Chapter 1  Complex Numbers

  1.1  Complex Numbers

  Exercises for 1.1

  Answers or Hints for Exercises 1.1

  1.2  Moduli and Conjugates

  Exercises for 1.2

  Answers or Hints for Exercises 1.2

  1.3  Exponential Form

  Exercises for 1.3

  Answers or Hints for Exercises 1.3

  1.4  Powers and Roots

  Exercises for 1.4

  Answers or Hints for Exercises 1.4

  1.5  Geometrically Application of Complex Numbers

  Exercises for 1.5

  1.6  Plane Topology

  Exercises for 1.6

  Answers or Hints for Exercises 1.6

  1.7  Curves

Chapter 2  Analytic Functions

  2.1  Complex-valued Functions of a Complex Variable

  Exercises for 2.1

  Answers or Hints for Exercises 2.1

  2.2  Limits and Continuity

  Exercises for 2.2

  Answers or Hints for Exercises 2.2

  2.3  The Extended Plane and Infinity

  Exercises for 2.3

  Answers or Hints for Exercises 2.3

  2.4  Complex Differentiability

  Exercises for 2.4

  Answers or Hints for Exercises 2.4

  2.5  Analytic Functions

  Exercises for 2.5

  Answers or Hints for Exercises 2.5

  2.6  Laplace's Equation and Harmonic Conjugates

  Exercises for 2.6

  Answers or Hints for Exercises 2.6

Chapter 3  Elementary Functions

  3.1  The Exponential Functions

  Exercises for 3.1

  Answers or Hints for Exercises 3.1

  3.2  Linear Fractional Transformations

  Exercises for 3.2

  Answers or Hints for Exercises 3.2

  3.3  Trigonometric Functions

  Exercises for 3.3

  Answers or Hints for Exercises 3.3

  3.4  The Radical Functions

  Exercises for 3.4

  Answers or Hints for Exercises 3.4

  3.5  The Logarithm Function

  Exercises for 3.5

  Answers or Hints for Exercises 3.5

  3.6  Complex Exponents

  Exercises for 3.6

  Answers or Hints for Exercises 3.6

  3.7  Inverse Trigonometric and Hyperbolic Functions

  Exercises for 3.7

  Answers or Hints for Exercises 3.7

Chapter 4  Complex Integrals

  4.1  Contour Integrals and Its Simple Properties

  Exercise for 4.1

  Answers or Hints for Exercises 4.1

  4.2  Antiderivatives

  Exercises for 4.2

  Answers or Hints for Exercises 4.2

  4.3  Cauchy Theorem

  Exercises for 4.3

  Answers or Hints for Exercises 4.3

  4.4  Cauchy Integral Formula

  Exercises for 4.4

  Answers or Hints for Exercises 4.4

  4.5  Maximum Modulus Principle

  Exercises for 4.5

  Answers or Hints for Exercises 4.5

Chapter 5  Power Series

  5.1  Complex Sequences, Series and Their Basic Properties

  Exercises for 5.1

  Answers or Hints for Exercises 5.1

  5.2  Series of Complex Functions and Its Basic Properties

  Exercises for 5.2

  Answers or Hints for Exercises 5.2

  5.3  Power Series

  Exercises for 5". 3

  Answers or Hints for Exercises 5.3

  5.4  Taylor Series for Analytic Functions

  Exercises for 5.4  ~

  Answers or Hints for Exercises 5.4

  5.5  Manipulation of Power Series

  Exercises for 5.5

  Answers or Hints for Exercises 5.5

  5.6  The Zeros of Analytic Functions

  Exercises for 5.6

  Answers or Hints for Exercises 5.6

Chapter 6  Laurent Series and Isolated Singularities

  6.1  Lanrent Decomposition

  Exercises for 6.1

  Answers or Hints for Exercises 6.1

  6.2  Isolated Singular Point and Its Types

  Exercises for 6.2

  Answers or Hints for Exercises 6.2

  6.3  Isolated Singularity at Infinity

  Exercises for 6.3

  Answers or Hints for Exercises 6.3

  6.4  Entire Functions and Meromorphic Functions

  Exercises for 6.4

  Answers or Hints for Exercises 6.4

Chapter 7  Residue

  7.1  Residue and Cauchy Residue Theorem

  Exercises for 7.1

  Answers or Hints for Exercises 7.1

  7.2  The Argument Principle, Rouche's Theorem

  Exercises for 7.2

  Answers or Hints for Exercises 7.2

Chapter 8  Evaluation of Real Integrals

  8.1  Integrals of Trigonometric Functions

  Exercises for 8.1

  Answers or Hints for Exercises 8.1

  8.2  Rational Functions over the Real Line

  Exercises for 8.2

  Answers or Hints for Exercises 8.2

  8.3  Rational and Trigonometric Functions over the Real Line

  E