Chapter 1 Complex Numbers

1.1 Complex Numbers

Exercises for 1.1

Answers or Hints for Exercises 1.1

1.2 Moduli and Conjugates

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1.3 Exponential Form

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1.4 Powers and Roots

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1.5 Geometrically Application of Complex Numbers

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1.6 Plane Topology

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1.7 Curves

Chapter 2 Analytic Functions

2.1 Complex-valued Functions of a Complex Variable

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2.2 Limits and Continuity

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2.3 The Extended Plane and Infinity

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2.4 Complex Differentiability

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2.5 Analytic Functions

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2.6 Laplace’ s Equation and Harmonic Conjugates

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Chapter 3 Elementary Functions

3.1 The Exponential Functions

Exercises for 3.1

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3.2 Linear Fractional Transformations

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3.3 Trigonometric Functions

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3.4 The Radical Functions

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3.5 The Logarithm Function

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3.6 Complex Exponents

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3.7 Inverse Trigonometric and Hyperbolic Functions

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Chapter 4 Complex Integrals

4.1 Contour Integrals and Its Simple Properties

Exercise for 4.1

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4.2 Antiderivatives

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4.3 Cauchy Theorem

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4.4 Cauchy Integral Formula

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4.5 Maximum Modulus Principle

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Chapter 5 Power Series

5.1 Complex Sequences， Series and Their Basic Properties

Exercises for 5.1

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5.2 Series of Complex Functions and Its Basic Properties

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5.3 Power Series

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5.4 Taylor Series for Analytic Functions

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5.5 Manipulation of Power Series

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5.6 The Zeros of Analytic Functions

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Chapter 6 Laurent Series and Isolated Singularities

6.1 Laurent Decomposition

Exercises for 6.1

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6.2 Isolated Singular Point and Its Types

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6.3 Isolated Singularity at Infinity

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6.4 Entire Functions and Meromorphic Functions

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Chapter 7 Residue

7.1 Residue and Cauchy Residue Theorem

Exercises for 7.1

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7.2 The Argument Principle，Rouche’s Theorem

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Chapter 8 Evaluation of Real Integrals

8.1 Integrals of Trigonometric Functions

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8.2 Rational Functions over the Real Line

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8.3 Rational and Trigonometric Functions over the Real Line，

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8.4 Principal Value Integrals，Indentation Round a Singularity

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8.5 Integrals with Branch Points

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