Complex analysis : an introduction to the theory of analytic functions of one complex variable / Lars V. Ahlfors.
Series International series in pure and applied mathematicsEditor: New York : McGraw-Hill, 1953Descripción: xi, 247 p. : il. ; 24 cmOtra clasificación: 30-01CHAPTER I COMPLEX NUMBERS 1 The algebra of complex numbers [1] 1.1. Arithmetic operations [1] 1.2. Square roots [2] 1.3. Justification [4] 1.4. Conjugation. Absolute value [6] 1.5. Inequalities [8] 2 The geometric representation of complex numbers [10] 2.1. Geometric addition and multiplication [11] 2.2. The binomial equation [13] 2.3. Definition of the argument [14] 2.4. Straight lines, half planes, and angles [18] 2.5. The spherical representation [20] 3. Linear transformations [22] 3.1. The linear group [23] 3.2. The cross ratio [25] 3.3. Symmetry [26] 3.4. Tangents, orientation, and angles [29] 3.5. Families of circles [31] CHAPTER II COMPLEX FUNCTIONS 1. Elementary functions [36] 1.1. Limits and continuity [36] 1.2. Analytic functions [38] 1.3. Rational functions [42] 1.4. The exponential function [46] 1.5. The trigonometric functions [49] 2. Topological concepts [51] 2.1. Point sets [51] 2.2. Connected sets [56] 2.3. Compact sets [59] 2.4. Continuous functions and mappings [61] 2.5. Arcs and closed curves [64] 3. Analytic functions in a region [66] 3.1. Definition and simple consequences [66] 3.2. Conformal mapping [69] 4. Elementary conformal mappings [72] 4.1. The use of level curves [72] 4.2. A survey of elementary mappings [75] 4.3. Elementary Riemann surfaces [79] CHAPTER III COMPLEX INTEGRATION 1. Fundamental theorems [82] 1.1. Line integrals [82] 1.2. Cauchy’s theorem for a rectangle [88] 1.3. Cauchy’s theorem in a circular disk [91] 2. Cauchy’s integral formula [92] 2.1. The index of a point with respect to a closed curve [92] 2.2. The integral formula [95] 2.3. Higher derivatives [96] 3. Local properties of analytic functions [99] 3.1. Removable singularities. Taylor’s theorem [99] 3.2. Zeros and poles [102] 3.3. The local mapping [105] 3.4. The maximum principle [108] 4. The general form of Cauchy’s theorem [111] 4.1. Chains and cycles [111] 4.2. Simple connectivity [112] 4.3. Exact differentials in simply connected regions [114] 4.4 Multiply connected regions [116] 5. The calculus of residues [119] 5.1. The residue theorem [120] 5.2. The argument principle [123] 5.3. Evaluation of definite integrals [125] CHAPTER IV INFINITE SEQUENCES 1. Convergent sequences [132] 1.1. Fundamental sequences [132] 1.2. Subsequences [134] 1.3. Uniform convergence [135] 1.4. Limits of analytic functions [137] 2. Power series [140] 2.1. The circle of convergence [140] 2.2. The Taylor series [141] 2.3. The Laurent series [147] 3. Partial fractions and factorization [149] 3.1. Partial fractions [149] 3.2. Infinite products [153] 3.3. Canonical products [155] 3.4. The gamma function [160] 3.5. Stirling’s formula [162] 4. Normal families [168] 4.1. Conditions of normality [168] 4.2. The Riemann mapping theorem [172] CHAPTER V THE DIRICHLET PROBLEM 1. Harmonic functions [175] 1.1. Definition and basic properties [175] 1.2. The mean-value property [178] 1.3. Poisson’s formula [179] 1.4. Hamack’s principle [183] 1.5. Jensen’s formula [184] 1.6. The symmetry principle [189] 2. Subharmonic functions [193] 2.1. Definition and simple properties [194] 2.2. Solution of Dirichlet’s problem [196] 3. Canonical mappings of multiply connected regions [199] 3.1. Harmonic measures [200] 3.2. Green’s function [205] 3.3. Parallel slit regions [206] CHAPTER VI MULTIPLE-VALUED FUNCTIONS 1. Analytic continuation [209] 1.1. General analytic functions [209] 1.2. The Riemann surface of a function [211] 1.3. Analytic continuation along arcs [212] 1.4. Homotopic curves [215] 1.5. The monodromy theorem [218] 1.6. Branch points [220] 2. Algebraic functions [223] 2.1. The resultant of two polynomials [223] 2.2. Definition and properties of algebraic functions [224] 2.3. Behavior at the critical points [226] 3. Linear differential equations [229] 3.1. Ordinary points [230] 3.2. Regular singular points [232] 3.3. Solutions at infinity [234] 3.4. The hypergeometric differential equation [235] 3.5. Riemann’s point of view [239]
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| 28 W832 Neuere Untersuchungen über eindeutige analytische Funktionen / | 28 Y68 The theory of integration / | 30 Ad244 Advances in the theory of Riemann surfaces : | 30 Ah285 Complex analysis : | 30 Ah285 Complex analysis : | 30 Ah285-2 Complex analysis : | 30 Ah285i Conformal invariants: topics in geometric function theory / |
CHAPTER I COMPLEX NUMBERS --
1 The algebra of complex numbers [1] --
1.1. Arithmetic operations [1] --
1.2. Square roots [2] --
1.3. Justification [4] --
1.4. Conjugation. Absolute value [6] --
1.5. Inequalities [8] --
2 The geometric representation of complex numbers [10] --
2.1. Geometric addition and multiplication [11] --
2.2. The binomial equation [13] --
2.3. Definition of the argument [14] --
2.4. Straight lines, half planes, and angles [18] --
2.5. The spherical representation [20] --
3. Linear transformations [22] --
3.1. The linear group [23] --
3.2. The cross ratio [25] --
3.3. Symmetry [26] --
3.4. Tangents, orientation, and angles [29] --
3.5. Families of circles [31] --
CHAPTER II COMPLEX FUNCTIONS --
1. Elementary functions [36] --
1.1. Limits and continuity [36] --
1.2. Analytic functions [38] --
1.3. Rational functions [42] --
1.4. The exponential function [46] --
1.5. The trigonometric functions [49] --
2. Topological concepts [51] --
2.1. Point sets [51] --
2.2. Connected sets [56] --
2.3. Compact sets [59] --
2.4. Continuous functions and mappings [61] --
2.5. Arcs and closed curves [64] --
3. Analytic functions in a region [66] --
3.1. Definition and simple consequences [66] --
3.2. Conformal mapping [69] --
4. Elementary conformal mappings [72] --
4.1. The use of level curves [72] --
4.2. A survey of elementary mappings [75] --
4.3. Elementary Riemann surfaces [79] --
CHAPTER III COMPLEX INTEGRATION --
1. Fundamental theorems [82] --
1.1. Line integrals [82] --
1.2. Cauchy’s theorem for a rectangle [88] --
1.3. Cauchy’s theorem in a circular disk [91] --
2. Cauchy’s integral formula [92] --
2.1. The index of a point with respect to a closed curve [92] --
2.2. The integral formula [95] --
2.3. Higher derivatives [96] --
3. Local properties of analytic functions [99] --
3.1. Removable singularities. Taylor’s theorem [99] --
3.2. Zeros and poles [102] --
3.3. The local mapping [105] --
3.4. The maximum principle [108] --
4. The general form of Cauchy’s theorem [111] --
4.1. Chains and cycles [111] --
4.2. Simple connectivity [112] --
4.3. Exact differentials in simply connected regions [114] --
4.4 Multiply connected regions [116] --
5. The calculus of residues [119] --
5.1. The residue theorem [120] --
5.2. The argument principle [123] --
5.3. Evaluation of definite integrals [125] --
CHAPTER IV INFINITE SEQUENCES --
1. Convergent sequences [132] --
1.1. Fundamental sequences [132] --
1.2. Subsequences [134] --
1.3. Uniform convergence [135] --
1.4. Limits of analytic functions [137] --
2. Power series [140] --
2.1. The circle of convergence [140] --
2.2. The Taylor series [141] --
2.3. The Laurent series [147] --
3. Partial fractions and factorization [149] --
3.1. Partial fractions [149] --
3.2. Infinite products [153] --
3.3. Canonical products [155] --
3.4. The gamma function [160] --
3.5. Stirling’s formula [162] --
4. Normal families [168] --
4.1. Conditions of normality [168] --
4.2. The Riemann mapping theorem [172] --
CHAPTER V THE DIRICHLET PROBLEM --
1. Harmonic functions [175] --
1.1. Definition and basic properties [175] --
1.2. The mean-value property [178] --
1.3. Poisson’s formula [179] --
1.4. Hamack’s principle [183] --
1.5. Jensen’s formula [184] --
1.6. The symmetry principle [189] --
2. Subharmonic functions [193] --
2.1. Definition and simple properties [194] --
2.2. Solution of Dirichlet’s problem [196] --
3. Canonical mappings of multiply connected regions [199] --
3.1. Harmonic measures [200] --
3.2. Green’s function [205] --
3.3. Parallel slit regions [206] --
CHAPTER VI MULTIPLE-VALUED FUNCTIONS --
1. Analytic continuation [209] --
1.1. General analytic functions [209] --
1.2. The Riemann surface of a function [211] --
1.3. Analytic continuation along arcs [212] --
1.4. Homotopic curves [215] --
1.5. The monodromy theorem [218] --
1.6. Branch points [220] --
2. Algebraic functions [223] --
2.1. The resultant of two polynomials [223] --
2.2. Definition and properties of algebraic functions [224] --
2.3. Behavior at the critical points [226] --
3. Linear differential equations [229] --
3.1. Ordinary points [230] --
3.2. Regular singular points [232] --
3.3. Solutions at infinity [234] --
3.4. The hypergeometric differential equation [235] --
3.5. Riemann’s point of view [239] --
MR, 14,857a
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