Teacher: Alexandre Eremenko
OFFICE: Math 600 OFFICE HOURS: by appointment and by e-mail. PHONE: (765) 494-1975 EMAIL: eremenko@math.purdue.edu GENERAL INFORMATION AND POLICIES
Pdf files contain additional material for reading (which is not in the book).
Homework is always due on Tuesday, before the class starts, and has to be submitted via Brightspace
Homework 1 (due January 26)
Read Chapter 1 and Appendices 1,2. Do: p.7: 1,3,5,6,7;
p. 11: 3, 4, p. 17: 1, 3, 5.
How can one keep
ice in a cellar, Complex numbers,
Linear ODE
(Second and third handounts are very important. They
contain minimal information
on complex numbers and ODE that you need to know for this course.
Please review them.
You can find more in Appendices 2, 5 of the book)
Homework 2 (due February 2)
pp. 26-28: derive 4, 7, 16, p. 37: 1a-d, 2, 3, 4, p. 42: 5,
6, 7.
Harmonic analysis, d'Alembert's formula (probl. 6 on p. 8, expanded). Fourier vs d'Alembert.
Homework 3 (due February 9) p. 47: 1,3,5; p. 56: 1,3,4,5; p. 67: 3,4,9.
Read p. 62-67, and Geometric interpretation of Fourier
series
Homework 4 (due February 16)
p. 71: 1,4,6,7, p. 79: 2,3,4,9, p. 85: 2,4,6,7.
Sturm-Liouville Problem.
Practice problems
Practice problems solved (Mistake corrected
on 2.28).
Read the handout
"Sturm-Liouville Problem".
Homework 5 (due February 23) Read pp.91-93 of the book: these are
important examples!
p. 85: 1,3,5, p. 93, 1, 2, 3,4, 5, 7, 8, 10;
Homework 6 (due Tuesday, March 9). p. 107: 1ab, 2, 5, 6, 7ab, p. 113: 2, 4, 5, 7. (Read Chap. IV carefully: it contains many examples. Derivation of formula (4.21) in the book is in the end of the handout "Linear ODE".)
Homework 7 (due March 16) P.119: 1,2,3,5,6abc, p. 126: 1, 2, 3, 4.
An interesting monument to Friedrich Wilhelm Bessel in Bremen:
Some useful integrals, Bessel functions, Plots of some Bessel functions
A book recommendation: Music: a mathematical offering, free. Chapters 1-2 cover about 1/2 of our course in a much more leisurely style then our textbook. Chapter 3 contains many solved examples of wave equation describing all kinds of musical instruments.
Homework 8 (due March 23) p. 132: 1, 2. p. 137: 1, 11, p. 143: 2, 8. p. 149: 1,3,6. p. 156: 1, 3.
Solution of Problem 3, on p. 157
Youtube movie showing zeros of Bessel functions (nodal lines of vibrations of a round plate). What you see in this movie are zeros of J_0 up to number 9. For other shapes of plates and membranes, you obtain larger variery These pictures are called "Chladni figures", or "Chladni plates".
Notice: in view of Purdue "reading days" some homework
deadlines have been changed to later dates.
More examples with Bessel functions
Homework 9 (due April 6) p. 173: 1,2,8, p. 183: 1, 4, 5, 8.
Orthogonal Polynomials,
Some applications of Legendre and Hermite polynomials
Problems in spherical coordinates
Fourier transform on the line.
Applications of Fourier transform
Kelvin's estimate of the age of the earth
Another
solution of the age of the Earth problem
Brief history of the first transatlantic cable (optional).
Fourier transform in higher dimensions and applications
Homework 11 (due April 20) p. 234: 1, 2, 3, 4, 5, 7, p. 240: 1, 4, 5, 6, 8.
Sample final exam with solutions
Homework 12 (due April 27) p. 247: 1, 2, 3, 5, 6, 10,
This was the last homework.
Perhaps you would like to see those people who discovered all this:
Pierre Simon Laplace,
Jean Baptiste Joseph Fourier,
Legendre
and Fourier (this is the only existing portrait of Legendre),
Simeon Denis Poisson,
Jacques Charles Francois Sturm,
Joseph Liouville,
Friedrich Wilhelm Bessel,
William Thomson (Lord Kelvin),
Frigyes Riesz,
Laurent Schwartz,
