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复变函数及应用(英文版·第9版)


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(美)James Ward Brown, Ruel V.Churchill 著
978-7-111-47087-8
85.00
472
2014年07月11日

数学 > 函数论 > 复变函数与积分变换
McGraw-Hill
1387
英文
16
Complex Variables and Applications
教材
华章数学原版精品系列








本书初版于20世纪40年代,是经典的本科数学教材之一,对复变函数的教学影响深远,被美国斯坦福大学、加州理工学院、加州大学伯克利分校、佐治亚理工学院、普度大学、达特茅斯学院、南加州大学等众多名校采用。.
本书阐述了复变函数的理论及应用,还介绍了留数及保形映射理论在物理、流体及热传导等边值问题中的应用。..
新版对原有内容进行了重新组织,增加了更现代的示例和应用,更加方便教学。
本书是复分析入门教材,内容丰富,写作精炼,论证严密。阐述了复变函数的理论及应用,还介绍了留数及保形映射理论在物理、流体及热传导等边值问题中的应用。第9版对第8版做了全面修订,重新组织了内容,增加了很多新的示例和习题,更加方便教学。
这本畅销全世界的经典教材初版于20世纪40年代,被国外众多名校广泛采用,如美国斯坦福大学、加州理工学院、加州大学伯克利分校、佐治亚理工学院、普度大学、达特茅斯学院、南加州大学等。前几版曾被译成日语、西班牙语、阿拉伯语、希腊语、韩语等众多版本,对复变函数的教学影响深远。
James Ward Brown 密歇根大学迪尔伯恩分校数学系荣休教授,美国数学会会士,入编《美国名人录》和《世界名人录》。1964年于密歇根大学获得博士学位,1971年至2011年任密歇根大学教授,并于1976年获得密歇根大学杰出教学奖。
Ruel V. Churchill(1899—1987) 生前是密歇根大学知名教授,于芝加哥大学取得理学学士学位,于密歇根大学取得物理学硕士及数学博士学位,自1922年起在密歇根大学执教。他从20世纪40年代开始编写一系列经典教材,除本书外,还与James Ward Rrown合著有《Fourier Series and Boundary Value Problems》。
CONTENTS
Preface
1 Complex Numbers 1
Sums and Products 1
Basic Algebraic Properties 3
Further Algebraic Properties 5
Vectors and Moduli 8
Triangle Inequality 11
Complex Conjugates 14
ExponentialForm 17
Products andPowersin ExponentialForm 20
Arguments of Products and Quotients 21
Roots of Complex Numbers 25
Examples 28
Regions in the Complex Plane 32
2 Analytic Functions 37
Functions and Mappings 37
The Mapping w = z2 40
Limits 44
Theorems on Limits 47
Limits Involving the Point at Infinity 50
Continuity 52
Derivatives 55
Rules for Differentiation 59
Cauchy–Riemann Equations 62
Examples 64
Sufficient Conditions for Differentiability 65
Polar Coordinates 68
Analytic Functions 72
Further Examples 74
Harmonic Functions 77
Uniquely Determined Analytic Functions 80
Reflection Principle 82
3 Elementary Functions 87
The Exponential Function 87
The Logarithmic Function 90
Examples 92
Branches and Derivatives of Logarithms 93
Some Identities Involving Logarithms 97
The Power Function 100
Examples 101
TheTrigonometric Functions sin z and cos z 103
Zeros and SingularitiesofTrigonometric Functions 105
Hyperbolic Functions 109
InverseTrigonometric and Hyperbolic Functions 112
4 Integrals 115
Derivatives of Functions w(t) 115
Definite Integrals of Functions w(t) 117
Contours 120
Contour Integrals 125
Some Examples 127
Examples Involving Branch Cuts 131
Upper Bounds for Moduli of Contour Integrals 135
Antiderivatives 140
Proof of the Theorem 144
Cauchy–Goursat Theorem 148
Proof of the Theorem 150
Simply Connected Domains 154
Multiply Connected Domains 156
CauchyIntegralFormula 162
An Extensionof the CauchyIntegralFormula 164
Verification of the Extension 166
Some Consequences of the Extension 168
Liouville’s Theorem and the Fundamental Theorem of Algebra 172
Maximum Modulus Principle 173
5 Series 179
Convergence of Sequences 179
Convergence of Series 182
Taylor Series 186
ProofofTaylor’s Theorem 187
Examples 189
NegativePowersof (z .z0) 193
Laurent Series 197
Proof of Laurent’s Theorem 199
Examples 202
Absolute and Uniform Convergence of Power Series 208
Continuity of Sums of Power Series 211
Integration and Differentiation of Power Series 213
Uniqueness of Series Representations 216
Multiplication and Division of Power Series 221
6 Residues and Poles 227
Isolated Singular Points 227
Residues 229
Cauchy’s Residue Theorem 233
Residue at Infinity 235
The ThreeTypesof Isolated Singular Points 238
Examples 240
Residues at Poles 242
Examples 244
Zeros of Analytic Functions 248
Zeros and Poles 251
Behavior of Functions Near Isolated Singular Points 255
7 Applications of Residues 259
Evaluation of Improper Integrals 259
Example 262
Improper Integrals fromFourier Analysis 267
Jordan’s Lemma 269
An IndentedPath 274
An Indentation Around a Branch Point 277
Integration Along a Branch Cut 280
Definite Integrals Involving Sines and Cosines 284
Argument Principle 287
Rouch′e’s Theorem 290
Inverse LaplaceTransforms 294
8 Mapping by Elementary Functions 299
LinearTransformations 299
TheTransformation w = 1/z 301
Mappingsby 1/z 303
Linear FractionalTransformations 307
An ImplicitForm 310
Mappings of the Upper Half Plane 313
Examples 315
Mappings by the Exponential Function 318
MappingVertical LineSegmentsby w = sin z 320
Mapping Horizontal Line Segments by w = sin z 322
Some Related Mappings 324
Mappingsby z2 326
Mappings by Branches of z1/2 328
Square Roots of Polynomials 332
Riemann Surfaces 338
Surfaces for Related Functions 341
9 Conformal Mapping 345
Preservationof Angles and ScaleFactors 345
Further Examples 348
Local Inverses 350
Harmonic Conjugates 354
Transformations of Harmonic Functions 357
Transformations of Boundary Conditions 360
10 Applications of Conformal Mapping 365
SteadyTemperatures 365
SteadyTemperaturesina Half Plane 367
ARelated Problem 369
Temperatures in a Quadrant 371
Electrostatic Potential 376
Examples 377
Two-Dimensional Fluid Flow 382
The Stream Function 384
Flows Around a Corner and Around a Cylinder 386
11 The Schwarz–Christoffel Transformation 393
Mapping the Real Axis onto a Polygon 393
Schwarz–ChristoffelTransformation 395
Triangles and Rectangles 399
Degenerate Polygons 402
Fluid Flow in a Channel through a Slit 407
Flow in a Channel with an Offset 409
Electrostatic Potential about an Edge of a Conducting Plate 412
12 Integral Formulas of the Poisson Type 417
Poisson IntegralFormula 417
Dirichlet Problem for a Disk 420
Examples 422
Related BoundaryValue Problems 426
Schwarz IntegralFormula 428
Dirichlet Problem for a Half Plane 430
Neumann Problems 433
Appendixes 437
Bibliography 437
TableofTransformationsofRegions 441
Index 451
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复变函数及应用(英文版·第8版)
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