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
- Cauchy-Rieman 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) Long-term capital management
- Some solutions
- Recipe for eggs Fourier: how long does it take to soft-boil 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.
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