Homework for Math 427, Autumn 2012

Assignments

• Homework Assignment #1; a hard copy will be given out at the first class.
  
• Homework Assignment 2, due W 10/3: Reading: For the homework problems, you should read carefully everything up through §15. Lectures in Week 2 will also cover §§16-26 and possibly §§29-30. (We will come back to §§27-28 later, because they use a theorem from chapter 6.)
  W problems for HW 2: (Don't panic, most of these are very short!) p. 5, #2; p. 8, #1ac (hint for c: polar form); p. 12, #2; pp. 14-15, #1bd,2b,5; p. 22, #1ab; pp. 29-31, #2b,8ab; p. 44, Make sure you understand the pictures in the appendix for #1 & 6.
  H problems for HW 2: p. 12, #3 - justify each step by reference to an equation in §4; p. 23, #6 and also give a counterexample to show the equation fails if Re(z) > 0 is replaced by Im(z) > 0 in the given conditions; p. 44, #7; p. 55, #5; and
  Problem E: Consider the mapping w = z².
  (a) Show that the image of a half line x ≥ 0, y = c > 0 is a half parabola.
  (b) Sketch the region x ≥ 0, 0 ≤ y ≤ 1 and its image under the mapping, labelling corresponding boundary points as in Appendix 2. Also include some of the half lines x ≥ 0, 0 < y = c < 1 and their images.
  
• Homework Assignment 3, due W 10/10: Reading: Ch. 2 through §26, then Ch. 3 §§29-33.
  p. 56: W #10c; H #11.
  pp. 62-62: W #4,8; H #9.
  pp. 71-72: W #1d,2d*,3b,4b,7; H Problem F below and #8.
  *To see why the function in 2d is interesting, see (14) on p. 106.
  Problem F: Follow the instructions for #3 on p. 71 for the function f(z) = x³ + i3x²y.
pp. 77-78: W #1b,2c,7; H #6.
pp. 81-82: W #2,4; H #1c,3.
p. 92: W #1,4,8a,10; H #8b,11.
pp. 97-98: W #1,2ab,4,8; H 2c,5.

• Homework Assignment 4, due W 10/17: Reading: Chapter 3 through §35, and Chapter 4 §§37-41. In Chapter 4 we will use line integrals and Green's Theorem. If you don't remember these, review in your vector calculus book, or check §1.6 in Fisher's Complex Variables (on reserve).
  p. 100: W #1,2,4; H #6.
  p. 104: W #1ab,2b; H #4.
  pp. 108-109: W #7 |sinz|²; H #7 |cosz|² only,14,15.
  pp. 111-112: W #8; H #15a.
  p. 121: W #1,2c,5; H #4.
• Homework Assignment 5, due W 10/24 (NOT 10/22, thanks to PW for the report of the typo!): Reading: rest of Chapter 4.
  pp. 125-126: W #1,4,5. On #5, one may use the book's suggestion, or write f' as a 2x2 matrix and use the multivariable chain rule.
  pp. 135-136: W #10a; H #7,10b.
  pp. 140-141: W #2,6; H #4.
  p. 149: W #2b; H #4.
  pp. 160-163: W #2a; H #1f,3,4,6. For problems in this section, if using the C-G Thm. or one of its consequences, explicitly list all the places an integrand fails to be analytic, to confirm that it is analytic on and inside the contour.
  pp. 170-172: H #1b,4,7.
• Homework Assignment 6, due W 10/31: Reading: Chapter 5 through § 59 (p. 195).
  H Problem G: Suppose f(z) = u + iv is analytic on a domain D, and the components of f satisfy the equation v = u² everywhere on D. Prove that f is constant.
  pp. 178-179: W #4; H #1,3,6.
  pp. 188-189: W #5,6,7,8; H #4.
  pp. 195-197: Note footnote on p. 195! W #1,2,11; H #3,6,7.
  Remark in class on 10/26: Basic series facts in §56 (including the W homework problems) need not be cited with page references when used in homework.
• Homework Assignment 7, due FRIDAY 11/16: Reading: Finish Chapter 5.
  pp. 205-207: W #4; H #2,5,6.
  H Problem H: For the function in #6, p. 207,
  (a) find a Laurent series that converges when |z - 1| > 2; and
  (b) find a Laurent series in centered at the origin that converges at 2i. What is the domain on which it converges?
  pp. 219-221: W #1,4,9; H #3,6,7.
  H Problem I: Let R be a positive real number and let D be the domain |z| < R. Suppose f(z) is analytic on D, and that f(0) = 0 and f '(0) = 1.
  (a) Show that you can define an analytic function g on D that is equal to f(z)/z for nonzero z. What is the value of g(0)?
  (b) Let M_r be the maximum of |f(z)| on the closed disk |z| ≤ r, where r < R (corrected from r ≤ R). Confirm that you can apply the MMP to g, and then use that to prove r ≤ M_r.
  (c) (Optional) What can you conclude if r = M_r for some r?
  POSTPONED to HW8: pp. 225-227: H #1,3,4.
  ANNOUNCEMENT: If your multipage homework paper is not STAPLED together, only the first sheet is sure to be graded (especially if you are a repeat offender for not stapling.)
• Homework Assignment 8, due W 11/28: Read Chapter 6 through §74 (p. 247).
  pp. 225-227, §67: H #1,3,4.
  Please note: Several of the problems in this assignment can be done in multiple ways (especially if you have already read or heard about the next section of the book). If the problem directs you to use a particular theorem or section, be sure to use the technique indicated.
  pp. 239-240, §71: W #1 (note different techniques used in different parts); H #2bd,3ab, and
  do #6 two ways: (a) using the §71 theorem, as directed in the book; (b) prove that the integral over C is equal to the integral over the circle z = R for large enough R, and prove the limit as R goes to infinity of the latter integral is zero. (Hint for (b): review the proof of the Fundamental Theorem of Algebra.)
  p. 243, §72: W #1ac,2a,3; H #2c,4.
  pp. 248-249, §74: W #7; H #3.
• Homework Assignment 9, due W 12/5.
  Reading: p. 249, §76,78-81, and 85.
  Regarding estimates of integrals along the semicircle C_R: Use the method on p. 266 for at least #6, p. 267. For the rest of the problems you may if you wish use the following Proposition.
  Proposition: If P(z) is a polynomial of order n, then there are real constants a, b, and c such that for |z| = R > c, we have aRⁿ < |P(z)| < bRⁿ. Consequently, for an integrand as in #6, p. 240, the limit as R goes to infnitiy of the integral along C_R vanishes.
  pp. 267-269, §79: H #3,5,6 (see note above),8; also
  H Problem J: Compute the integral from 0 to infinity of 1/(1+x⁵).
  pp. 275-276, §81: H #3,6.
  pp. 290-291, §85: H #1,6,7, (3 - optional, maybe we'll do it in class).

Solutions and comments on homework

• For HW 2.
• For HW 4.
• Some comments on HW 6 (included with sample midterm solutions).
• For HW 7.

Guidelines for writing up and handing in homework

• Homework assignments will have two kinds of problems, "W" problems and "H" problems. W problems are problems you should work, sometimes because they will be a warmup for a later problem, but they will not be collected or graded, so you don't need to write them up nicely to hand them in. H problems should be written up nicely and handed in on the due date.
• I realize that with all the demands on your time, it's tempting to skip the W problems. Don't do it! It's OK to do them partly in your head, or not finish a computation if you are sure you know how to do it, or to do some of the W problems later, say when studying for a quiz or test; but doing the homework is essential for doing well in the course. One student wrote on a 427 course evaluation, "Thanks for assigning the homework sets. You were right about the fact that we need to do that many problems to have a decent understanding of the subject."
• Expect to rewrite! Your first draft usually includes some distracting, unnecessary steps, and is not as well organized as you should want for work you are to be graded on. This course is about mathematical communication as well as mathematical content.
• Don't be stingy with space. Leave at least one blank line between problems, and leave margins on at least two and preferably all four edges of the page. If your writing shows through the paper, use only one side of the paper.
• Label every problem clearly (page and/or section number as well as problem number).
• When a topic has just been introduced, cite the theorems and other major results when you use them in your work. Also check that the hypotheses of any theorem you use are satisfied. Later on, you don't need to justify steps we've used several times. But if you have to look up something in the book while doing the problem, cite the page (and if relevant, the display number) in your solution.
• You may use the results of problems previously assigned (including W problems); again, if you have to look up the result, cite it.
• Write your name, the class, the assignment number, and the due date in the upper right corner of at least the first page of your paper.
• STAPLE your pages together. The "fold and tear" method of fastening pages is not acceptable.
• Homework should be turned in on the due date in class, or after class in my office hours.
• Late homework usually will not be accepted; turn in the work you have gotten done on time. You may ask me about possibly turning in all or part of your homework late if you have exceptional circumstances (e.g., serious illness).

Return to the Math 427 Homepage.
Most recently updated on December 3, 2012