《AUTOMORPHIC FUNCTIONS》

CHAPTER ⅠLINEAR TRANSFORMATIONS1

1．The Linear Transformation1

2．Symbolic Notation4

3．The Fixed Points of the Transformation6

4．The Linear Transformation and the Circle8

5．Inversion in a Circle10

6．The Multiplier，K15

7．The Hyperbolic Transformation，K=A18

8．The Elliptic Transformation，K=eiθ19

9．The Loxodromic Transformation，K=Aeiθ20

10．The Parabolic Transformation21

11．The Isometric Circle23

12．The Unit Circle30

CHAPTER ⅡGROUPS OF LINEAR TRANSFORMATIONS33

13．Definition of a Group．Examples33

14．Properly Discontinuous Groups35

15．Transforming a Group36

16．The Fundamental Region37

17．The Isometric Circles of a Group39

18．The Limit Points of a Group41

19．Definition of the Region R44

20．The Regions Congruent to R44

21．The Boundary of R47

22．Example．A Finite Group49

23．Generating Transformations50

24．Cyclic Groups51

25．The Formation of Groups by the Method of Combination56

26．Ordinary Cycles59

27．Parabolic Cycles62

28．Function Groups64

CHAPTER ⅢFUCHSIAN GROUPS67

29．The Transformations67

30．The Limit Points67

31．The Region R and the Region Ro69

32．Generating Transformations71

33．The Cycles72

34．Fuchsian Groups of the First and Second Kinds73

35．Fixed Points at Infinity．Extension of the Method75

36．Examples78

37．The Modular Group79

38．Some Subgroups of the Modular Group81

CHAPTER ⅣAUTOMORPHIC FUNCTIONS83

39．The Concept of the Automorphic Function83

40．Simple Automorphic Functions86

41．Behavior at Vertices and Parabolic Points88

42．The Poles and Zeros91

43．Algebraic Relations94

44．Differential Equations98

CHAPTER ⅤTHE POINCARE THETA SERIES102

45．The Theta Series102

46．The Convergence of the Series104

47．The Convergence for the Fuchsian Group of the Second Kind106

48．Some Properties of the Theta Functions108

49．Zeros and Poles of the Theta Functions112

50．Series and Products Connected with the Group115

CHAPTER ⅥTHE ELEMENTARY GROUPS117

Ⅰ．The Finite Groups117

51．Inversion in a Sphere117

52．Stereographic Projection119

53．Rotations of the Sphere120

54．Groups of the Regular Solids123

55．A Study of the Cube124

56．The General Regular Solid127

57．Determination of All the Finite Groups129

58．The Extended Groups136

Ⅱ．The Groups with One Limit Point139

59．The Simply and Doubly Periodic Groups139

60．Groups Allied to the Periodic Groups140

61．The Automorphic Functions144

Ⅲ．The Groups with Two Limit Points146

62．Determination of the Groups146

CHAPTER ⅦTHE ELLIPTIC MODULAR FUNCTIONS148

63．Certain Results from the Theory of Elliptic Functions148

64．Change of the Primitive Periods150

65．The Function J(τ)151

66．Behavior of J(τ) at the Parabolic Points153

67．Further Properties of J(τ)155

68．The Function λ(τ)157

69．The Relation between λ(τ) and J(τ)159

70．Further Properties of λ(τ)160

CHAPTER ⅧCONFORMAL MAPPING164

71．Conformal Mapping164

72．Schwarz＇s Lemma165

73．Area Theorems167

74．The Mapping of a Circle on a Plane Finite Region169

75．The Deformation Theorem for the Circle171

76．A General Deformation Theorem175

77．An Application of Poisson＇s Integral177

78．The Mapping of a Plane Simply Connected Region on a Circle．The Iterative Process179

79．The Convergence of the Process183

80．The Behavior of the Mapping Function on the Boundary187

81．Regions Bounded by Jordan Curves198

82．Analytic Arcs and the Continuation of the Mapping Function across the Boundary201

83．Circular Arc Boundaries202

84．The Mapping of Combined Regions203

85．The Mapping of Limit Regions205

86．The Mapping of Simply Connected Finite-sheeted Regions213

87．Conformal Mapping and Groups of Linear Transformations216

CHAPTER ⅨUNIFORMIZATION．ELEMENTARY AND FUCHSIAN FUNCTIONS220

88．The Concept of Uniformization220

89．The Connectivity of Regions221

90．Algebraic Functions of Genus Zero．Uniformization by Means of Rational Functions229

91．Algebraic Functions of Genus Greater than Zero．Uniformization by Means of Automorphic Functions233

92．The Genus of the Fundamental Region of a Group238

93．The Cases p=1 and p>1239

94．More General Fuchsian Uniformizing Functions241

95．The Case p=0245

96．Whittaker＇s Groups247

97．The Transcendental Functions249

CHAPTER ⅩUNIFORMIZATION．GROUPS OF SCHOTTKY TYPE256

98．Regions of Planar Character256

99．Some Accessory Functions258

100．The Mapping of a Multiply Connected Region of Planar Character on a Slit Region262

101．Application to the Uniformization of Algebraic Functions266

102．A Convergence Theorem267

103．The Sequence of Mapping Functions271

104．The Linearity of Tn273

105．An Extension278

106．The Mapping of a Multiply Connected Region of Planar Character on a Region Bounded by Complete Circles279

CHAPTER ⅪDIFFERENTIAL EQUATIONS284

107．Connection with Groups of Linear Transformations284

108．The Inverse of the Quotient of Two Solutions287

109．Regular Singular Points of Differential Equations293

110．The Quotient of Two Solutions at a Regular Singular Point296

111．Equations with Rational Coefficients299

112．The Equation with Two Singular Points303

113．The Hypergeometric Equation303

114．The Riemann-Schwarz Triangle Functions305

115．Equations with Algebraic Coefficients308

A BIBLIOGRAPHY OF AUTOMORPHIC FUNCTIONS311

AUTHOR INDEX325

SUBJECT INDEX327