第1章　復(fù)數(shù) 1
1.1　復(fù)數(shù)代數(shù) 1
1.2　復(fù)數(shù)的點表示 7
1.3　向量與極式 14
1.4　復(fù)指數(shù) 26
1.5　冪與根 33
1.6　平面集 39
1.7　黎曼球面與球極射影 44
小結(jié) 51
第2章　解析函數(shù) 53
2.1　復(fù)變函數(shù) 53
2.2　極限與連續(xù)性 58
2.3　解析性 65
2.4　柯西–黎曼方程 73
2.5　調(diào)和函數(shù) 79
*2.6　調(diào)和函數(shù)的一個實例—恒溫 87
*2.7　迭代映射—茹利亞集與芒德布羅集 91
小結(jié) 95
第3章　初等函數(shù) 99
3.1　多項式與有理函數(shù) 99
3.2　指數(shù)函數(shù)、三角函數(shù)與雙曲函數(shù) 110
3.3　對數(shù)函數(shù) 118
3.4　墊、楔與壁 125
3.5　復(fù)冪函數(shù)與復(fù)反三角函數(shù) 131
*3.6　在振蕩系統(tǒng)中的應(yīng)用 138
小結(jié) 145
第4章　復(fù)積分 149
4.1　周線 149
4.2　周線積分 161
4.3　積分與路徑的無關(guān)性 173
4.4　柯西積分定理 180
4.4.a　周線形變法 180
4.4.b　向量分析法 191
4.5　柯西積分公式及其推論 204
4.6　解析函數(shù)的界 214
*4.7　在調(diào)和函數(shù)中的應(yīng)用 221
小結(jié) 230
第5章　解析函數(shù)的級數(shù)表示 235
5.1　序列與級數(shù) 235
5.2　泰勒級數(shù) 242
5.3　冪級數(shù) 252
*5.4　收斂的數(shù)學(xué)理論 262
5.5　洛朗級數(shù) 269
5.6　零點與奇點 277
5.7　無窮遠點 287
*5.8　解析延拓 292
小結(jié) 304
第6章　留數(shù)理論 307
6.1　留數(shù)定理 307
6.2　[0,2π]上三角函數(shù)的積分 314
6.3　(–∞,+∞)上某些函數(shù)的反常積分 318
6.4　涉及三角函數(shù)的反常積分 328
6.5　凹周線 337
6.6　關(guān)于多值函數(shù)的積分 345
6.7　輻角原理與儒歇定理 355
小結(jié) 367
第7章　共形映射 369
7.1　拉普拉斯方程的不變性 369
7.2　幾何性質(zhì) 377
7.3　默比烏斯變換 383
7.4　默比烏斯變換（續(xù)） 395
7.5　施瓦茨–克里斯托費爾變換 407
7.6　在靜電學(xué)、熱流與流體力學(xué)中的應(yīng)用 419
7.7　共形映射在物理中的進一步應(yīng)用 432
小結(jié) 443
第8章　應(yīng)用數(shù)學(xué)的變換 445
8.1　傅里葉級數(shù)（有限傅里葉變換） 446
8.2　傅里葉變換 464
8.3　拉普拉斯變換 476
8.4　z變換 486
8.5　柯西積分與希爾伯特變換 495
小結(jié) 509
附錄A　共形映射的數(shù)值結(jié)構(gòu) 513
附錄B　共形映射表 531
奇數(shù)練習(xí)答案 539


Contents
Contents
1 Complex Numbers 1
1.1 TheAlgebraofComplexNumbers ................... 1

1.2 Point Representation of Complex Numbers . ............. 7

1.3 VectorsandPolarForms ........................ 14

1.4 TheComplexExponential ....................... 26

1.5 PowersandRoots ............................ 33

1.6 PlanarSets ............................... 39

1.7 The Riemann Sphere and Stereographic Projection . ......... 44

Summary ................................ 51

2 Analytic Functions 53
2.1 FunctionsofaComplexVariable .................... 53

2.2 LimitsandContinuity.......................... 58

2.3 Analyticity ............................... 65

2.4 TheCauchy-RiemannEquations .................... 73

2.5 HarmonicFunctions........................... 79

2.6 *Steady-State Temperature as a Harmonic Function . ......... 87

2.7 *IteratedMaps:JuliaandMandelbrotSets . . ............. 91

Summary ................................ 95

3 Elementary Functions 99
3.1 PolynomialsandRationalFunctions . . . . . ............. 99

3.2 The Exponential, Trigonometric, and Hyperbolic Functions . . . . . . 110
3.3 TheLogarithmicFunction ....................... 118

3.4 Washers,Wedges,andWalls ...................... 125

3.5 Complex Powers and Inverse Trigonometric Functions . . . . . . . . 131
3.6 *ApplicationtoOscillatingSystems . . . . . ............. 138

Summary ................................ 145

4 Complex Integration 149
4.1 Contours................................. 149

4.2 ContourIntegrals ............................ 161

4.3 IndependenceofPath .......................... 173

4.4 Cauchy’sIntegralTheorem ....................... 180

4.4a Deformation of Contours Approach . . . ........... 180

4.4b VectorAnalysisApproach ................... 191

4.5 Cauchy’s Integral Formula and Its Consequences ........... 204

4.6 BoundsforAnalyticFunctions ..................... 214

4.7 *ApplicationstoHarmonicFunctions . . . . . . ........... 221

Summary ................................ 230

5 Series Representations for Analytic Functions 235
5.1 SequencesandSeries .......................... 235

5.2 TaylorSeries .............................. 242

5.3 PowerSeries .............................. 252

5.4 *MathematicalTheoryofConvergence. . . . . . ........... 262

5.5 LaurentSeries.............................. 269

5.6 ZerosandSingularities ......................... 277

5.7 ThePointatIn.nity........................... 287

5.8 *AnalyticContinuation ......................... 292

Summary ................................ 304

6 Residue Theory 307
6.1 TheResidueTheorem.......................... 307

6.2 Trigonometric Integrals over [0, 2π] .................. 314

6.3 Improper Integrals of Certain Functions over (.∞, ∞) ........ 318

6.4 Improper Integrals Involving Trigonometric Functions . . . . .... 328
6.5 IndentedContours............................ 337

6.6 Integrals Involving Multiple-Valued Functions . . ........... 345

6.7 The Argument Principle and Rouch′e’s Theorem . ........... 355

Summary ................................ 367

7 Conformal Mapping 369
7.1 InvarianceofLaplace’sEquation .................... 369

7.2 GeometricConsiderations ....................... 377

7.3 M¨obiusTransformations ........................ 383

7.4 M¨obiusTransformations,Continued .................. 395

7.5 The Schwarz-Christoffel Transformation . . . . . ........... 407

7.6 Applications in Electrostatics, Heat Flow, and Fluid Mechanics .... 419
7.7 Further Physical Applications of Conformal Mapping . . . . . .... 432
Summary ................................ 443
8 The Transforms of Applied Mathematics 445
8.1 FourierSeries(TheFiniteFourierTransform) ............. 446

8.2 TheFourierTransform ......................... 464

8.3 TheLaplaceTransform......................... 476

8.4 Thez-Transform ............................ 486

8.5 CauchyIntegralsandtheHilbertTransform .............. 495

Summary ................................ 509

A Numerical Construction of Conformal Maps 513
B Table of Conformal Mappings 531
Answers to Odd-Numbered Problems 539