Homework for Math 428, Winter 2013

Assignments

• Homework Assignment 1 (revised), due W 1/16; hard copy available at the first class.

• HW 2 due F 1/25: Reading: §§ 86-89.
  §84, pp. 286-288: [W] correction #8; [H] #5, but instead of doing it "formally" with "integration along a branch cut," split into two contours with a clearly described branch choice analytic on each one, as in correction #8 (discussed in class).
  Problem E. [H] Let f be a non-constant analytic function with an isolated singularity at z₀. Suppose there's a complex number C such that for every ε > 0 there is a z satisfying both |z - z₀| < ε and f(z) = C. Prove that f is singular at z₀. (Note that f is not defined at z₀, so there is a point not equal to z₀ where f(z) = C.)
  §87, pp. 296-297: [W] #1,2,7; [H] #8,9.
  Problem F. [W] Assume the Argument Principle Conditions on a contour C and a function f. Prove that the number of zeros and number of poles of f inside C are both finite. (This result is a reformulation of problems 10 and 11, §76, p. 257, and problem 4, §87, p. 297.)
  Problem G. [H] Suppose an analytic function f has a zero of (positive) order m at z₀.
  (a) Prove that there is a neighborhood N of z₀ and a positive real number ε so that for any complex number w₀ with |w₀| < ε, the equation f(z) = w₀ has exactly m solutions in N, counting multiplicity.
  (b) Show that correction if w₀ is not zero, the m solutions will, in fact, be distinct; that is, each is order one. (It may be necessary to shrink the neighborhood N and therefore also the bound ε to ensure this.) Hint: What can you say about the derivative of a function at a simple zero?
  §89, pp. 306-309: [H] #3.

• HW 3 due F 2/8. Reading: §§90-93, 101-102. Also review §17.
  Optional: Read §94. Problems in our class probably will be as easy or easier to do with the "direct" method of finding LFT's used in §93 and in class. If you do use the method in §94, be sure you understand the special cases in Examples 1 & 2 and #8, p. 324.
  §18, p. 56 [W] #11 (review, you did this in 427).
  §89, pp. 306-309: [H] #8. You may do this "formally" (as the book instructs), that is, ignoring the questions of convergence of the integral along C_R to zero, and of convergence of the sum in the answer.
  §90, p. 313 [W] #6: contrast the answer in 6 to the description of w = Az + B in class.
  §92, pp. 317-19 [H] #9 and include in your sketches a few lines y = c for c in (0,1) and their images.
  If you missed class on 1/29 and/or 2/1, also work [W] #3 & 11, and draw sketches of curves and regions discussed, and their images.
  §94, pp. 324-325 [W] If you want practice problems with the answer given, try #1 & 3. [H] #2*,6.
  *In #2 and H (below), sketch on the (u,v)-plane the (finite) images of the points 1, i, 0, ∞, the real and imaginary axes, and the unit circle. Also indicate the images of quadrants I, II, III, & IV.
  Problem H*. [H] Find the LFT that fixes 1 and i and maps 0 to 1+i (and do sketch as described* above).
  Problem I. [W] Appendix 2, pp. 452-460, has pictures of regions and their images under various transformation. Look at the pictures (but not the formulas) in Figures 2, 3, 11, 9-11, and 16-18. How can you tell just from the pictures (not the formulas) that these are not LFT's?
  Problem J. [H] Notation: Let (x,y)^T indicate a 2-dimensional column vector, and let L be a linear transformation of R² to itself that is conformal. Prove that the magnitudes of L(1,0)^T and L(0,1)^T are equal by considering their dot products.
  §103, pp. 362-363: [W] #2,4; [H] #1,5 - in #5, "Note" means "study the pictures and notice where the mapping shown is and is not conformal;" nothing to write down.

• HW 4 due F 2/15. Reading: §§95-97 & 103-108. Also review §§13-14 & 26. Optional: §§99-100 on Riemann Surfaces.
  Problem K. [H] The purpose of this problem is to find all the LFTs that transform the first quadrant to the upper half of the unit disk; that is, that transform the domain D: x > 0, y > 0, to the domain D': v > 0, u² + v² < 1.
  (a) Explain why such an LFT must map the origin to either 1 or -1.
  (b) Find the general formula for an LFT T that maps the origin to 1 and D to D'. Sketch the (u,v)-plane, showing the images of 1, infinity, and the four quadrants.
  Hints: What must the images of the coordinate axes be? Keeping in mind that LFTs preserve orientation, what must the image of the second quadrant be?
  (c) Find the general formula for an LFT T that maps the origin to -1 and D to D'. Sketch the (u,v)-plane, showing the images of 1, infinity, and the four quadrants.
  §26, pp. 82-83: [W] #8; [H] #7 - hint, remember that perpendicular lines have negative reciprocal slopes.
  §96, pp. 334-336: [W] #4; [H] #2 (here "Note" = "Explain") and #5 (here you don't need to write anything in response to "Note", just think about it).
  Problem L. [H] Let f(z) = cosh(z), and let D be the region x > 0, 0 < y < π/2. Find the image f(D) as follows. Because cosh(z) = cos(iz) = sin(π/2 - iz), f is the composition of the three transformations: Z =g(z) = iz, W = h(Z) = π/2 - Z, w = sin(W). Use this to find the image f(D) by sketching (a) D in the z-plane, then (b) g(D) in the Z-plane, then (c) h(g(D)) in the W-plane, and finally (d) f(D) in the w-plane.
  Indicate in each sketch the image of the points z = 0 and z = iπ/2. (Hint: In (d) you should find #2 from §96 helpful.)
  §97, pp. 340-341: [W] #1 (&/or review Math 427 problem E); [H] #4.
  §103, pp. 362-363: #6-7 are part of the proof in §103, read if interested.
  §106, pp. 370-372: [W] (or at least read) #6,7; [H] #5.
  §110, pp. 382-385: [W] #2; [H] #4.

• HW 5 due F 2/22. Reading: Chapter 10 through §112.
  Problem M. [H] Find the steady state temperature in a semicircular plate x² + y² < 4, y > 0, if the temperature is zero on the straight edge and 10 on the circular edge. (Insulated faces as usual so temperature is only a function of x and y.)
  §110, pp. 382-385: [W] #13; [H] #6,8,12*.
  *For 12, analyze isotherms as well as heat flow lines, and draw a sketch showing both. §96 may be more helpful than the figure in the appendix.
  §112, pp. 387-390: [H] #5. The "bounded electrostatic potential" is harmonic, so mathematically this problem is the same as the steady state temperature problems.

• Extra Credit problem: Hand in no later than Monday, 2/25, to me directly (not with homework). Up to 4 points added to your course total.
  Rules: Work alone, not with other students. Use no resources except our text and me - it's OK to come ask me questions about this problem.
  The problem: Suppose 0 < a < b, C_a is the circle of radius a centered at a (on the real axis), and C_b is the circle of radius b centered at b. Thus C_a is inside C_b and the two circles are tangent at the origin. Let C₀ be any circle inside C_b, outside C_a and tangent to both of them. Prove using conformal mapping that starting with C₀, in each direction "around" C_a there is an infinite sequence of circles, each tangent to the previous one and to both C_a and C_b.
  (Connection to current work in the course: As in Chapter 10 problems, the idea is to find a conformal map that transforms this problem into one that has an "obvious" solution.)

• HW 6 due F 3/8. Reading: §§113-115.
  §110, p. 384, [H] #11.
  §112, pp. 387-390: [H] #3, #7.
  §115, pp. 398-401: [W] #8; [H] #2, #6 - omit sketch, it's in the book, and #7 - also explain where pressure is greatest,
  and #18 - "Note" = "show" = "explain" (so the first, third, and fourth sentences say what you have to do, while the second and fifth are just observations).

• HW 7 due F 3/8 by 3:30. Note earlier deadline (3:30, not 4). Reading: §§116-119; §§120-121 optional.
  Problem N. [H] This problem is both a review for the final exam and an independent proof of part of one step in the construction of the Schwarz-Christoffel Transformation.
  Let f '(z) be the function in eqn. (1), §117, p. 405, with branch choices as in (2), same page, and A = 1. Also assume |k_j| < 1, and (changed condition) 1 < k₁ + k₂ + ... + k_n-1 < 3. Using the technique in §83, prove that the integral of f '(z) over the entire real line vanishes.
  (b) Why is it necessary to make the indentations in the contour to avoid the points x_j? What step in the justification of our computations fails if the contour includes these points?
  §115, p. 399: [W] #9.
  §119, pp. 414-415: [H] #1 & #5.
  Problem O. [H] (a) Show that an LFT maps the real axis to itself if and only if the constants a, b, c, and d, may be chosen so that they are all real.
  (b) If an LFT with a, b, c, and d all real maps the upper half-plane y > 0 to itself, what additional restriction is needed on a, b, c, and d?
  (c) Let C be either a vertical ray x = constant, y > 0, or the upper half of a circle centered on the x-axis. Show that there is an LFT that maps the upper half-plane y > 0 to itself and maps C to the upper half of the imaginary axis.
  
  Here are some more problems that are both homework and final exam review.
  Problem P. [H] Show that the equation z + 3 = 2e^z has only one solution in the left half-plane. Hint: Consider a closed contour consisting of an interval on the imaginary axis and the left half of a circle of arbitrarily large radius centered at the origin.
  Problem Q. [W] How many roots of z⁶ - 5z⁵ + 10 = 0 are in the annulus 1 < |z| < 2?
  Problem R. [H] Suppose f(z) is an entire function and has real values for real values of z and pure imaginary values for imaginary values of z. Show that f is an odd function; that is, f(-z) = -f(z).
  NOTE this assignment due in class, or to my office by 3:30.

• Extra Credit problem: Hand in no later the start of the final exam on Monday, 2/18, to me directly (not with homework). You must do at least one of parts (b) & (c); you may do both, and may also add part (a). Maximum extra credit: 7 points.
  Rules: Work alone, not with other students. Use no resources except our text and me. (This includes not consulting any websites.) It's OK to come ask me questions about this problem.
  (a) Prove that a non-constant analytic function f is an open map. This means that if w₀ = f(z₀), then f is surjective onto some neighborhood of w₀.
  (b) Let C be a simple closed contour. Assume that f is analytic on and inside C, and is one-to-one on C. (Thus f(C) is also a simple closed contour.) Prove f maps the domain inside C to the domain inside f(C), and is one-to-one and onto between these domains. Hint: Use part (a) (which you may assume even if you didn't do it) and the Argument Principle.
  (c) Assume the result of part (b). Suppose f is analytic on the closed upper half-plane y ≥ 0, and one-to-one on the real axis. Further suppose that f has a finite limit as z goes to infinity. Prove that f is one-to-one on the upper half-plane. Extend your result to the case in which f goes to infinity as z goes to infinity. Give a counterexample in the case f has no limit as z goes to infinity.

Solutions and comments on homework

• HW 1: Includes more than one version of proofs. If you are having trouble with writing proofs, reading the various versions and thinking about how you might have gotten to these ideas may be helpful.
• HW 2: Revised, solution for Problem G added.
• HW 3 typed and handwritten comments and solutions.
• Problems K and M.

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).

Class schedule: planned lecture topics, homework due dates, test dates.
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Most recently updated on March 12, 2013