15 Homework sets on CHAPTER 1: Vectors.

CHAPTER 3: Complex Calculus

• Homework 30, Fri. 4/10/09 : STUDY SECTIONS 1.1, 1.2, 1.3
  • Must know: what is a complex number, Real and Imaginary parts, conjugate, modulus=magnitude=norm, addition, subtraction, multiplication, division.
  • What is the binomial formula? Why is it true? Explain/prove it to your peers (supposedly, high school material).
  • Exercises 1.1. 1,2,3 (#3 essentially done in class).
  • Exercises 1.2. 1,2 (can also use the geometric sum formula (9))
  • Exercises 1.3. 1,2 (The mother of all series).
  • If you deposit D dollars every month in a bank account that pays you r % per year compounded monthly: what is the interest rate per month? how much money will you have after n months? (interest rate is constant).
  • Determine your loan payments P: If you borrow L₀ dollars at annual rate r compounded monthly, what should be your monthly payment P in order to pay back the loan after n months?
• Homework 31, Fri. 4/17/09 : STUDY SECTIONS 1.1, 1.2, 1.3, 1.4, 1.5, 1.6
  • 1.4. Prove the ratio test, don't just believe it because someone told you!
  • 1.5. 1--7 (#1--5 done in class). Must know general Taylor Series formula, and also why it is the way it is.
  • 1.6. 1--6. Must know Taylor Series formula for exp(z), cos(z), sin(z). Must know formula (31) and (32).
  • For Dominick only!
• Homework 32, Mon. 4/20/09 : STUDY SECTIONS 1.7, 1.8
  • Page 9 of the notes rewritten to fit lecture on 2009/04/20 (you may want to reprint it).
  • Explore complex numbers with Matlab's help
  • Yaroslav and Svyatoslav both figured out that z^½ = |z|^½ e^{i arg(z)/2} using the polar form z = |z| e^{i arg(z)} and the standard rules for exponents. Yaroslav defined 0 ≤ arg(z) < 2 π but Svyat chose -π/2 ≤ arg(z) < 3 π/2. What are i^½, (-i) ^½ according to both of them?
  • Matlab picks -π < arg(z) ≤ π. What can you say about the real part of Matlab's z^½?
  • What is z^1/3? How many possible definitions are there? What is i^1/3? What can you say about Matlab's z^1/3? Can its real part ever be negative? can it ever be pure imaginary?
  • What are all the solutions w of w³=z? where z is a known complex number.
  • What are all the possible values of (-1)ⁱ? What is Matlab's value and why?
  • What are all the possible values of iⁱ? What is Matlab's value and why?
  • 1.8. 1, 2, 3
• Homework 33, Wed. 4/22/09 : STUDY SECTIONS 1.7, 1.8, 2.1
  • What is the fundamental theorem of algebra?
  • What are all the solutions of z³=-8? z³=-8i? Show all the solutions in the complex plane.
  • Find a function f(z) such that d²f/dz² = z f with f(0)=1, df/dz(0)=0 using a Taylor series expansion f(z) = Σ c_n zⁿ about z=0. What is the radius of convergence of your series?
  • State and derive the Cauchy-Riemann equations.
  • Can the complex function F(x,y)= x² + y² + 2 i x² y of 2 real variables x, y be written as a differentiable function f(z) of the complex variable z=x + iy?
  • Can x - i y be written as a differentiable function f(z) of z=x + iy? [Done in notes and in class 4/24]
  • 2.1. 2
  • Since you know that e^ln(z) = z by definition of ln(z), and you know how to differentiate the exponential and you're a certified chain rule master, show that the derivative of ln(z) is what you know it is (don't you know?).
• Homework 34, Fri. 4/24/09 : STUDY SECTIONS 2.1, 2.2
  • What are the Cauchy-Riemann equations? Must be able to explain to a student who took Math 234. Why and when do they hold?
  • Show that both the real and imaginary part of a differentiable complex function f(z) of a single complex variable z = x + i y satisfy Laplace's equation. (Yep, done in class, but can you do it without looking at your notes?). Example: Verify that e^z = e^{x + i y} = e^x cos y + i e^x sin y ≡ u(x,y) + i v(x,y) is such that u(x,y) and v(x,y) both satisfy Laplace's equation.
  • Show that u(x,y)=e^{k y} cos( kx) satisfies Laplace's equation for any real k. Do this directly by checking, and also by considering a certain complex function. Find another function that satisfies Laplace's equation and whose contours are everywhere orthogonal to those of u(x,y).
  • Derivative of z^b for general complex b. From the definition z^b = e^{b ln(z)} and your brilliant knowledge of the derivatives of the exponential and the log and of the chain rule, deduce that the derivative of z^b with respect to z is ... (suspense!).
  • Verify the answer to Ali's question: is z^½ differentiable everywhere? From the previous problem deduce that the derivative is ½ z^-½, looks good except of course at z=0. Any other bad places? Use Matlab's definition: - π < arg(z) ≤ π , to show that d/dz (z^½) does not exist for real, negative z (i.e. on the negative real axis). Go back to limit definition of derivative [What the heck is that?!: df/dz = lim_{a → 0} [f(z+a)-f(z)]/a ] and use z = -R < 0 (real negative z), a= R+R e^{i θ} with θ → -π (from above), so , a → 0
    Moral: watch out! There is a problem with d/dz (z^½) and with d/dz ln(z) and lots of other functions not just at some points but along entire "branch cuts" where the arg(z) jumps by 2 π.
  • What is the geometric meaning of the Cauchy-Riemann equations? (Hint: study section 2.2).
  • Sketch the curves u(x,y) = u_n and v(x,y) = v_n, where u_n and v_n, n=1,2,3,... are a set of constant values, for the functions (1) f(z) = z², (2) f(z) = z^½ (Basically done in the notes below, but you may have to switch (x,y) and (u,v) around).
  • 2.1. 1, 2. If your OK with all this: try 3 and 4 also.
  • Study the (beautiful) complex maps (not as hard as it seems, just got to sit down and try to digest it).
• Homework 35, Mon. 4/27/09 : STUDY SECTIONS 3.0, 3.1
  • What do we mean by a complex integral such as ∫ f(z) dz? Must digest equations (61), (62) in the notes.
  • Parametrize the complex curve that goes straight from the complex number a to complex b. [Done in class]
  • Parametrize the complex curve that goes from (complex) a to (complex) b following a parabolic arc with dz/dt=1 at a. Sketch the arc we're talking about.
  • Parametrize the complex curve that goes from a to b following a clockwise circular arc centered at the midpoint between a and b.
  • Compute the integral of (1) z², (2) z^3/2, (3) 1/z, (4) 1/z², (5) ln(z), around the unit circle by using an explicit parametrization. (Ask John Chiang for help if necessary). [Several of those done in class]
  • When is ∫_C f(z) dz = ∫_C' f(z) dz where C and C' are two distinct curves with the same endpoints? (i.e. they both go from point a to point b).
  • What is "Cauchy's theorem"? Why is it true?
  • What does Cauchy's theorem tell you about the integral around the unit circle of (1) z², (2) z^3/2, (3) 1/z, (4) 1/z², (5) ln(z) ?
• Homework 36, Wed. 4/29/09 : STUDY SECTIONS 3.0, 3.1
  • What are all the possible values of the integral over a closed loop of (z-c)ⁿ where c is an arbitrary complex number and n is an arbitrary integer. Explain your answers in detail. [Done in class and in notes, but can you do it?]
• Homework 37, Fri. 5/1/09 : STUDY SECTIONS 3.1, 3.2 Seemed like quite a few students followed John Wieting on an early summer break today... The rest of us:
  • Emphasized how powerful Cauchy's theorem (eqn (66) in notes) and the result (67) are. They allows us to deal with lots of integrals in relatively simple ways...
  • ...like all the integrals in Exercises 3.1. 1--5, 9 + page 23, eqn (89) and the integral over the unit circle of 1/[z³(1+ z²/2)] and of 1/[z³(1+ z²/2 + z⁴/4! )]
  • We used those power tools (pun intended) to deduce Cauchy's formula (eqn(69)) and the super-duper Cauchy formula (eqn (79)). We did that by cheating a bit: assuming that f(z) could be expanded in a convergent Taylor series. But that's a good cheat to known and understand because it works!
  • We introduced some of the jargon in this area: "singularity", "pole", "simple pole", "n-th order pole", "essential singularity". (There's also the important "branch point")
  • We went back and deduced Cauchy's formula (69) without assuming that the Taylor series exist, using a limit argument as explained in the notes (page 18, eqn (70) and surrounding discussion). For those of you who want to dig deeper (don't you all?!?), the amazing result here is that an f(z) that is differentiable once in a neighborhood of z (e.g. in a disk centered at that z) is infinitely differentiable in that disk, AND its Taylor series converges!. That's why differentiable (in a neighborhood) = holomorphic (infinitely differentiable) = analytic (Taylor series converges). The proof of this is on page 19. It starts from Cauchy's formula and uses the mother-of-all-series! Beautiful deep stuff. Relatively easy proof (eqns (71), (75) and (76), just a couple of lines really).
• Homework 38, Mon. 5/4/09 : STUDY SECTION 4
  • Solved (83) for n=1 today. Discussed related integrals. Digest that method laid out in (81), (82).
  • Detailed discussion of Exam 4, #5 Direct application of eqn (67) in the notes, HWK 36 above.
  • Exercise 4.1. 1
  • Understand the (simple) `complex method' to calculate integral (84). Read on: how to compute integrals (85), (87). (same as (84) but a bit more algebra).
  • Vocabulary: the Cauchy theorem-based method we are using (isolating each `pole' and computing its contributions by using a tight little circle around it) is called ``calculating residues''
• Homework 39, Wed. 5/6/09 : STUDY SECTION 4
  • Must understand in depth the complex method to compute (84), (85), (86), (87), including proof that integral over the big circular arc goes to zero as R → ∞.
    (85), (87) Done in class (and in notes). Both of these require some digesting.
  • What was this business about `MLB'? That's not in the notes, is it? (well, it is, twice on page 22 really. I did it differently in class).
  • 4.1. 2, 3, 5
  • we discussed something like (95) before (we did the integral of z^3/2).
  • Can you do the integral on page 26?
• Homework 40, Fri. 5/8/09 :
  • Understand and appreciate all subtleties involved in calculating the real integral (90).
  • check out these solutions
  • check out this site (but remember that this is written by anyone, there are mistakes some time, also don't let the horrendous algebra on some of these problems scare you!).
  • Get ready for cumulative final exam: Monday, May 11, 2:45-4:45 in Soc. Sci. 6102! Final will cover all three parts of the material, with perhaps a slight bias toward function of a complex variable.