MAA 4402/5404 homework

MAA 4402/5404 homework Homework Assignments
MAA 4402/5404: Functions of a Complex Variable
Fall 2009

Last update made by D. Groisser Mon Dec 14 13:55:21 EST 2009

Homework problems and due dates (not the dates the problems are assigned) are listed below. This list, especially the due dates, will be updated frequently, usually in the late afternoon or evening the day of class or the next morning. Due dates, and assignments more than one lecture ahead, are estimates; in particular, due dates may be moved either forward or back, and problems not currently on the list from a given section may be added later (but prior to their due dates, of course).

Exam dates and some miscellaneous items may also appear below.

If one day’s assignment seems lighter than average, it’s a good idea to read ahead and start doing the next assignment, which may be longer than average.

Unless otherwise indicated, problems are from our textbook (Brown and Churchill, Complex Variables and Applications,, 8th ed.). Read the corresponding section of the book before working the problems. Don’t read only the examples, and don’t try the homework problems first and refer to the text only if you get stuck.

Date due page # / problem #s
W 8/26/09
  • Read the homepage and syllabus webpages.
  • p. 5/ 1-4,9. I recommend that you read ahead and do the remaining problems on p. 5, which are due Friday.
  • p. 12/1
    • F 8/28/09
  • p. 5/ 5-8
  • p. 8/ 1-3,5-8
  • p. 12/ 2,4
  • p. 14/ 1ab,2a
    • M 8/31/09
  • p. 12/ 3,5,6
  • p. 14/ 1cd,2b,4,5-7,11-15
  • pp. 22-24/ 1
    • W 9/2/09
  • pp. 22-24/ 3-4,5cd,6,9,10b
    • F 9/4/09
  • pp. 29-31/ 1-4,6-9
    • W 9/9/09
  • p. 33/ 1-10. Note: the definition of “open set” that I gave in class is the conclusion of problem 6. For the sake of doing problem 6, use the book’s definition: a set is open if and only if it contains none of its boundary points. Then, what you’re showing in #6 is that the two definitions are equivalent.
    • F 9/11/09
  • pp. 37-38/ 1-4
  • pp. 44-45/ 1-3, 8
    		•
    M 9/14/09 pp. 44-45/ 4-7. #6 tells you to verify that Figure 7 in Appendix 2 is correct. Before doing doing this, verify that Figure 8 is correct. Figure 7 is not clearly presented. In this figure, for the part of the diagram in the xy plane, it is intended that the shaded region extend infinitely far to the left. For the part of the diagram in the uv plane, the little hash-mark at the origin is intended to mean that the origin is not part of the shaded region. In class, I’ve been indicating the absence of the origin by drawing a tiny circle there, as I would do in Calculus 1 for a “missing point” on a graph.
    W 9/16/09
    hand-in date
    (old HW)     		Before redoing or writing up these problems (which had due-dates from 8/31 to 9/11), read the homework rules.

    1. p. 12/ 5b
    2. (a) First half of p. 23/ 9 (just establish the identity with the z‘s; I’m not asking you to hand in the part about the trig identity). (b) p. 30/ 7.
    3. p. 37/ 1b,d
    4. MAA 5404 student only: p. 33/8.

    The numbers above are from the 8th edition of the textbook. If you have the 7th edition, the corresponding numbers are:

    1. p. 11/ 4b
    2. (a) First half of p. 22/10. (b) p. 29/ 7.
    3. p. 36/ 1b,d
    4. p. 32/ 8
    W 9/16/09
    (new HW)    		 pp. 55-56/ 1,2,5,7-9
    F 9/18/09
  • pp. 55-56/ 10-13
  • Derive the formulas I gave at the end of Wednesday’s class for the stereographic-projection map sterN and its inverse (sterN)-1.
    • M 9/21/09
  • p. 62/ 3
  • MAA5404 student only: p. 63/ 4
    • W 9/23/09
  • pp. 62-63/ 1,2,8,9
  • p. 71/ 1
    		•
    F 9/25/09 pp. 71-72/ 2cd,3,5
    M 9/28/09 p. 71/ 4
    W 9/30/09
  • pp. 77-78/ 1-4, 5 (first part only), 6 (first part only).
  • Study for Wednesday’s exam. This includes doing the problems above.
    • F 10/2/09
  • pp. 77-78/ 5 (second part), 6 (second part)
  • MAA 5404 student only: pp. 72-73/ 10.
    • M 10/5/09
  • pp. 81-82/ 1-10. In #8, the term “in agreement with” is used in a mathematically precise way. It means “not in disagreement with”.
  • MAA5404 student only: p. 82/ 11. Figure out how to use your sketches from #9 to get the sketches for this problem, without doing any significant amount of new work.
    • W 10/7/09 p. 97/ 1-3,7 (“roots” should be “solutions”)
    General information The grade-scale page for your class is now functional. (So far it reflects only the first exam, of course.) Your exams have been graded and will be returned in class on Wednesday.

    Please remember that I will not communicate grades by email, or discuss them by email.

    F 10/9/09
  • p. 97/ 4,6,8,9. In #4 and #8, the author’s “log” is the indicated branch of log, what I denoted in class by “log with a tilde over it” (I don’t know how to achieve that effect in HTML).
  • Do (or re-do) all the problems on the exam on which you did not get a perfect score.
    • M 10/12/09
    hand-in date
    (old HW)     		Hand in the following problems (which had due-dates from 9/4 to 9/28):

    1. p. 30/ 8a
    2. p. 71/ 4b
    3. p. 72/ 5

    The numbers above are from the 8th edition of the textbook. If you have the 7th edition, the corresponding numbers are:

    1. p. 29/ 8a
    2. p. 68/ 4b
    3. p. 69/ 5

    Students with the 7th edition have until Wednesday to hand these in, since the 7th-edition numbers were not posted here till Sunday night.

    M 10/12/09 No new homework
    W 10/14/09
  • p. 100/ 1,2
  • p. 104/ 1-8
    • M 10/19/09
  • p. 104/ 9
  • pp. 108-109/ 2-8
  • p. 111/1-5
    • W 10/21/09
  • p. 109/ 16
  • p. 112/ 14-16. You do not have to do #15 using the identities referred to in the book; you may use the formulas we derived in class.
  • pp. 114-115/ 2,3
    • F 10/23/09 p. 92/ 1-7,10-13. This is material from a few weeks ago, but I originally did not notice there were exercises on this page.
    M 10/26/09
    hand-in date
    (old HW)     		Hand in the following problems (which had due-dates from 10/5 to 10/19):

    1. p. 81/ 2,9
    2. p. 97/ 3b
    3. p. 104/ 9

    The numbers above are from the 8th edition of the textbook. If you have the 7th edition, the corresponding numbers are:

    1. pp. 78-79/ 2,9
    2. p. 94/ 3b
    3. p. 100/ 9
    M 10/26/09
    (new HW)    		 p. 121/ 1-5. In #4, first do the integral the way the book suggests. Then do it by evaluating the integrals the way you learned in Calculus 2 (via two successive integrations by parts for each real integral). See how much easier the first method is, now that you know how to work with complex exponentials.
    W 10/28/09 pp. 125-126/ 1-6

    We need to talk about the date of the next midterm. Please remind me on Wednesday. The date will not be the originally-estimated Nov. 4.

    F 10/30/09 p. 135/ 1-5.
    M 11/2/09 pp. 135-136/ 6-8, 10
    W 11/4/09
  • pp. 140-142/ 1,2,4,6,8
  • p. 149/ 1-3. In #3, note that the exponent in the integral is any integer other than –1.
    • F 11/6/09 pp. 149/ 4,5
    M 11/9/09
  • p. 163/ 7
  • Let a be an arbitrary real number not equal to –1, and let f(z) be the principal value of za. Let r0 > 0 and let C be the circle of radius r0 centered at the origin, oriented counterclockwise. Compute ∫C f(z) dz. As always, simplify your answer.

    There is an obvious consistency check on your answer: what must it reduce to when a is an integer (other than –1)?

    Here are two special cases to check your answer against: (i) If r0=1 and a=1/2, your answer should reduce to –4i/3. (ii) If r0=1 and a=-1/2, your answer should reduce to 4i. If your formula for general r0 and a does not yield these answers, it’s wrong.

    Take your formula for general r0 and a and compute its limit as a→–1. What value would you expect for this limit? (Answer: 2πi. Why?) Is that the value you are getting? If not, there’s a mistake somewhere in your work.

    • F 11/13/09 pp. 160-163/ 1,4,5,6
    M 11/16/09 Second midterm exam (assignment is to study for it)
    W 11/18/09
  • p. 161/ 2,3
  • pp. 170-171/ 1abc,3,7. In #3, s is a complex variable of integration, not an arclength variable. The “z0″ in the Cauchy Integral Formula on p. 164 is the z in Exercise 3; the z in the Cauchy Integral Formula on p. 164 is s in Exercise 3. Just as for real definite integrals, the variable of integration in a contour integral is a “dummy variable” for which any letter that doesn’t have some other meaning may be used. So, for example, in a situation in which the letters z, w, and ζ are not being used to represent variables that have life outside the integral, ∫C f(z) dz = ∫C f(w) dw =∫C f(ζ) dζ.
    • F 11/20/09 no new homework
    M 11/23/09 pp. 170-172/ 1de,2,4,6,9
    General information The grade-scale page for your class has been updated to include the second exam.

    Please remember that I will not communicate grades by email, or discuss them by email.

    W 11/25/09 pp. 178-180/ 1,9 (note that #9 continues on p. 180). After doing #1, show, by the same reasoning, that if -u has an upper bound, then u is constant. Deduce that if the real part u of an entire function f is bounded above or below, then u is constant, and deduce from this that f is constant. Does the same argument work for the imaginary part of f?
    M 11/30/09
  • p. 179/ 3-8
  • pp. 188-189/ 1-3, 5-9
    		•
    W 12/2/09
  • We’ve now covered as much of Sections 55, 56, 57, and 58 (8th edition numbers) as we’re going to in class, but you’re still responsible for anything in there that we didn’t touch on in class, so read carefully. (If you have the 7th edition, these are the first four sections of the chapter entitled “Series”, probably chapter 5.) We have begun Section 59, and will do more of it in class on Wednesday, but you will want to read the entire section before attempting the problems below from pp. 195-196.

    Today we also covered parts of Sections 63, 64, 65, and 66. You’re won’t be able to understand everything in these sections yet, because the author assumes you’ve covered Sections 60-62 first. However, based on what we did in class (and, in one case, the author’s reference to an example in Section 65), you should be able to do the problems below from pp. 219-220. You may postpone reading Sections 63-66 till I tell you it’s time to read them.

  • p. 188/ 4
  • pp. 195-196/ 1-7, 11-13. Most of these rely on the uniqueness theorem that we covered at the end of class: if you have a power series based at z0 that you can show, by any valid means, converges to f(z) on some disk centered at z0, then this series is automatically the Taylor series of f based at z0, whether or not you found the coefficients by computing all the derivatives of f at z0.
  • pp. 219-220/ 1, 3, 4, 8. In #4, note that you can’t do this problem by taking the corresponding limit for a real variable x, blindly substituting z for x, and thinking that’s all there is to it. The fact that a limit exists when you approach a point a along the real axis doesn’t mean that it exists for all paths in the complex plane approaching a.
    • F 12/4/09
    hand-in date
    (old HW)     		Hand in the following problems (which had due-dates from 11/9 to 11/25):

    1. pp. 161-163/ 2a,7
    2. p. 171/ 3,4
    3. p. 178/ 1

    The numbers above are from the 8th edition of the textbook. If you have the 7th edition, the corresponding numbers are:

    1. pp. 153-156 / 2a,7
    2. p. 163/ 3,4
    3. p. 170/ 2
    F 12/4/09
    (new HW)
  • pp. 205-206/ 1-3,7
  • pp. 219-220/ 2
    • M 12/7/09
  • Read Sections 63-66 (8th edition numbers). In Section 63 you will be held responsible for the content (i.e. statement and understanding) of Theorem 1, but not the proof, and you will not responsible for the material on uniform convergence. In Sections 64-66 you will be held responsible for the content of the theorems and corollary, but not the proofs.
  • Read Section 67. You are responsible for everything in this section. I touched on this material briefly at the end of class Friday, but did not have time to show you as many examples as you should see before starting the exercises for this section.
  • pp. 205-206/ 4,5
  • p. 225/ 1-3
    • W 12/9/09 In Monday’s class, the sections of the 8th edition that we covered were 68: Isolated Singular Points; 69: Residues; 70: Cauchy’s Residue Theorem; and 72: The Three Types of Isolated Singular Points. (We skipped Section 71.) The 7th edition has the same material, but organized differently; the sections are 62 (= new 68+69), 63, and 65. In both editions they are part of Chapter 6, Residues and Poles.

    I did not have time to state the fact in the next-to-last paragraph of Section 72 of the 8th edition (the paragraph after Example 3 in Section 65 of the 7th edition). Read this on your own; it’s referred to in one of the exercises below.

    Two things I did not have time to define are the principal part of a function at an isolated singular point, and the terminology simple pole. Both of these are defined near the beginning of Section 72 (65 in the 7th edition), and are also referred to in the exercises below.

    • p. 239/ 1,2 (in 7th edition these are p. 230/ 1,2)
    • p. 243/ 1,2,3 (in 7th edition these are p. 233/ 1,2,3)
    • Show that if f has a simple pole at z0, then Resz0(f) = limz → z0 (z-z0)f(z).
      This fact often gives a quick and easy way to evaluate residues at simple (!) poles. For example, you can apply it to the integrand in the example in Section 70 (63 in the 7th edition), which I did in class, to calculate the residues without computing the partial-fractions decomposition or the Laurent series of this function.

    Once we cover Section 77 (70 in 7th edition) on Wednesday, it will follow that the converse of the stated in the previous problem is also true: f has a simple pole at z0 only if limz → z0 (z-z0)f(z) exists and is ≠ 0. Assuming this fact, do the following problem which has now been corrected from the original:

    • Let f(z)=1/sin(πz)
      1. Show that the set of singular points of f is exactly the set of all integers.
      2. Show that the singularity of f at each integer n is a simple pole with residue (–1)n/π.
      3. Let C be a positively-oriented circle, centered on the x-axis, such that neither of the points at which C intersects the x-axis is an integer. Show that ∫C f(z) dz must be either 2i, 0, or –2i. Determine, completely, under which conditions on C the integral has each of these values.

    g

    Before final exam The 8th-edition sections we covered in class on Wednesday were 73 (Residues at Poles), 74 (Examples, although I used different examples), 75 (Zeros of Analytic Functions), 76 (Zeros and Poles; we went over only Theorem 1), and 77 (you’re only responsible for Theorems 1 and 2; note that I packaged the content of Theorem 2 slightly differently from the way it’s given in the book.)

    For another shortcut to finding residues at simple poles, read Theorem 2 in the section “Zeroes and Poles” (76 in 8th edition) and the examples following the theorem. This shortcut is useful in some of the exercises below.

  • p. 248/ 1,3,4,5. (In the 7th edition, these are on p. 238.) The book’s answer to 4a is correct, contrary to what I originally said on this page.
  • p. 255/ 1,2a,4a. (In the 7th edition, these are on p. 245.)
    • W 12/16/09 FINAL EXAM begins at 7:30 a.m. in our usual classroom.
    After the exam, please do not email me with questions about your grade for the class, your performance on the exam, etc. I will not email any information relating to the final exam or grades. Course grades should be available from ISIS shortly after I submit them, which will be a few days after your final. I will post some exam statistics, and perhaps some other statistics, on your grade scale page.

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