Office: Math 227

Email: packer@colorado.edu

URL: http://spot.colorado.edu/~packer

The material to be covered includes the Appendix, most of Chapters 1 and 3, and parts of Chapters 2, 5, and 6, in the book "Fourier Analysis" by Eric Stade. Topics include:

A review of the definition and arithmetic of complex numbers, periodic functions on the real line

- Homework will be assigned every week. Some, but not all, of the problems will be graded. Please note that the assigments for Math 5330 will include extra problems. The assessment of homework performance will count for 20% of the final grade.
- In-class mid-term exam - Wednesday, Feb. 19, 2014, 9 a.m. - 10 a.m., DUAN G2B21: 25 % of final grade.
- Solutions to first midterm exam, Feb. 19, 2014.
- Take-home mid-term exam - given out Wednesday, April 2, 2014, 10 a.m., due Wednesday April 9, 2014, 4:30 p.m. (please note that takehome exams will be different for Math 4330 and Math 5530, and students in Math 5330 will also need to do a 15 minure oral presentation): 25 % of final grade.
- Here is the take-home midterm, handed out at 9:50 a.m. Wednesday April 2, 2014.
- Solutions to take-home midterm exam, handed in April 9, 2014.
- In-class final exam - Wednesday, May 7, 2014, 7:30 p.m. - 10 p.m., in DUAN G2B21 - note the evening hour! : 30% of final grade.
- Example of past-year final exam for Math 4330, given in May 2010.

- Assignment 1, due Wednesday, January 22, 2014.
- Selected solutions to Assignment 1.
- Assignment 2, due Monday, January 27, 2014.
- Selected solutions to Assignment 2.
- Assignment 3, due Monday, February 3, 2014.
- Selected solutions to Assignment 3.
- Assignment 4, due Monday, February 10, 2014.
- Selected solutions to Assignment 4.
- Assignment 5, due Monday, February 24, 2014.
- Assignment 6, due Monday, March 3, 2014.
- Selected solutions to Assignment 6.
- Assignment 7, due Monday, March 10, 2014.
- Selected solutions to Assignment 7
- Assignment 8, due Monday, March 17, 2014.
- Selected solutions to Assignment 8.
- Assignment 9, due Monday, March 31, 2014.
- Solution to Exercise 5.1.2.
- Assignment 10, due Monday, April 21, 201.4.
- Selected solutions to Assignment 10
- Assignment 11, due Monday, April 28, 2014.
- Selected solutions to Assignment 11.

- D. Bernoulli
- L. Carleson
- J. Fourier
- G.H. Hardy
- D. Hilbert
- J. Littlewood
- M.A. Parseval
- S. Poisson
- B. Riemann
- M. Riesz
- K. Weierstrass

- Section 1.6: Gibbs' phenomenon and the sawtooth function
- Section 1.6: Gibbs' phenomenon in 3D (a complex-valued, piecewise smooth, discontinuous function and partial sums of its Fourier series)
- Section 2.3: "Thermal imaging" of heat flow in a bar, with Neumann boundary conditions (cf. Example 2.3.1. The initial position function is f(x)=x. Red portions of the bar are warmest; blue portions are coolest. See equation (2.64) for the analytic solution)
- Section 2.3: "Thermal imaging" of heat flow in a bar, with mixed boundary conditions (cf. Example 2.3.2. Same set-up, color coding, and initial temperature function as in the previous video. See equation (2.74) for the analytic solution)
- Section 2.4: "Thermal imaging" of heat flow in a bar, with periodic boundary conditions (cf. Example 2.4.1. Same set-up, color coding, and initial temperature function as in the previous video. See equation (2.87) for the analytic solution)
- Section 2.7: A solution to the wave equation, illustrating the principle (as reflected by the differential equation (2.149)) that acceleration is proportional to concavity
- Section 2.9: D'Alembert's solution the wave equation. We assume the initial velocity g to be zero for simplicity, so that d'Alembert's formula (2.187) reads y(x,t)=(f_odd(x+ct)+f_odd(x-ct))/2. That is, we have the graph of f_odd/2 moving left with velocity c, plus the graph of f_odd/2 moving right with velocity c. To make the interaction of these two waves more apparent, we have dashed them in beyond the actual, physical boundaries x=0 and x=ell of the string itself (which appears as a solid curve)
- Section 3.2: The Cau chy sequence of Figure 3.2: this sequence converges in mean square norm to zero, but does not converge pointwise on any set of real numbers.
- Section 3.2.: The Cauchy sequence of Figure 3.3: (f_1 through f_22 in red, the function f to which the f_N's converge pointwise in black.)
- Section 5.7: Convolution with approximate identities I. A discontinuous function f (in BLACK), and f*g_epsilon (in RED), with epsilon decreasing as time evolves. Here f and g are as in the top portion of Figure 5.3
- Section 5.7: Convolution with approximate identities II. A continuous function f (in BLACK), and f*g_epsilon (in RED), with epsilon decreasing as time evolves. Here f and g are as in the bottom portion of Figure 5.3
- Section 5.7: Limits of some approximate identities. The approximate identity g_epsilon of Figure 5.4, with epsilon decreasing as time evolves.

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Last modified January 10, 2014.