MAA 4402/5404 homework

MAA 4402/5404 homework Homework Assignments
MAA 4402/5404: Functions of a Complex Variable
Fall 2011

Last update made by D. Groisser Wed Dec 7 17:56:26 EST 2011

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). Due dates are not the same as hand-in dates. Every homework problem has a due-date by which you are expected to have done the problem (or tried your hardest to do it). Not every problem has a hand-in date. I will tell you in class which problems are to be handed in, and what the hand-in date is; make sure you read the rules for hand-in homework. For most problems, I will not tell you whether the problem is going to be a hand-in problem until after the due-date. Waiting until I announce which homework problems are to be handed in, then doing only those problems, is a prescription for failure.

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 8/24/11
  • Read the home, syllabus, and rules for hand-in homework webpages.
  • p. 5/ 1-11
  • p. 12/ 1
    • F 8/26/11
  • p. 8/ 1-8
  • p. 12/ 2
    • M 8/29/11
  • p. 12/ 3-6
  • pp. 14-15/ 1-7, 11-15
    • W 9/7/11

    (old HW)

    		Write up and hand in the problems below from the HW that had due-dates from 8/24/11 to 8/29/11. Before redoing or writing up these problems, read the homework rules.

  • p. 5/ 6b
  • p. 12/ 3
  • p. 14/ 2
    • W 8/31/11
  • pp. 22-24/ 1-6. I did #2 in class, but I was rushing and it would be good for you to do it again anyway.
    • F 9/2/11
  • pp. 22-24/ 9,11. After #11, specifically write out the formula that 11(a) gives you for cos(4θ) (without Σ-notation or binomial-coefficient notation).
  • pp. 29-31/ 1-4,6-9. Read Section 10 for terminology and notation I did get to in class on Wednesday (specifically, the fractional-exponent notation and the terminology “root of unity” and “principal root”). I will go over this on Friday, but you should be able to do these problems easily based on the reading and our classwork to date.
    • W 9/7/11
  • pp. 29-31/ 9
  • p. 33/ 1-3,5,6. Make sure you read Section 11 first; in Wednesday’s class, we did not get to all the terminology used in these problems. Note that when you see the word “domain” in the book, it does not mean the same thing as “domain of a function” that you learned in precalc or calculus. (For the latter, you’ll see in Section 12 that the book uses “domain of definition”.) However, purely using the book’s terminology, there is a synonym for the book’s “domain”, which is “open region”. In class, in order to avoid confusion or misunderstanding, I’ll use the term “open region” instead of “domain”.
  • If you feel you understand the book’s definition of “accumulation point” and “closure” well enough, get started on the next assignment (the one due Friday 9/9).
    • F 9/9/11
  • p. 33/ 4,7-10
  • pp. 37-38/ 1-4
    • M 9/12/11
  • pp. 44-45/ 1-8. Note: #6 tells you to verify that Figure 7 in Appendix 2 is correct. Before doing doing this, verify that Figure 8 (yes, 8) is correct. Figure 7 is not clearly presented. In this figure, for the part of the diagram in the xy plane, it’s 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. I usually indicate the absence of the origin by drawing a tiny circle there, which is probably how you’ve seen “missing points” in graphs indicated in your prior math classes.
    • W 9/14/11
  • pp. 55-56/ 1,2,5,7-9. In #2, make sure you follow the instructions to use the definition of limit, not the theorems in Section 16, to show the various limit formulas.
    • W 9/21/11

    (old HW)

    		Write up and hand in the problems below from the HW that had due-dates from 8/31/11 to 9/2/11.

  • p. 23/ 9
  • p. 30/ 3a,7. NOTE CORRECTION: In class I wrote #6; I meant #7.
    • F 9/16/11
  • Derive the formulas I gave in Wednesday’s class for the stereographic-projection map “ster” and its inverse ster-1.
    • M 9/19/11
  • pp. 55-56/ 10-13
    • W 9/21/11
  • Read Section 20 and do these problems: pp. 62-63/ 1-3
    • F 9/23/11
  • pp. 62-63/ 8,9
  • pp. 71-72/ 1
    • M 9/26/11
  • pp. 71-72/ 2abc,3,5.
  • Read Section 23 and do p. 71/ 4.
    • W 9/28/11 Read Section 24 (it’s short) and Example 1 in Section 25, and do the problems below. I’ll be going over this material Wednesday, but I want to give you a head start so that you’ll have more experience with the material before next week’s exam.

  • pp. 77-78/ 1acd,2-4, 5 (first part only), 6 (first part only)
    • General information I have updated the schedule of lectures and exams. Everything but the date of the first exam (and final exam) is subject to change, but it’s unlikely that the second exam will be any earlier than what’s on the new schedule. The pace of section-coverage in the new schedule is probably over-optimistic.
    F 9/30/11
  • p. 71/ 2d
  • p. 77/ 1b
    • M 10/3/11
  • pp. 81-82/ 1-6. (Note that problems 7-10 are now the assignment due Fri. Oct. 7, the first class after the midterm. Sorry, no homework holiday after the exam.)
    • W 10/5/11 First midterm exam (assignment is to study for it)
    F 10/7/11
  • pp. 81-82/ 7-10. In #8, the term “in agreement with” is used in a mathematically precise way. It means “not in disagreement with”.
    • 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 Monday.

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

    M 10/10/11

  • p. 97/ 1-5,7 (“roots” should be “solutions”).
    • W 10/12/11
  • Do (or re-do) all the problems on the exam on which you did not get a perfect score.
  • p. 92/ 1-7,10-13.
  • p. 97/ 8. The author’s “log” is the indicated branch of log, what I denoted in class by “Logα with a tilde over it” (I don’t know how to achieve that effect in HTML). A similar comment applies to #4 (from the previous assignment): in 4(a), the author’s “log” would have been my “Logπ/4 with a tilde over it”, and in 4(b), the author’s “log” would have been my “Log3π/4 with a tilde over it”.
    • W 10/19/11

    (old HW)

    		Write up and hand in the problems below from previously assigned HW.

  • p. 55/ 5
  • p. 72/ 5
  • p. 82/ 8
    • F 10/14/11
  • p. 97/ 6,9. Note: #6 gives you a second way to deduce what the derivative of log(z) is, but only once you know that this function is differentiable. You can’t use the method in this problem to show that log(z) is differentiable.
  • p. 100/ 1,2
  • p. 104/ 1-8
    • M 10/17/11 Read Sections 34 and 35 (pp. 104-111). If you feel ready to start doing exercises on this material, start the problems that are listed below as being due on Wednesday.
    W 10/19/11
  • p. 104/ 9
  • pp. 108-109/ 2, 3, 5-9
  • pp. 111-112/ 1-8, 14-16. Wherever you see the term “root(s) of the equation” in the book, the correct term is “solution(s)” of the equation.
    • F 10/21/11
  • pp. 114-115/ 2,3,4,6,7
    • M 10/24/11
  • 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 10/26/11
  • pp. 125-126/ 1-6
    • F 10/28/11 p. 135/ 1-5
    M 10/31/11 pp. 135-136/ 8-10. In #8, don’t confuse “Exercise 3, Sec. 38″ (p. 121) with Example 3, Sec. 38 (p. 120). It’s Exercise 3 that he’s telling you to use. In #10, although Section 42 is called “Examples with Branch Cuts”, and I’ve done no such example in class, the branch-cut technique in Section 42 is not needed for this exercise. You can deduce the results at the top of p. 135 from exercise #8 (just set n=0; the n in equation (5) on p. 135 is the m in exercise 8.
    W 11/2/11
  • p. 149/ 1-3. In #3, note that the exponent in the integral is any integer other than –1.
  • 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.

    • M 11/7/11
  • p. 149/ 4,5
    • W 11/9/11
  • Read Section 46.
  • Get started on pp. 160-163/ 1,4,5,6. If not for the upcoming midterm, I would have made Monday 11/14 the due-date for this assignment, since (as of Mon. 11/7) I haven’t yet talked about the subject of Section 46, the Cauchy-Goursat Theorem. I’ll be covering this in class on Wednesday, but in order for you to get as comfortable as possible with this topic before the midterm, it’s better that you get an early start on it. Don’t be alarmed if you find it difficult to understand what’s going on in problems 4-6.
    • M 11/14/11
  • pp. 140-142/ 1,2,4,6,8.
  • Complete pp. 160-163/ 1,4,5,6, if you haven’t already done so.
    • W 11/16/11 Second midterm exam (assignment is to study for it)
    F 11/18/11 No new homework
    M 11/21/11
  • p. 161/ 2,3
    • 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 11/23/11
  • 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.
        In #7, when you rewrite the integral in terms of θ, you won’t immediately get the integration formula in the book; you’ll get a similar-looking formula but with 2π in each of the two places the book has π. You then have to do a little work (not too much) to get the book’s formula. An interesting observation (that won’t help you to do the problem) is that the value of this integral is independent of a. By considering some large values of a, think about how this could possibly be the case.
        To help you appreciate the power of the Cauchy Integral Formula, try to compute the trig integral in #7 any other way you’ve ever learned.
    • M 11/28/11 pp. 170-172/ 1de,2,4,6,9
    W 11/30/11
  • Read the lemma on p. 175, the theorem on p. 176, and the corollary on p. 178. I won’t hold you responsible for knowing the proofs in Section 54, but I will hold you responsible for knowing the facts that are proven there.
  • pp. 178-180/ 1-8. 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?
    • F 12/2/11

    		We’ve now covered as much of Sections 55, 56, 57, and 58 as we’re going to in class, but you’re still responsible for anything in there that we didn’t touch on in class (other than the proof in Section 58), so read carefully. We have begun Section 59, and will do more of it in class on Friday, but you will want to read the entire section before attempting the problems below from pp. 195-196.

  • p. 188/ 4
  • pp. 195-196/ 1-7, 11-13. Problems 1,3,5a, and 6 are intended to be done using the Taylor series for cosh(z), the geometric series, ez, and the following “uniqueness of power-series representations” statement that we’ll cover in class on Friday (and which you should assume in order to do these problems):
      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.
    • M 12/5/11
  • Read Sections 63-66. What I did in class on Friday is a summary of this material. 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.
       Section 63 also defines some terminology I didn’t use in class on Friday, circle of convergence, that you’ll need to know for some later sections. “Circle of convergence” is a term applicable only to power series with a positive, finite radius of convergence. The circle of convergence of a power series centered at z0 is the circle of radius R centered at z0, where R is the radius of convergence (assumed positive and finite).
       In the theorems and corollary in Sections 64-66, if the radius of convergence of the power series is ∞, the phrases like “at each point inside [or interior to] the/its circle of convergence …” should be interpreted as “at each point in the complex plane.”
       Note: the theorem in Section 64 is a logical consequence of the Corollary on p. 215. However, had I wanted to prove the Corollary on p. 215, I would have needed to prove the theorem in Section 64 first (which is why Section 64 comes before Section 65).
  • pp. 219-220/ 1, 3, 5, 8.
  • Read Sections 60 and 62, so that I can cover them quickly in class on Monday.
  • Read Section 67 through Example 1. I don’t expect to have time to cover this in class, but you are responsible for it.

    Note: In Monday’s and Wednesday’s lectures, I want to cover Sections 60 and 62, the portions of Sections 63-66 that address Laurent series, and Sections 68, 69, 70, 72, and 73. If time permits, I will also cover Sections 76 and 77. I recommend that you read as much of this material as you can before Monday, and get a head-start on the problems due Wednesday.

    • W 12/7/11
  • pp. 205-206/ 1-7
  • pp. 219-220/ 2
  • p. 225/ 1-3
    • before final exam
  • p. 239/ 1
  • p. 243/ 1,2,3
  • p. 248/ 1 (interpret “any” as “every”),3,4,5
  • pp. 255/ 1b,3,4
  • 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. (“Simple” means “order 1”. For poles of higher order, and for essenital singularities, the limit above does not exist.) For example, you can apply it to the integrand in p. 248/3, and to the Example on pp. 235-236, to calculate the residues without computing Laurent series or partial-fractions decompositions.
  • Since I raced through the definition of “a zero of order m” in class:
    • Example: z5(z-7i)2(z-3) has a zero of order 5 at 0, a zero of order 2 at 7i, a zero of order 1 at 3, and no other zeroes. Similar statements apply to any polynomial, once you’ve factored it.
    • z2sin z has a zero of order 3 at 0, since its Taylor series centered at 0 is z3 -z5/3! + z7/5! … = z3(1-z2/3! +z4/5! + …).
    • In general, if an analytic function f has a zero at z0, then the order of the zero is the degree of the first nonzero term in the Taylor series of f centered at z0. Using this, you can show that the function z2sin z has a zero of order one at π, and at any integer multiple of π other than 0.
    • Read the portion of Section 75 on pp. 249-250, and Theorem 1 on p. 76.
  • Read Theorem 1 on p. 257. You are responsible for knowing the fact stated in this theorem. You are not responsible for the proof of Theorem 1, or anything else in Section 77. (In particular, you are not responsible for knowing how a function behaves near an essential singularity, one of the facts I reeled off at the end of the last class.)
    • General information In Chapter 6, the material you’re responsible for is:

  • Everything in Sections 68, 69, 70, 72.
  • Nothing in Sections 71 or 73. Note: the examples in Section 74 can be done in many ways; you’ve learned techniques for doing them that would not require you to use the theorem in Section 73. So these types of examples are still fair game. I just don’t want you memorizing more formulas for computing residues than we saw in class or homework (there are many such formulas). I’d rather you know one reliable way of computing residues (i.e. a way that you can consistently get the right answer with, and not mis-apply), than several ways that you might mix up.
  • In Section 75: Know what “zero of order m” means, and the content of Theorem 1 on p. 249. Be able to do problems like Examples 1 and 2. You’re not responsible for Theorems 2 and 3.
  • In Section 76: Know Theorem 1. Be able to do problems like Example 1, and the problems I assigned on p. 255. You’re not responsible for anything else in this section.
  • In Section 77: Know Theorem 1. You’re not responsible for anything else in this section.
    • R 12/15/11 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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