near Singularities	p. 496
The Residue Calculus	p. 502
Residues and the Residue Theorem	p. 502
Applications to Real Improper Integrals	p. 507
Supplementary Exercises	p. 512
Complex Functions as Mappings	p. 515
The Extended Complex Plane	p. 515
Liner Fractional Transformation	p. 519
Basic LFTs	p. 519
General LFTs	p. 521
Conformal Mappings	p. 524
Motivation and Definition	p. 524
More Examples of Conformal Mappings	p. 527
The Schwarz Lemma and the Riemann Mapping Theorem	p. 530
Supplementary Exercises	p. 534
Bibliography	p. 537
Index	p. 539
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Real and Complex Analysis

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