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复变函数及其应用(翻译版 原书第9版)

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Content Description

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,增加了例题和图、更加方便教学。

Author Description

詹姆斯·沃德·布朗(James Ward Brown),密歇根大学迪尔本分校数学系荣誉教授.取得哈佛大学理学学士学位和密歇根大学科学技术研究院数学硕士和博士学位.他与丘吉尔博士合著了《傅里叶级数和边值问题》,目前刊印到第9版.他曾获美国国家科学基金和密歇根大专院校董事会协会杰出教师奖,被列入世界名人录.


鲁埃尔V.丘吉尔(Ruel VChurchill),密歇根大学数学系荣誉教授, 从1922年开始在密歇根大学任教,1987年去世,曾取得芝加哥大学理学学士学位和密歇根大学物理硕士学位以及密歇根大学数学博士学位.他和布朗博士合著了《傅里叶级数和边值问题》,这是一部经典著作,大约起草于75年前.他还编写了《运算数学》一书.他曾在美国数学学会和其他数学协会或委员会担任过多种职务

Catalogue

译者序

作者序

前言

第1章复数1

1.和与积1

2.基本代数性质2

3.其他代数性质4

4.向量和模6

5.三角不等式8

6.共轭复数11

7.指数形式13

8.指数形式的乘积与幂16

9.乘积与商的辐角17

10.复数的根20

11.例子22

12.复平面中的区域26

第2章解析函数30

13.函数与映射30

14.映射w=z232

15.极限35

16.关于极限的定理37

17.涉及无穷远点的极限39

18.连续性41

19.导数44

20.导数的运算法则46

21.柯西黎曼方程49

22.例子50

23.可微的充分条件51

24.极坐标53

25.解析函数的定义及性质56

26.其他例子58

27.调和函数60

28.唯一确定的解析函数63

29.反射原理64

第3章初等函数67

30.指数函数67

31.对数函数70

32.例子71

33.对数函数的分支和导数72

34.一些涉及对数的恒等式75

35.幂函数77

36.例子78

37.三角函数sinz和cosz80

38.三角函数的零点和奇点82

39.双曲函数85

40.反三角函数与反双曲函数87

第4章积分90

41.函数w(t)的导数 90

42.函数w(t)的定积分91

43.围线94

44.围线积分98

45.一些例子100

46.涉及支割线的例子103

47.围线积分的模的上界107

48.原函数111

49.定理的证明114

50.柯西–古萨定理117

51.定理的证明119

52.单连通区域123

53.多连通区域124

54.柯西积分公式129

55.柯西积分公式的推广130

56.推广的柯西积分公式的证明133

57.推广的柯西积分公式的一些

结果134

58.刘维尔定理与代数基本定理137

59.最大模原理138

第5章级数143

60.序列的收敛性143

61.级数的收敛性145

62.泰勒级数148

63.泰勒定理的证明149

64.例子151

65.(z-z0)的负次幂154

66.洛朗级数157

67.洛朗定理的证明159

目录复变函数及其应用(翻译版·原书第9版)68.例子161

69.幂级数的绝对收敛和一致收敛167

70.幂级数的和函数的连续性169

71.幂级数的积分与求导171

72.级数展开式的唯一性173

73.幂级数的乘法和除法177

第6章留数和极点182

74.孤立奇点182

75.留数184

76.柯西留数定理187

77.无穷远点处的留数188

78.三种类型的孤立奇点191

79.例子193

80.极点处的留数194

81.例子196

82.解析函数的零点199

83.零点和极点201

84.函数在孤立奇点附近的性质205

第7章留数的应用208

85.广义积分的计算208

86.计算广义积分的例子210

87.傅里叶分析中的广义积分214

88.若尔当引理216

89.缩进路径221

90.绕分支点的缩进路径223

91.沿着支割线的积分225

92.涉及正弦和余弦的定积分229

93.辐角原理232

94.儒歇定理234

95.拉普拉斯逆变换237

第8章初等函数的映射240

96.线性变换240

97.变换w=1/z242

98.1/z的映射242

99.分式线性变换246

100.隐式分式线性变换248

101.上半平面的映射251

102.例子253

103.指数函数的映射255

104.垂线段在w=sinz映射下的象256

105.水平线段在w=sinz映射下

的象258

106.与正弦函数相关的映射259

107. z2的映射262

108. z1/2的分支的映射263

109.多项式的平方根266

110.黎曼曲面271

111.相关函数的曲面273

第9章共形映射276

112.保角性和伸缩因子276

113.两个例子278

114.局部逆变换280

115.调和共轭282

116.调和函数的映射285

117.边界条件的映射287

第10章共形映射的应用292

118.稳定温度292

119.半平面上的稳定温度293

120.一个相关问题295

121.在象限内的温度297

122.静电势301

123.求解电势问题的例子302

124.二维的流体流动306

125.流函数308

126.沿拐角和柱面的流动310

第11章施瓦茨克里斯托费尔

映射316

127.实轴到多边形的映射316

128.关于施瓦茨克里斯托费尔

映射317

129.三角形和矩形320

130.退化的多边形323

131.管道内通过狭缝的流体流动327

132.有支管的管道内的流动329

133.导电板边缘的静电势331

第12章泊松型积分公式335

134.泊松积分公式335

135.圆盘的狄利克雷问题337

136.例子339

137.相关的边值问题342

138.施瓦茨积分公式344

139.半平面的狄利克雷问题345

140.诺伊曼问题348

部分习题解答352

第1章复数352

2.基本代数性质352

3.其他代数性质353

5.三角不等式353

6.共轭复数355

9.乘积与商的辐角357

11.例子360

12.复平面上的区域363

第2章解析函数365

14.映射w=z2365

18.连续性366

20.导数的运算法则367

24.极坐标368

26.其他例子371

27.调和函数371

第3章初等函数372

30.指数函数372

33.对数函数的分支和导数375

34.一些涉及对数的恒等式377

36.例子378

38.三角函数的零点和奇点379

39.双曲函数382

40.反三角函数与反双曲函数384

第4章积分384

42.函数w(t)的定积分384

43.围线385

46.涉及支割线的例子386

47.围线积分的模的上界389

49.定理的证明392

53.多连通区域393

57.推广的柯西积分公式的一些

结果395

第5章级数399

61.级数的收敛性399

65.(z-z0)的负次幂400

68.例子402

72.级数展开式的唯一性406

73.幂级数的乘法和除法407

第6章留数和极点411

77.无穷远点处的留数411

79.例子416

81.例子419

83.零点和极点423

第7章留数的应用428

86.广义积分计算的例子428

88.若尔当引理438

91.沿着支割线的积分445

92.涉及正弦和余弦的定

积分451

94.儒歇定理452

95.拉普拉斯逆变换454

附录A参考文献459

附录B区域映射图(见

第8章)462

Introduction

本书是单复变函数理论及应用的一本教科书,供一学期使用.本书保持了之前版本的基本内容和风格,最初两版是由已故的Ruel VChurchill独自编写而成.

本书有两个主要目标.第一个目标是发展那些在应用中表现突出的理论部分.第二个目标是介绍留数和共形映射的应用.留数的应用包括用它来计算实数反常积分, 求拉普拉斯逆变换和函数的零点.共形映射可以用来解热传导和流体流动中产生的边值问题.作者的另一著作《傅里叶变换和边值问题》讲解了一种解偏微分方程边值问题的另一种经典方法,因此本书可以看作是该书的姊妹篇.

本书前9章在密歇根大学作为必修的课程已经有很多年了.后3章有一些变动主要是用来自学和参考.本书主要适用于数学.工程或物理专业的高年级学生.学习本书之前,应该至少完成三学期的微积分课程和一个学期的常微分方程课程的学习.如果想在本书中提前学习初等函数的映射,读者可以在完成第3章后直接跳到第8章学习初等函数,然后再回来学习第4章的积分.

我们介绍一些此版本的变动,其中一些变动是使用过本书的学生和教师提出的.首先移动了很多内容.例如,虽然在第2章仍然介绍调和函数,但是共轭调和函数挪到了第9章,因为第9章更需要共轭调和函数.另外,证明代数基本定理的一个重要不等式的推导从第4章移到了第1章, 因为第1章介绍了与其密切相关的不等式.这样做的优点在于把这些不等式放在一起可以使读者关注这些不等式,而且使得代数基本定理的证明更加简明,不让读者分心.第2章对映射定义的介绍有所缩短,只强调了映射w=z2.这是上一版的读者提出的建议,因为他们觉得在第2章用这一个例子阐明映射的定义就足够了.最后,因为第5章学习的大多数泰勒级数和洛朗级数依赖于读者对6个麦克劳林级数的熟悉程度,我们把它们放在一起方便读者查询.另外,第5章在泰勒定理之后包含单独的一节,主要致力于涉及zz0的负次幂的级数表达式.经验表明,这使得从泰勒级数到洛朗级数的转变显得很自然.


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Jingdong book