1550251515-Classical_Complex_Analysis__Gonzalez_

• 0.1 Sets


• 0.2 Mappings


• 0.3 Notations

• 1.1 The Complex Number System


• 1.2 Real and Imaginary Complex_ Numbers. The Complex Units


• 1.3 Complex Conjugates
  
  
  • Exercises 1.1
  
  
  
  


• 1.4 Ordering of the Complex Numbers


• 1.5 The Complex System as a Linear System and as an Algebra


• 1.6 Absolute Value of a Complex Number
  
  
  • Exercises 1.2
  
  
  
  


• 1.7 Geometric Representation of Complex Numbers


• 1.8 Polar Form of the Complex Number


• 1.9 Exponential Form of the Complex Number
  
  
  • Exercises 1.3
    
    
    • Bibliography
    
    
    
    
  
  
  
  


• 3 Functions. Limits and Continuity. Arcs and Curves
  
  
  • 3.1 Complex Functions
  
  
  • 3.2 The Components of a Complex Function
  
  
  • 3.3 Geometric Representation of Complex Functions
  
  
  • 3.4 Functions Associated with a Given Function
  
  
  • 3.5 Limit of a Complex Function at a Point
  
  
  • 3.6 Finite Limit at Infinity
  
  
  • 3.7 Properties of Finite Limits
  
  
  • 3.8 Infinite Limit at a Finite Point
  
  
  • 3.9 Infinite Limit at Infinity
  
  
  
  


• 3.10 Infinitely Small and Infinitely Large Functions
  
  
  • Exercises 3.1
  
  
  
  

• 3.12 Properties of Continuous Functions
  
  
  • Exercises 3.2
  
  
  
  


• 3.13 Oriented Arcs and Curves
  
  
  • Exercises 3.3
  
  
  
  


• 3.14 Chains and Cycles

• 3.16 The Winding Number of a Curve


• 3.17 Homology. The Connectivity of a Region i
  
  
  • Exercises 3.4
  
  
  • Bibliography
  
  
  
  


• 4 Sequences and Series
  
  
  • 4.1 Sequences of Complex Numbers
    
    
    • 4.2 Convergence of Sequences
    
    
    
    
  
  
  • 4.3 Properties of Convergent Sequences
  
  
  • 4.4 Limit of a Real Sequence. Limit Superior and Limit Inferior
  
  
  • 4.5 Cauchy Condition for Convergence
    
    
    • Exercises 4.1
    
    
    • 4.6 Series of Complex Numbers
    
    
    • 4.7 Criteria for Convergence of Series of Complex Numbers
    
    
    • 4.8 Some Properties of Series of Complex Terms
    
    
    • 4.9 Absolute Convergence Tests
    
    
    
    
  
  
  
  


• 4.10 Sequences and Series of Functions


• 4.11 Power Series
  
  
  • Exercises 4.2
  
  
  • Bibliography
  
  
  
  

• 5.1 The Translation w = z + b


• 5.2 The Similitude w = az (a I 0)


• 5.3 The Linear FUnction w = az + b (a I 0)
  
  
  • 5.4 The Reciprocal Function w = 1/ z
  
  
  • 5.5 The Bilinear Function w = (az + b)/(cz + d)
  
  
  • 5.6 Fixed Points of the Bilinear Transformation
  
  
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    5. 7 Multiplier of the Bilinear Transformation
    
    
    
    
  
  
  
  


• 5.8 Classification of the Bilinear Transformations


• 5.9 Symmetry with Respect to a Circle

• Geometry 5.11 The Poincare Model of Lobachevsky Non-Euclidean


• Exercises 5.1

• Exercises 5.2

• Exercises 5.3

• Exercises 5 .4


• Surfaces 5.20 The Function w = * Viz. Introduction of the Riemann

• Exercises 5.5


• Bibliography


• 6 Differentiation
  
  
  • Functions 6.1 The Concept of the Derivative. Monogenic and Analytic
  
  
  • 6.2 Continuity and Differentiability
  
  
  • 6.3 Differentiation Rules
  
  
  • 6.4 Differentiability of a Real Function of Two Real Variables
  
  
  
  


• 7.5 Other Types of Complex Integrals
  
  
  • the Limit of a Sum 7.6 Complex Integrals Along a Rectifiable Arc. The Integral as
  
  
  
  


• 7.7 Elementary Properties of the Complex Integral


• 7.8 Further Properties of the Complex Integral


• 7.9 Cauchy's Fundamental Theorem
  
  
  • Exercises 7 .1
  
  
  
  

• Applications 7.13 Analytical Representation of the Winding Number.

• Evaluation of Some Real Improper Integrals 7.15 Application of the Cauchy-Goursat Theorem to the


• Region 7.16 Primitives of an Analytic Function in a Multiply Connected


• Exercises 7.2

• Half-Plane 7.19 Cauchy's Integral Formula for Functions Analytic in a


• Exercises 7.3

• Exercises 7.4

• Exercises 7 .5

• Bibliography


• Representations. Some Special Functions 8 Sequences and Series of Functions. Series


• Functions 8.1 Integration and Differentiatio~ of Sequences and Series of


• 8.2 Analytic Functions Defined by Real Improper Integrals Contents xiii


• 8.3 The Cauchy-Taylor Expansion Theorem


• 8.4 Operations with Power Series
  
  
  • Bernoulli and Euler Numbers 8.5 Further Series Expansions. The Symbolic Method.
  
  
  • Exercises 8.1
  
  
  
  


• 8.6 Taylor Series for Nonanalytic Functions of Class C^00


• 8.7 Behavior of a Power Series on the Circle of Convergence
  
  
  • Exercises 8.2
  
  
  
  


• 8.8 Zeros of Continuous Functions
  
  
  • Functions 8.9 Zeros of Analytic Functions. Identity Principle for Analytic
  
  
  
  

• Exercises 8.3


• Analytic (or Conjugate Analytic) Functions 8.11 The Maximum and Minimum Modulus Principles for


• Functions 8.12 Maximum and Minimum Principles for Real Harmonic


• Exercises 8.4

• Exercises 8.5

• Exercises 8.6

• Exercises 8.

• Exercises 8.8

• Exercises 8.9

- Exercises 8.10

- Bibliography

      - 9.1 Regular and Singular Points

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• 9.2 Isolated Singularities xiv Contents


• 9.3 Behavior at a Pole


• 9.4 Behavior at an Essential Singularity


• 9.5 Nonisolated Singularities. Cluster Points
  
  
  • Exercises 9 .1
  
  
  • of Their Singularities 9.6 Characterization of Some Simple Functions by the Nature
  
  
  
  


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  9. 7 Residues
  
  
  
  


• 9.8 Some Special Rules for the Computation of Residues
  
  
  • Exercises 9.2
  
  
  
  


• 9.9 The Residue Theorem
  
  
  • Exercises 9.3
  
  
  
  

• Theorem 9.11 Evaluation of Real Improper Integrals by Using the Residue


• Exercises 9.4


• Exercises 9.5


• Exercises 9.6


• Exercises 9.

• Exercises 9.8

• Exercises 9.9


• Theorems 9.16 Mapping Properties of Analytic Functions. Inverse Function


• Exercise 9.10


• Bibliography