Complex Analysis
 Complex Numbers
 Real and imaginary parts, polar form, conjugates
 Operations: addition, subtraction, multiplication, division
 Euler's formula
 Complex Functions
 Powers
 Polynomials
 Roots
 Trig functions
 Exponential
 Logarithm
 Hyperbolic trigonometric functions
 Differentiation
 Derivatives
 Analytic functions
 CauchyRieman Equations
 Harmonic functions
 Singularities
 Isolated singularities
 Poles
 Removable singularities
 Line integrals
 Definitions and examples
 Cauchy's Theorem
 Path independence
 Generalizations
 Residues
 Applications
Differential Equations
 What are PDEs and how do the arise?
 Physical motivation of some of the classical equations
 Separation of variables: turning a PDE into ODEs.
 Some other approaches
 Fourier series
 Some basics of power series
 Orthogonality of sines and of cosines
 Convergence of Fourier series
 A glimpse of wavelets: another way to represent functions
 Laplace's equation
 Where does this equation arise?
 Boundary value problems in general
 Some solutions of Laplace's equation
 The heat and diffusion equations
 Motivation, including Brownian motion and (possibly) Longterm capital management
 Some solutions
 Recipe for eggs Fourier: how long does it take to softboil an egg?
 The wave equation
 Some solutions
 Effects of boundary geometry
 Can you hear the shape of a drum?
 Drums with fractal boundaries
See the course calendar for details concerning exams and quizzes.

