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 : Tokyo : McGraw-Hill ; Kogakusha, c1966Edición: 2nd ed; International student edDescripción: xiii, 317 p. : il. ; 23 cmTema(s): Analytic functionsOtra clasificación: 30-01Preface vii CHAPTER 1: COMPLEX NUMBERS [1] 1. The Algebra of Complex Numbers [1] 1.1. Arithmetic Operations [1] 1.2. Square Roots [3] 1.3. Justification [4] 1.4. Conjugation, Absolute Value [6] 1.5. Inequalities [9] 2. The Geometric Representation of Complex Numbers [12] 2.1. Geometric Addition and Multiplication [12] 2.2. The Binomial Equation [15] 23. Analytic Geometry [17] 2 4. The Spherical Representation [18] CHAPTER 2: COMPLEX FUNCTIONS [21] 1. Introduction to the Concept of Analytic Function [21] 1.1. Limits and Continuity [22] 1.2. Analytic Functions [24] 1.3. Polynomials [28] 1.4. Rational Functions [30] 2. Elementary Theory of Power Series [33] 2.1. Sequences [34] 2.2. Series [35] 23. Uniform Convergence [36] 2.4. Power Series [38] 2.5. Abel's Limit Theorem [42] 3. The Exponential and Trigonometric Functions [43] 3.1. The Exponential [43] 3.2. The Trigonometric Functions [44] 3.3. The Periodicity [45] 3.4. The Logarithm [46] CHAPTER 3: ANALYTIC FUNCTIONS AS MAPPINGS [49] 1. Elementary Point Set Topology [50] 1.1. Sets and Elements [50] 1.2. Metric Spaces [51] 1.3. Connectedness [54] 1.4. Compactness [59] 1.5. Continuous Functions [64] 1.6. Topological Spaces [67] 2. Conformality [68] 2.1. Arcs and Closed Curves [68] 2.2. Analytic Functions in Regions [69] 2.3. Conformal Mapping [73] 3. Linear Transformations [76] 3.1. The Linear Group [76] 3.2. The Cross Ratio [78] 3.3. Symmetry [80] 3.4. Oriented Circles [83] 3.5. Families of Circles [84] 4. Elementary Conformal Mappings [89] 4.1. The Use of Level Curves [89] 4.2. A Survey of Elementary Mappings [93] 4.3. Elementary Riemann Surfaces [97] CHAPTER 4: COMPLEX INTEGRATION [101] J. Fundamental Theorems [101] 1.1. Line Integrals [101] 1.2. Rectifiable Arcs [104] 1.3. Line Integrals as Functions of Arcs [105] 1.4. Cauchy's Theorem for a Rectangle [109] 1.5. Cauchy's Theorem in a Circular Disk [112] 2. Cauchy’s Integral Formula [114] 2.1. The Index of a Point with Respect to a Closed Curve [114] 2.2. The Integral Formula [118] 2J. Higher Derivatives [120] 3. Local Properties of Analytic Functions [124] 3.1. Removable Singularities. Taylor’s Theorem [124] 3.2. Zeros and Poles [126] 3.3. The Local Mapping [130] 3.4. The Maximum Principle [133] 4. The General Form of Cauchy’s Theorem [137] 4.1. Chains and Cycles [137] 4.2. Simple Connectivity [139] 4.3. Exact Differentials in Simply Connected Regions [141] 4.4. Multiply Connected Regions [144] 5. The Calculus of Residues [147] 5.1. The Residue Theorem [147] 5.2. The Argument Principle [151] 5.3. Evaluation of Definite Integrals [153] 6. Harmonic Functions [160] 6.1. Definition and Basic Properties [160] 6.2. The Mean-value Property [163] 6.3. Poisson’s Formula [165] 6.4. Schwarz’s Theorem [167] 6.5. The Reflection Principle [170] CHAPTER 5: SERIES AND PRODUCT DEVELOPMENTS [173] 1. Power Series Expansions [173] 1.1. Weierstrass’s Theorem [173] 1.2. The Taylor Series [177] 1.3. The Laurent Series [182] 2. Partial Fractions and Factorization [185] 2.1. Partial Fractions [185] 2.2. Infinite Products [189] 2.3. Canonical Products [192] 2.4. The Gamma Function [196] 2.5. Stirling’s Formula [199] 3. Entire Functions [205] 3.1. Jensen’s Formula [205] 3.2. Hadamard’s Theorem [206] 4. Normal Families [210] 4.1. Equicontinuity [210] 4.2. Normality and Compactness [211] 4.3. Arzela’s Theorem [214] 4.4. Families of Analytic Functions [215] 4.5. The Classical Definition [217] CHAPTER 6: CONFORMAL MAPPING. DIRICHLET’S PROBLEM [221] The Riemann Mapping Theorem [221] 1.1. Statement and Proof [221] 1.2. Boundary Behavior [224] 1.3. Use of the Reflection Principle [226] 1.4. Analytic Arcs [226] 2. Conformal Mapping of Polygons [227] 2.1. The Behavior at an Angle [227] 2.2. The Schwarz-Christoffel Formula [228] 2.3. Mapping on a Rectangle [230] 2.4. The Triangle Functions of Schwarz [233] 3. A Closer Look at Harmonic Functions [233] 3.1. Functions with the Mean-value Property [234] 3.2. Hamack’s Principle [235] 4. The Dirichlet Problem [237] 4.1. Subharmonic Functions [237] 4.2. Solution of Dirichlet’s Problem [240] 5. Canonical Mappings of Multiply Connected Regions [243] 5.1. Harmonic Measures [244] 5.2. Green’s Function [249] 5.3. Parallel Slit Regions [251] CHAPTER 7: ELLIPTIC FUNCTIONS [255] 1. Simply Periodic Functions [255] 1.1. Representation by Exponentials [255] 1.2. The Fourier Development [256] 1.3. Functions of Finite Order [256] 2, Doubly Periodic Functions [257] 2.1. The Period Module [257] 2.2. Unimodular Transformations [258] 2.3. The Canonical Basis [260] 2.4. General Properties of Elliptic Functions [262] 3. The Weierstrass Theory [264] 3.1. The Weierstrass ؏-function [264] 3.2. The Functions ζ (z) and σ(z) [265] 3.3. The Differential Equation [267] 3.4. The Modular Function λ(r) [269] 3.5. The Conformal Mapping by λ(r) [271] CHAPTER 8: GLOBAL ANALYTIC FUNCTIONS [275] 1. Analytic Continuation [275] 1.1. General Analytic Functions [275] 1.2. The Riemann Surface of a Function [277] 1.3. Analytic Continuation along Arcs [278] 1.4. Homotopic Curves [281] 1.5. The Monodromy Theorem [285] 1.6. Branch Points [287] 2. Algebraic Functions [291] 2.1. The Resultant of Two Polynomials [291] 2.2. Definition and Properties of Algebraic Functions [292] 2.3. Behavior at the Critical Points [294] 3. Picard's Theorem [297] 3.1. Lacunary Values [297] 4. Linear Differential Equations [299] 4.1. Ordinary Points [300] 4.2. Regular Singular Points [302] 4.3. Solutions at Infinity [304] 4.4. The Hypergeometric Differential Equation [305] 4.5. Riemann’s Point of View [309] Index [313]
| Item type | Home library | Shelving location | Call number | Materials specified | Status | Date due | Barcode | Course reserves |
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Instituto de Matemática, CONICET-UNS | Libros ordenados por tema | 30 Ah285-2 (Browse shelf) | Available | A-9269 |
Preface vii --
CHAPTER 1: COMPLEX NUMBERS [1] --
1. The Algebra of Complex Numbers [1] --
1.1. Arithmetic Operations [1] --
1.2. Square Roots [3] --
1.3. Justification [4] --
1.4. Conjugation, Absolute Value [6] --
1.5. Inequalities [9] --
2. The Geometric Representation of Complex Numbers [12] --
2.1. Geometric Addition and Multiplication [12] --
2.2. The Binomial Equation [15] --
23. Analytic Geometry [17] --
2 4. The Spherical Representation [18] --
CHAPTER 2: COMPLEX FUNCTIONS [21] --
1. Introduction to the Concept of Analytic Function [21] --
1.1. Limits and Continuity [22] --
1.2. Analytic Functions [24] --
1.3. Polynomials [28] --
1.4. Rational Functions [30] --
2. Elementary Theory of Power Series [33] --
2.1. Sequences [34] --
2.2. Series [35] --
23. Uniform Convergence [36] --
2.4. Power Series [38] --
2.5. Abel's Limit Theorem [42] --
3. The Exponential and Trigonometric Functions [43] --
3.1. The Exponential [43] --
3.2. The Trigonometric Functions [44] --
3.3. The Periodicity [45] --
3.4. The Logarithm [46] --
CHAPTER 3: ANALYTIC FUNCTIONS AS MAPPINGS [49] --
1. Elementary Point Set Topology [50] --
1.1. Sets and Elements [50] --
1.2. Metric Spaces [51] --
1.3. Connectedness [54] --
1.4. Compactness [59] --
1.5. Continuous Functions [64] --
1.6. Topological Spaces [67] --
2. Conformality [68] --
2.1. Arcs and Closed Curves [68] --
2.2. Analytic Functions in Regions [69] --
2.3. Conformal Mapping [73] --
3. Linear Transformations [76] --
3.1. The Linear Group [76] --
3.2. The Cross Ratio [78] --
3.3. Symmetry [80] --
3.4. Oriented Circles [83] --
3.5. Families of Circles [84] --
4. Elementary Conformal Mappings [89] --
4.1. The Use of Level Curves [89] --
4.2. A Survey of Elementary Mappings [93] --
4.3. Elementary Riemann Surfaces [97] --
CHAPTER 4: COMPLEX INTEGRATION [101] --
J. Fundamental Theorems [101] --
1.1. Line Integrals [101] --
1.2. Rectifiable Arcs [104] --
1.3. Line Integrals as Functions of Arcs [105] --
1.4. Cauchy's Theorem for a Rectangle [109] --
1.5. Cauchy's Theorem in a Circular Disk [112] --
2. Cauchy’s Integral Formula [114] --
2.1. The Index of a Point with Respect to a Closed Curve [114] --
2.2. The Integral Formula [118] --
2J. Higher Derivatives [120] --
3. Local Properties of Analytic Functions [124] --
3.1. Removable Singularities. Taylor’s Theorem [124] --
3.2. Zeros and Poles [126] --
3.3. The Local Mapping [130] --
3.4. The Maximum Principle [133] --
4. The General Form of Cauchy’s Theorem [137] --
4.1. Chains and Cycles [137] --
4.2. Simple Connectivity [139] --
4.3. Exact Differentials in Simply Connected Regions [141] --
4.4. Multiply Connected Regions [144] --
5. The Calculus of Residues [147] --
5.1. The Residue Theorem [147] --
5.2. The Argument Principle [151] --
5.3. Evaluation of Definite Integrals [153] --
6. Harmonic Functions [160] --
6.1. Definition and Basic Properties [160] --
6.2. The Mean-value Property [163] --
6.3. Poisson’s Formula [165] --
6.4. Schwarz’s Theorem [167] --
6.5. The Reflection Principle [170] --
CHAPTER 5: SERIES AND PRODUCT DEVELOPMENTS [173] --
1. Power Series Expansions [173] --
1.1. Weierstrass’s Theorem [173] --
1.2. The Taylor Series [177] --
1.3. The Laurent Series [182] --
2. Partial Fractions and Factorization [185] --
2.1. Partial Fractions [185] --
2.2. Infinite Products [189] --
2.3. Canonical Products [192] --
2.4. The Gamma Function [196] --
2.5. Stirling’s Formula [199] --
3. Entire Functions [205] --
3.1. Jensen’s Formula [205] --
3.2. Hadamard’s Theorem [206] --
4. Normal Families [210] --
4.1. Equicontinuity [210] --
4.2. Normality and Compactness [211] --
4.3. Arzela’s Theorem [214] --
4.4. Families of Analytic Functions [215] --
4.5. The Classical Definition [217] --
CHAPTER 6: CONFORMAL MAPPING. DIRICHLET’S PROBLEM [221] --
The Riemann Mapping Theorem [221] --
1.1. Statement and Proof [221] --
1.2. Boundary Behavior [224] --
1.3. Use of the Reflection Principle [226] --
1.4. Analytic Arcs [226] --
2. Conformal Mapping of Polygons [227] --
2.1. The Behavior at an Angle [227] --
2.2. The Schwarz-Christoffel Formula [228] --
2.3. Mapping on a Rectangle [230] --
2.4. The Triangle Functions of Schwarz [233] --
3. A Closer Look at Harmonic Functions [233] --
3.1. Functions with the Mean-value Property [234] --
3.2. Hamack’s Principle [235] --
4. The Dirichlet Problem [237] --
4.1. Subharmonic Functions [237] --
4.2. Solution of Dirichlet’s Problem [240] --
5. Canonical Mappings of Multiply Connected Regions [243] --
5.1. Harmonic Measures [244] --
5.2. Green’s Function [249] --
5.3. Parallel Slit Regions [251] --
CHAPTER 7: ELLIPTIC FUNCTIONS [255] --
1. Simply Periodic Functions [255] --
1.1. Representation by Exponentials [255] --
1.2. The Fourier Development [256] --
1.3. Functions of Finite Order [256] --
2, Doubly Periodic Functions [257] --
2.1. The Period Module [257] --
2.2. Unimodular Transformations [258] --
2.3. The Canonical Basis [260] --
2.4. General Properties of Elliptic Functions [262] --
3. The Weierstrass Theory [264] --
3.1. The Weierstrass ؏-function [264] --
3.2. The Functions ζ (z) and σ(z) [265] --
3.3. The Differential Equation [267] --
3.4. The Modular Function λ(r) [269] --
3.5. The Conformal Mapping by λ(r) [271] --
CHAPTER 8: GLOBAL ANALYTIC FUNCTIONS [275] --
1. Analytic Continuation [275] --
1.1. General Analytic Functions [275] --
1.2. The Riemann Surface of a Function [277] --
1.3. Analytic Continuation along Arcs [278] --
1.4. Homotopic Curves [281] --
1.5. The Monodromy Theorem [285] --
1.6. Branch Points [287] --
2. Algebraic Functions [291] --
2.1. The Resultant of Two Polynomials [291] --
2.2. Definition and Properties of Algebraic Functions [292] --
2.3. Behavior at the Critical Points [294] --
3. Picard's Theorem [297] --
3.1. Lacunary Values [297] --
4. Linear Differential Equations [299] --
4.1. Ordinary Points [300] --
4.2. Regular Singular Points [302] --
4.3. Solutions at Infinity [304] --
4.4. The Hypergeometric Differential Equation [305] --
4.5. Riemann’s Point of View [309] --
Index [313] --
MR, 32 #5844
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