ESAM 311-3 Complex Variables

Hermann Riecke
Spring 2007

1. Introduction
2. Algebraic Properties of Complex Numbers
   1. Addition etc.
   2. Geometrical Representation
3. Functions
   1. Polynomials
   2. Exponential function
   3. Trigonometric and hyperbolic functions
   4. Logarithm
   5. Fractional powers
   6. Inverse trigonometric functions
4. Riemann Surfaces
5. Derivatives. Analytic Functions
   1. Limits of complex functions
   2. Derivatives
   3. Cauchy-Riemann equations
   4. Harmonic functions
   5. Applications: fluid flow
6. Complex Integration
   1. Contour integrals
   2. Cauchy integral theorem
   3. Cauchy integral formula
7. Series Expansions
   1. Taylor series
   2. Laurent series
   3. Convergence, differentiation and integration of power series
8. Singularities and Residus
   1. Singularities
   2. Residues
   3. Improper real integrals
   4. Definite integrals involving trigonometric functions
   5. Improper integrals involving trigonometric functions
      1. Fourier transformation
   6. Cauchy principal value integral
   7. Integrals with branch cuts
   8. Laplace transformation
9. Conformal Mapping and Harmonic Functions
   1. Conformal mappings of domains
   2. Transformations of harmonic functions
      1. Cusps
      2. Joukowski airfoil
   3. Schwarz-Christoffel transformation

Assignments:
HW 1
HW 2
HW 3
HW 4
Practice Midterm
HW 5
HW 6
Review for Final

The midterm is Friday May 4, 12-1, in the usual class room. It is closed-book.

Office Hours: M 1-2 W 4-5 F 1-2 M458
Contact: h-riecke@northwestern.edu, 491-8316
Teaching Assistant:
Richard Kublik
Office Hours: Tu 10-11 & 1-3 Th 1-3 M451