MAT 129 (Spring 2008) Homework Page
Homeworks will be graded according to two criterion: completeness and correctness.
A subset of the assigned problems will be graded for correctness.
You must show your work and make sure that your homework is legible and logical.  The reader is instructed to penalize you severely if your work cannot be followed.
Homework Assignment # 1, Due Wednesday April 9

  • Download the following article and read it.
E. A. Gonzales-Velasco: "Connections in mathematical analysis: the case of Fourier series," Amer. Math. Monthly, vol.99, no.5, pp.427-441, 1992.

  • Read Chapter 1.
  • Do Exercises 1.1.1, 1.1.3
  • Use the results in the text (or in Lecture 1) to solve the following problem
ut = (1/3) uxx ; u(0,t) = u(,t) = 0; u(x,0) = sin(2x) - sin(3x).

  • Read Section 2.1
  • Derive the Fourier Series shown in Table 1. #6 and #8  (pg. 26 in the text)
  • Read Section 2.2
  • Do Exercise 2.2.1

Solution to selective problems


Homework Assignment # 2, Due Wednesday April 16
  • Do Exercises 2.2.2, 2.2.4, 2.2.5 (in Section 2.2)
  • Read Section 2.3
  • Do Exercises 2.3.1, 2.3.3, 2.3.4, 2.3.5, and 2.3.7 (Hint for 2.3.4: Equation (2.17) and Entry 6 of Table 1 are useful)

  • Read Section 2.4
  • Do Exercies 2.4.6, 2.4.9, 2.4.12

Solutions pdf
 

Homework Assignment # 3, Due Wednesday April 23
  • Read Section 2.6
  • Do Exercise 2.6.1
    • (You will find that the paper The Gibbs-Wilbraham Phenomenon: An Episode in Fourier Analysis by Hewitt & Hewitt is extremely illustrative and helpful in understanding this exercise.
    You can download it from http://www.springerlink.com/content/nlnh6n8054317063/)
(More Hint for 2.6.1: After change of variable phi = (N+1/2)*theta, the new integral is from 0 to pi, the numerator in the integrand is sin(phi), and the denominator in the integrand is (N+1/2)*sin[phi/(2N+1)]. Write the denominator in the form (sin x)/x for some x dependent on phi. We also know that the ratio (sin x)/x ---> 1 as x--->0.)
  • Read Section 3.1
  • Do Exercises 3.1.1, 3.1.7, 3.1.9
    (For the first part of problem 3.1.9, you may assume without loss of generality that <a,b>  is a real number, and show that a - tb = 0 for a correctly chosen real number t)

  • Read Section 3.2
  • Do Exercises 3.2.1, 3.2.4


Solutions (pdf)


Extra Credit (25pts): Find a basis for the vector space PC(a,b). You can have until Wed. 04/30 to submit this.
(My notes on definition of a vector space are here).

Homework Assignment # 4, Due Wednesday April 30
  • Do exercises 3.3.1, 3.3.2, 3.3.4, 3.3.5, 3.3.6, 3.3.8 (only with respect to the basis in exercise 6), 3.3.9, 3.3.10b,d (Hint for 3.3.10b,d: entries 11 and 17 Table 1)
  • Do the following problem: (A1) Let T be a self-adjoint linear operator from the k-dimensional complex vector space Ck to Ck. Let {u1, u2, ..., uk} be eigenvectors of T forming an orthonormal basis of Ck, and {lambda1, lambda2, ..., lambdak} be the corresponding eigenvalues. Let b be a fixed vector in Ck. The solution of Tx = b can be easily found by exploiting the property of the basis of eigenvectors. Derive the formulas for the solution x. (Hint: continue where I left off in lecture today).

Solutions (pdf)

Homework Assignment # 5, Due Friday May 9

  • Do the following exercise : (A1) Suppose {phin(x)}, n=1,2,..., is an orthonormal basis for L2(a,b), where [a,b] is an interval of finite length. Show that {phin(x)/sqrt(w(x))} is an orthonormal basis for L2w(a,b), where w(x) > 0 is continuous on [a,b].
  • Read Section 3.5
  • Do the following exercises : 3.5.1, 3.5.3, 3.5.8 (for 3.5.8 skip part b, the statement is obvious but cumbersome to write down)

  • Notes from Lectures 14,15: PDF  (Notice that there is an error in page 1.
Under Step 1, should read :
T '/ T = [r(x) X']' / [w(x)X] + p(x)/w(x) = - lambda
==>     T ' + lambda T = 0,
             L(X) + lambda w(x) X = 0, B1(X) = 0, B2(X) = 0, where L(X) = [r(x) X']' + p(x)X .
Under Step 2, should read :
..... associated regular S-L problem L(X) + lambda w(x) X = 0, B1(X) = 0, B2(X) = 0. )
  • Read Section 4.2
  • Do exercise 4.2.5, 4.2.8
  • Do the following exercise: (A2) Consider a vibrating string, such as a guitar string, of length l pinned at both ends x=0, x=l. Suppose the string is plucked in the middle at x= ½l so that the initial displacement u(x,0) is 2mx/l for 0<= x <= ½l  and  2m(l-x)/l  for ½l <= x < l, where m >0 a fixed constant (so the maximum displacement, at x= ½l, is m), and the initial velocity is ut(x,0) = 0. The displacement of the string satisfies the wave equation utt = c2 uxx , where c2 > 0. Find the displacement u(x,t) as a Fourier series. (Hint: The eigenbasis to consider for this case is discussed in lecture today. To find the expansion of u(x,0) in this basis, just do direct integration.)
Solutions (PDF)

Homework Assignment # 6, Due Wednesday May 21

  • Read pg 204-205 and Section 7.1. Additional Notes (pdf)
  • Do exercise 7.1.1, 7.1.3, 7.1.4, 7.1.5
  • Show that if f is piecewise continuous and g is bounded and vanishes outside a finite interval [a,b] then the convolution of f and g, (f*g)(x), exists for all x.

Solutions (pdf)

Homework Assignment # 7, Due Friday May 23

  • Do exercise 7.1.8
  • Read Section 7.2
  • Do exercises 7.2.1, 7.2.2, 7.2.3, 7.2.4, 7.2.5, 7.2.8, 7.2.13.

Solutions (pdf)

Homework Assignment # 8, Due Friday May 30

  • Do exercises 7.2.7, 7.2.12
  • Read the subsection Partial Differential Equations (pg 226-228)
  • Do exercises 7.3.1., 7.3.2
  • Read the subsection on Sampling Theorem (bottom half of pg 230 and pg 231)
  • Do exercises 7.3.6, 7.3.8
  • Do exercise (A1) (download in pdf format)
  • Read the subsection on Heisenberg's Inequality (pg 232-233)
  • Do exercise 7.3.9

Solutions (pdf)

Homework Assignment # 9, Due Monday June 9 in my office (MSB2113)

  • Read pages 241-244
  • Do exercises 7.5.1, 7.5.6(a,b)
  • Read section 7.6, pages 249-252
  • Read the notes from Gonzalez & Woods book
  • Do the following exercises (AE) (download in pdf format)

Hints and errata for the problems (pdf)

Solutions (pdf)


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