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Covered so far
PART I Fourier Analysis
Lecture 1 (29.9.09): Introduction to the course. A motivating example: \dot{x}=Ax solved using the eigenvectors of A.
Lecture 2 (1.10.09): \dot{x}=Ax solved using the eigenvectors of A continued (Note 1). Outline of the link to Fourier series. Fourier series and the ear.
Lecture 3 (2.10.09): Definition of a vector space. The vector space of periodic functions.
Lecture 4 (6.10.09): Vector spaces: linear independence, dimension, basis. The dimension of the space of periodic functions. Aleph-null. Start of inner products.
Lecture 5 (8.10.09): Inner product spaces. Example of the dot product. Orthonormal bases. Projection.
Lecture 6 (9.10.09): Inner products for functions. The Fourier modes as a basis. Discussion.
Lecture 7 (13.10.09): Calculating the Fourier coefficient. The periodic step function example.
Lecture 8 (15.10.09): Odd/even. The Fourier expansion of the periodic step fxn: t=pi/2, t=0, interpolating discontinuities, Dirichlet's theorem.
Lecture 9 (16.10.09): Calculating cos n\pi, sin n\pi/2. The Fourier series of the rectified sine wave. Start of complex Fourier series.
Lecture 10 (20.10.09) Complex Fourier series, c_n=c_{-n}^* for real f(t). Complex inner products. Calculating the formula for c_n.
Lecture 11 (22.10.09) Complex Fourier series for the periodic step function.
Lecture 12 (23.10.09) Recap of PS3 q1, complex fourier series for f(t)=e^t for t in (-pi,pi) and periodic.
Lecture 13 (27.10.09) Parcevals theorem outline proof and example, start of Fourier transform, motivation and formulas.
Lecture 14 (29.10.09) More Fourier transform: motivation and general discussion, example.
Lecture 15 (30.10.09) PS4 q2, L^1 fxns have a Fourier transform, it closes on the Schwartz space. Gaussian curve example.
Lecture 16 (3.11.09) Fourier transform of the Gaussian, recap. Plancherel formula. Motivation for Dirac delta, definition and approx as thin narrow spike.
Lecture 17 (5.11.09) More on the Dirac delta. Different approaches. Fourier transform of the Dirac delta.
Lecture 18 (6.11.09) PS5 Q1. Dirac delta recap. The Dirac delta and the Fourier transform.
PART II Differential Equations
Lecture 19 (10.11.09) Start of differential equations. Definitions. First order linear equations.
Lecture 20 (12.11.09) First order homogeneous. Particular and complementary functions. Start of second order.
Lecture 21 (13.11.09) The forced damped harmonic oscillator. Two distinct real eigenvalues.
Lecture 22 (17.11.09) The damped harmonic oscillator, under, over and critically damped cases.
Lecture 23 (19.11.09) The under damped case, getting a real answer. The force oscillator, exponential forcing.
Lecture 24 (20.11.09) More on the forced damped oscillator; sum of two exponentials, etc.
Lecture 25 (26.11.09) The Euler equation. Start of series solutions.
Lecture 26 (27.11.09) Series solutions, an example and discussion.
Lecture 27 (1.12.09) Series solution \ddot{y}-ty=0 [a_0=y(0) and a_1=\dot{y}(0)]
Lecture 28 (3.12.09) Series solutions Hermite's equation, Hermite polynomials.
Lecture 29 (4.12.09) Series solutions, a long and messy example.
Lecture 30 (8.12.09) The method of Frobenius.
Lecture 31 (10.12.09) Bessel's equation, solved by Frobenius.
Lecture 32 (11.12.09) Bessel's eqn, v=\pm 1/2 solutions, general remarks.
Lecture 33 (15.12.09) More on Frobenius.
Lecture 34 (16.12.09) Series solutions, the theorem. A second solution, outline of method.
Lecture 35 (17.12.09) Christmas Quiz!