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Course Outline:

Complex Analysis

  1. Complex Numbers
    1. Real and imaginary parts, polar form, conjugates
    2. Operations: addition, subtraction, multiplication, division
    3. Euler's formula

  2. Complex Functions
    1. Powers
    2. Polynomials
    3. Roots
    4. Trig functions
    5. Exponential
    6. Logarithm
    7. Hyperbolic trigonometric functions

  3. Differentiation
    1. Derivatives
    2. Analytic functions
    3. Cauchy-Rieman Equations
    4. Harmonic functions

  4. Singularities
    1. Isolated singularities
    2. Poles
    3. Removable singularities

  5. Line integrals
    1. Definitions and examples
    2. Cauchy's Theorem
    3. Path independence
    4. Generalizations
    5. Residues
    6. Applications

Differential Equations

  1. What are PDEs and how do the arise?
    1. Physical motivation of some of the classical equations
    2. Separation of variables: turning a PDE into ODEs.
    3. Some other approaches

  2. Fourier series
    1. Some basics of power series
    2. Orthogonality of sines and of cosines
    3. Convergence of Fourier series
    4. A glimpse of wavelets: another way to represent functions

  3. Laplace's equation
    1. Where does this equation arise?
    2. Boundary value problems in general
    3. Some solutions of Laplace's equation

  4. The heat and diffusion equations
    1. Motivation, including Brownian motion and (possibly) Long-term capital management
    2. Some solutions
    3. Recipe for eggs Fourier: how long does it take to soft-boil an egg?

  5. The wave equation
    1. Some solutions
    2. Effects of boundary geometry
    3. Can you hear the shape of a drum?
    4. Drums with fractal boundaries

See the course calendar for details concerning exams and quizzes.



[HOME] Math 19 (Spring 2001) web pages
Created: 29 Mar 2001
Last modified: Apr 4, 2001 11:38:31 AM
Comments to: dpvc@union.edu
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