Basic Complex Analysis

    1.1 Introduction to Complex Numbers

    1.2 Properties of Complex Numbers

    1.3 Some Elementary Functions

    1.4 Continuous Functions

    1.5 Basic Properties of Analytic Functions

2. Cauchy's Theorem

    2.2 Cauchy's Theorem-A First Look

    2.3 A Closer Look at Cauchy's Theorem

    2.4 Cauchy's Integral Formula

    2.5 Maximum Modulus Theorem and Harmonic Functions

3. Series Representation of Analytic Functions

    3.1 Convergent Series of Analytic Functions

    3.2 Power Series and Taylor's Theorem

    3.3 Laurent Series and Classification of Singularities     

4. Calculus of Residues

    4.1 Calculation of Residues

    4.2 Residue Theorem

    4.3 Evaluation of Definite Integrals

5. Conformal Mappings

    5.1 Basic Theory of Conformal Mappings

    5.2 Fractional Linear and Schwarz-Christoffel Transformations

    5.3 Applications of Conformal Mappings to Laplace's Equation, Heat  Conduction, Electrostatics, and Hydrodynamics

6. Further Development of the Theory

    6.1 Analytic Continuation and Elementary Riemann Surfaces

    6.2 Rouche Theorem and Principle of the Argument

    6.3 Mapping Properties of Analytic Functions

7. Asymptotic Methods

    7.1 Infinite Products and the Gamma Function

    7.2 Asymptotic Expansions and the Method of Steepest Descent

    7.3 Stirlings Formula and Bessel Functions    

 8. Laplace Transform and Applications

  Answers to Odd-Numbered Exercises

  Index