Functions of a complex variable with applications

                                           COURSE OUTLINE

The material covered in the course is roughly Chapters 1-4 of Marsden and Hoffman's "Basic Complex Analysis". More explicitly, here is the list of topics. The numbering does *not* correspond to sections in Marsden and Hoffman.

1. The complex number system.
    1.1 The complex numbers as a field extension.
    1.2 The complex plane, C as a vector space over R, absolute value, complex conjugate.
    1.3  More vector stuff, the polar form, multiplication in C as rotations in the plane, Euler's and de Moivre's
            formulae.
    1.4-5 The complex exponential and trigonometric functions, powers and roots.
    1.6 Topology of the plane.

2. Functions of a complex variable.
    2.1 Limits and continuity for complex sequences and functions of a complex variable.
    2.2 Differentiability and analyticity.
    2.3 The Cauchy-Riemann equations.
    2.4 Harmonic functions.

3. More on elementary complex functions, including the logarithmic function.
     3.1 The complex exponential, trigonometric, and hyperbolic functions. definitions, relations,
             derivatives.
     3.2  Definition of the logarithmic function and its analyticity properties.

4. Complex integration and contour integrals.
    4.1 Smooth curves in the plane and contours.
    4.2 Definition of contour integrals.
    4.3 Basic properties of contour integrals.

5. Cauchy's integral theorem and its consequences.
    5.1 Continuous deformation of contours, homotopy, simply connected domains.
    5.2  Cauchy's Integral Theorem.

-- Midterm Material Ends Here --

    5.3 Cauchy's Integral Formula
    5.4 Cauchy's Intergal Formula generalized.
    5.5 Cauchy Estimates, Liouville's Theorem, the Fundametal Theorem of Algebra.
    5.6 The Maximum Modulus Principle.

6. Series representations of analytic functions .
    6.1 Sequences and series, a review.
    6.2 Power series, radius of convergence.
    6.3 Convergence of sequences and series of functions.
    6.4 Tayler series.
    6.5  Laurent series.
    6.6 Classification of zeros and singularities of analytic functions.
    6.7  The point at infinity and singularities at infinity.
 
7. Residue theory.
   7.1 Residues and ways to calculate them.
   7.2 Cauchy's residue theorem.
   7.3 Applications to real integrals.