Complex Analysis, Vol. 1 Book

	Preface	xiii
	Symbols and Terms	xix
1	Preliminaries	1
1.1	Preview	1
A	It Takes Two Harmonic Functions	3
B	Heat Flow	6
C	A Geometric Rule	9
D	Electrostatics	10
E	Fluid Flow	13
F	One Model, Many Applications	14
	Exercises	15
1.2	Sets, Functions, and Visualization	18
A	Terminology and Notation for Sets	18
B	Terminology and Notation for Functions	20
C	Functions from R to R	25
D	Functions from R[superscript 2] to R	27
E	Functions from R[superscript 2] to R[superscript 2]	29
	Exercises	30
1.3	Structures on R[superscript 2], and Linear Maps from R[superscript 2] to R[superscript 2]	34
A	The Real Line and the Plane	34
B	Polar Coordinates in the Plane	36
C	When Is a Mapping M: R[superscript 2] [right arrow] R[superscript 2] Linear?	38
D	Visualizing Nonsingular Linear Mappings	40
E	The Determinant of a Two-by-Two Matrix	44
F	Pure Magnifications, Rotations, and Conjugation	45
G	Conformal Linear Mappings	46
	Exercises	48
1.4	Open Sets, Open Mappings, Connected Sets	51
A	Distance, Interior, Boundary, Openness	51
B	Continuity in Terms of Open Sets	55
C	Open Mappings	56
D	Connected Sets	57
	Exercises	58
1.5	A Review of Some Calculus	61
A	Integration Theory for Real-Valued Functions	61
B	Improper Integrals, Principal Values	63
C	Partial Derivatives	66
D	Divergence and Curl	68
	Exercises	70
1.6	Harmonic Functions	71
A	The Geometry of Laplace's Equation	71
B	The Geometry of the Cauchy-Riemann Equations	72
C	The Mean Value Property	73
D	Changing Variables in a Dirichlet or Neumann Problem	76
	Exercises	77
2	Basic Tools	83
2.1	The Complex Plane	83
A	The Definition of a Field	83
B	Complex Multiplication	84
C	Powers and Roots	87
D	Conjugation	89
E	Quotients of Complex Numbers	90
F	When Is a Mapping L : C [right arrow] C Linear?	91
G	Complex Equations for Lines and Circles	92
H	The Reciprocal Map, and Reflection in the Unit Circle	93
I	Reflections in Lines and Circles	96
	Exercises	97
2.2	Visualizing Powers, Exponential, Logarithm, and Sine	102
A	Powers of z	103
B	Exponential and Logarithms	104
C	Sin z	106
D	The Cosine and Sine, and the Hyperbolic Cosine and Sine	110
	Exercises	111
2.3	Differentiability	115
A	Differentiability at a Point	115
B	Differentiability in the Complex Sense: Holomorphy	119
C	Finding Derivatives	122
D	Picturing the Local Behavior of Holomorphic Mappings	124
	Exercises	126
2.4	Sequences, Compactness, Convergence	128
A	Sequences of Complex Numbers	128
B	The Limit Superior of a Sequence of Reals	131
C	Implications of Compactness	133
D	Sequences of Functions	134
	Exercises	135
2.5	Integrals Over Curves, Paths, and Contours	138
A	Integrals of Complex-Valued Functions	138
B	Curves	138
C	Paths	144
D	Pathwise Connected Sets	147
E	Independence of Path and Morera's Theorem	148
F	Goursat's Lemma	150
G	The Winding Number	153
H	Green's Theorem	155
I	Irrotational and Incompressible Fluid Flow	158
J	Contours	161
	Exercises	162
2.6	Power Series	166
A	Infinite Series	166
B	The Geometric Series	167
C	An Improved Root Test	171
D	Power Series and the Cauchy-Hadamard Theorem	172
E	Uniqueness of the Power Series Representation	174
F	Integrals That Give Rise to Power Series	178
	Exercises	180
3	The Cauchy Theory	187
3.1	Fundamental Properties of Holomorphic Functions	188
A	Integral and Series Representations	188
B	Eight Ways to Say "Holomorphic"	193
C	Determinism	193
D	Liouville's Theorem	196
E	The Fundamental Theorem of Algebra	196
F	Subuniform Convergence Preserves Holomorphy	197
	Exercises	198
3.2	Cauchy's Theorem	204
A	Cerny's 1976 Proof	205
B	Simply Connected Sets	208
C	Subuniform Boundedness, Subuniform Convergence	209
3.3	Isolated Singularities	212
A	The Laurent Series Representation on an Annulus	212
B	Behavior Near an Isolated Singularity in the Plane	216
C	Examples: Classifying Singularities, Finding Residues	219
D	Behavior Near a Singularity at Infinity	225
E	A Digression: Picard's Great Theorem	229
	Exercises	229
3.4	The Residue Theorem and the Argument Principle	236
A	Meromorphic Functions and the Extended Plane	236
B	The Residue Theorem	239
C	Multiplicity and Valence	242
D	Valence for a Rational Function	243
E	The Argument Principle: Integrals That Count	243
	Exercises	249
3.5	Mapping Properties	251
	Exercises	259
3.6	The Riemann Sphere	260
	Exercises	264
4	The Residue Calculus	267
4.1	Integrals of Trigonometric Functions	268
	Exercises	270
4.2	Estimating Complex Integrals	273
	Exercises	276
4.3	Integrals of Rational Functions Over the Line	277
	Exercises	280
4.4	Integrals Involving the Exponential	282
A	Integrals Giving Fourier Transforms	286
	Exercises	290
4.5	Integrals Involving a Logarithm	293
	Exercises	301
4.6	Integration on a Riemann Surface	302
A	Mellin Transforms	306
	Exercises	307
4.7	The Inverse Laplace Transform	309
	Exercises	315
5	Boundary Value Problems	317
5.1	Examples	318
A	Easy Problems	318
B	The Conformal Mapping Method	323
	Exercises	326
5.2	The Mobius Maps	327
	Exercises	338
5.3	Electric Fields	341
A	A Point Charge in 3-Space	341
B	Uniform Charge on One or More Long Wires	342
C	Examples with Bounded Potentials	347
	Exercises	350
5.4	Steady Flow of a Perfect Fluid	350
	Exercises	354
5.5	Using the Poisson Integral to Obtain Solutions	355
A	The Poisson Integral on a Disk	355
B	Solutions on the Disk by the Poisson Integral	358
C	Geometry of the Poisson Integral	361
D	Harmonic Functions and the Mean Value Property	363
E	The Neumann Problem on a Disk	364
F	The Poisson Integral on a Half-Plane, and on Other Domains	365
	Exercises	366
5.6	When Is the Solution Unique?	368
	Exercises	370
5.7	The Schwarz Reflection Principle	370
5.8	Schwarz-Christoffel Formulas	374
A	Triangles	375
B	Rectangles and Other Polygons	385
C	Generalized Polygons	389
	Exercises	390
6	Lagniappe	393
6.1	Dixon's 1971 Proof of Cauchy's Theorem	394
6.2	Runge's Theorem	398
	Exercises	403
6.3	The Riemann Mapping Theorem	404
	Exercises	405
6.4	The Osgood-Taylor-Caratheodory Theorem	406
	References	413
	Index	419