Spring 2011, Math 321 TA page

Exercises on PART 1: VECTOR ALGEBRA

Exercises on PART 2: VECTOR CALCULUS

PART 3: Complex Calculus

• REVIEW YOUR EXAM 2: do the failure analysis or the improvement analysis. What basic concepts are you not understanding and why? How could you have studied better? Does the homework correspond to exams? Which homework problems were most related to the exam problems? How closely were they related? How did you study for exam 2 and how could you adjust your studying in preparation for exam 3 and the final?

• Homework 27, Mon. 4/4/2011 : Study Section 1.1 of `complex' notes. Binomial formula. Factorials and Binomial coefficients. Definition of a complex number. Algebra of complex numbers. `Review' (?) of some high school algebra.
  1. Derive the binomial formula. (Derive≡ deduce the formula from scratch so that a reasonably intelligent person would follow and believe you).
     Derive is more than `prove'. `Prove' means provide a rigorous verification of a known formula. Derive means find the formula using (correct) reasoning.
  2. Binomial theorem (formula) and Pascal's triangle. Good Wikipedia page, READ/STUDY IT! (You should understand 99% of that page. Newton's extension does come up in physics and engineering, e.g. Taylor series approximation of (1+ε)^α for small ε and non-integer α.)
  3. You known the formula for the quadratic equation, i.e. what is x if a x² + b x + c = 0? (don't you?!), but how did people obtain that formula? i.e. how did they derive that formula? Is there a formula for the cubic equation a x³ + b x² + c x + d = 0? YES, there is, and those formula are the origins of `imaginary' and `complex' numbers.
     • Roots of cubic equations (FYGMC=For Your General Math Curiosity)
     • Another good page on solving the cubic. (FYGMC)
  4. What is a complex number? What are the real and imaginary parts of a complex number? What is our notation for complex numbers, real part, imaginary part?
  5. How do we add, subtract, multiply and divide complex numbers?
  6. What is a complex conjugate? What is our notation for complex conjugate?
  7. Exercises: 1.1. 1, 2, 3
  8. Find the first 5 terms in the power series expansion of (1+ε)^α for small ε, where α is an arbitrary real number. (done in class on 4/8/11,... and in the Wikipedia links in #2 above)

• Homework 28, Wed. 4/6/2011 : Study Sections 1.1, 1.2, 1.3, 1.4, 1.5 of `complex' notes. Complex plane, modulus, argument, geometric interpretation of addition, subtraction, multiplication and division of complex numbers.
  1. What did we mean/denote by z and i in 3D vector calculus?
  2. What do we mean/denote by z and i in complex calculus?
  3. When did people start the systematic study and use of complex numbers?
  4. Who was Cardano ? What is he known for in mechanical engineering? Do you use his mechanical inventions?
  5. When was the geometric interpretation of complex numbers introduced?
  6. Who was Argand ?
  7. What is the `Argand diagram'?
  8. What is the `complex plane'?
  9. What are |z| and arg(z)? Give and show a geometric interpretation and specify in terms of x and y if z = x + i y, where `i²=-1'.
  10. What are the geometric visualizations of z₁ + z₂ and z₁ - z₂? Sketch them.
  11. Show geometrically that the algebraic definition of the complex product z= z₁ z₂ of two complex numbers z₁ and z₂, is a complex number z whose magnitude is the product of the magnitudes |z| = |z₁| |z₂| and whose argument is the sum of the arguments arg(z)=arg(z₁)+arg(z₂).
  12. What are all the z's such that z³= 8? Draw and solve explicitly (z=?).
  13. What is the geometric interpretation of the ratio of two complex numbers z= z₁ / z₂?
  14. What is the geometric interpretation of |z-a|=R where z and a are arbitrary complex numbers, and R is a positive real number? Draw and solve explicitly (z=?).
  15. What is the geometric interpretation of |z-a|< R where z and a are arbitrary complex numbers, and R is a positive real number?
  16. What are all the z's such that |z-a| = |z+a| where a is an arbitrary complex number? Draw and solve explicitly (z=?).
  17. What are all the z's such that |z-a| = |z-a*| where a is an arbitrary complex number? Draw and solve explicitly (z=?).
  18. What are all the z's such that |z-a| + |z+a| = R where a is an arbitrary complex number and R is a positive real number? Draw and solve explicitly (z=?).
  19. What is a series?
  20. What is a power series?
  21. What is a Taylor series?

• Homework 29, Fri. 4/8/2011 : Study Sections 1.1, 1.2, 1.3, 1.4, 1.5 of `complex' notes. Geometric sum, Geometric series. `Review' (?) of Math 222.
  1. What is a geometric sum? What is a compact notation for the sum? What is the formula for the value of the sum?
  2. What is the idea of the formula? How do you prove the formula?
  3. If you have a loan that you are paying back monthly, make sure the bank computed your monthly payments correctly. Otherwise, ask your parents how much they borrowed to buy a house (or car, or...), at what rate, for what term, and verify their monthly payments.
  4. What is a geometric series? When does the series converge (for arbitrary complex number)? Geometric series
  5. Experiment with the geometric series for various complex q's. Visualize the partial sums for various q's in the complex plane.
  6. Show that 0.999999.... =1
  7. What is a general series?
  8. Prove the validity of the ratio test. (What is the ratio test? what do we use it for?)
  9. What is a general power series? What is the radius of convergence of a series?
  10. What is the radius of convergence of the binomial series ? Derive it yourself, don't just read then state the result. Think of x and α as complex numbers. We're not just on the real line anymore, we've moved on to the entire complex plane.
  11. Explain what the picture on page 4 of the `complex' notes is about.
  12. Prove formula (8) for complex z and integer n do this for both n>0 and n<0 using both (1) the binomial formula, (2) the geometric sum.
  13. Exercises: 1.2. 1, 2 (same as #12 (1) above), 1.3. 1, 2. 1.5. 1, 2, 3, 4, 5, 6, 7. 1.5. 7 is easy for people who have read the notes and understood what is going on in formula (18--21). All this is Math 222 material, although now we're doing it for complex numbers.

• Homework 30, Fri. 4/11/2011 : Study Sections 1.6, 1.7, 1.8 of `complex' notes. `Elementary functions,' exponential, sine, cosine, log, general power functions, roots. `Review' (?) of Math 222 still? Now for complex, but algebra and concepts are the same.
  1. Use Taylor series to find complex functions f(z), such that (1) d²f/dz²= f,   [A+:](2) d²f/dz²= z f. What is the radius of convergence of your series?
  2. What is the Taylor series of 1/z about z=2+3i? What is its radius of convergence? (this is a repeat of 1.5.5 in hwk 29 problem)
  3. What are the Taylor series for exp(z), cos(z) and sin(z) about z=0? (Must know them) What is their radius of convergence?
  4. [A+:] Show from the Taylor series definition of exp(z) that exp(z+a)=exp(z) exp(a) for any z and a. (Did you study the notes?) It is thanks to that result that we can write exp(z) = e^z where e=exp(1)=1+1+1/2+1/6+1/24+1/120+... = 2.71828...
  5. Show that cos(z) = [exp(iz)+exp(-iz)]/2 for any z (where i²=-1). (Must know this result)
  6. Show that sin(z) = [exp(iz)-exp(-iz)]/(2i) for any z (where i²=-1). (Must know this result)
  7. What is Euler's formula? (Must know this result).
  8. Everyone must know the formula (22), (23), (24), (26), (31), (32), (33) in the `complex' notes. Intelligent people will know and understand where these formula come from.
  9. Exercises: 1.6. 1, 3, 5, 6
  10. Exercises: 1.8. 1, 2

• Homework 31, Wed. 4/13/2011 :
  • WANT SOLUTIONS TO HOMEWORK PROBLEMS? Lots of what we did in class today should help you out with many of the problems. Including problems 1.6. 5, 6 for instance.
  • Derivation in eqn (34) was done in class. Must know and understand.
  • 1.6. 3 done in class. (OK, not quite, cos(3+2i) was done in class.)
  • 1.6. 4 discussed in class (in passing).
  • Hwk 28, #12, done in class. Must know and understand.
  • Derivation in (44) leading to (46) done in class. Must know and understand.
  • Derivation in (51) done in class. Must know and understand. Then should be OK with 1.8. 1, and be fascinated by 1.8. 2 (PLOT SOLUTIONS in complex plane!)
  • 1.8. 3, effectively done in class.
  1. All by yourself now: formula (47), you should know it from Math 221, but now we're using it also for complex numbers since we know exponential and log of complex numbers.
  2. What is ln(-1)? Figure it out and try it in your calculator or on a computer with expensive Matlab or free Octave or Python with >>> from cmath import * or from numpy import *). If you are using a linux machine, Octave and Python may already be installed on it.
  3. what is iⁱ?
  4. what is √i?
  5. what is i^1/3?
  6. what is √z? z^1/3?
  7. Calculate √(- 4+i 0.01) and √(- 4-i 0.01) (yes with a calculator or computer). Are the results close to each other? Why?
  8. Why do the graphs of w= z² and w= z^1/2 look like this:
     respectively, where u=Re(w), v=Im(w) are real.
  9. Why do the graphs of w= z³ and w= z^1/3 look like this:
     respectively, where u=Re(w), v=Im(w) are real.
  10. What does the graph of w= z^1/n look like?
  11. What does the graph of w= ln z look like?
  12. Visualizing w=z² and w=z^1/2 using expensive Matlab (yep, expensive with an `e', not expansive with an `a'.)

• Homework 32, Fri. 4/15/2011 : Study Section 2 of the `complex' notes. Plotting complex functions, surface plots, contour plots. Derivative of f(z), Cauchy-Riemann equations, geometric implications of the Cauchy-Riemann equations.
  1. What do the graphs of w= z² and w= z^1/2 look like? Do you understand those graphs? [Hint1: look above. Hint2: don't just look, think.]
  2. What do the contour plots of u and v look like for w= z^1/2 and w= z²? Do you understand those plots? [Hint1: look above, study the notes. Hint2: look below!].
  3. Study these complex map examples: w= z², hence z= ± w^1/2 (hence w= ± z^1/2 by renaming variables), w= e^z (hence w= ln z), w=cosh z.
  4. Prove that the function f(z) = z^* = conj(z) is continuous but not z-differentiable! Wow. Stop, think.
  5. Prove that the function f(z) = |z| = abs(z) is continuous but not z-differentiable!
  6. Consider the complex numbers z₀, z₁ and z₂. For instance, z₀=2+3i, z₁=3+4i and z₂=4+5i. What is a quick way to compute the angle between the segments (z₀, z₁) and (z₀, z₂)? Can you write your algorithm as a complex function of z₀, z₁ and z₂? [Hint: study the notes].
  7. Exercises: 2.2.1,2,3,4.

• Homework 33, Mon. 4/18/2011 : Study Sections 2.2, 3 of the `complex' notes. Conformal mapping. Integration of complex functions.
  1. Study these complex map examples: w= z², hence z= ± w^1/2 (hence w= ± z^1/2 by renaming variables), w= e^z (hence w= ln z), w=cosh z. If you have not already done so in HWK 32. You should be able to reconsruct these pictures from scratch. Page 1 was done in class.
  2. What is a `conformal map' (or `conformal mapping')?
  3. What is the Mercator map and what is it used for? Is that map conformal? If yes, is it conformal everyhwere?
  4. When and where does a complex function provide a conformal map? Could you prove your statement or do we just have to believe everything you say?
  5. What happens to angles at points z such that df/dz=0 but d²f/dz² ≠ 0? Give an example.
  6. What happens to angles at points z such that df/dz=d²f/dz² =0 but d³f/dz³ ≠ 0? Give an example.
  7. Consider w=cosh z. What happens to angles at z= i π ? Are there other points where the same thing happens?
  8. Consider w= z^1/2. Is that map conformal? Are there points where it is not conformal? What happens at those points?
  9. Consider 2 arbitrary complex numbers a and b. Parametrize the straight segment from a and b. What does that mean?
  10. Sketch z= a (t-1)²+ 2 b t (t-1) + c t² where a, b, c are arbitrary complex numbers and t=0 → 1, is real. Show dz/dt at t=0, 1/2, 1 on your sketch.
  11. If φ = arg(b-a) and R=|b-a|/2 where a, b, are arbitrary complex numbers. Sketch z= (a+b)/2 + R exp(i (φ+t)) for t=-π → 0.
  12. Calculate ∫ z² dz from a to b along (1) a straight line from a to b and (2) a circular arc centered at the midpoint of the segment (a, b), clockwise. Compare to the result obtained using the fundamental theorem of calculus.
  13. Calculate ∫ z^-1 dz from 1 to i along a circular arc centered at 0, (1) clockwise, (2) counterclockwise. Compare to the result obtained using the fundamental theorem of calculus.

• Homework 34, Wed. 4/20/2011 : Study Sections 3, 3.1 of the `complex' notes. Integration of complex functions. Cauchy's Theorem.
  1. What is Cauchy's theorem? (Yep, you are `responsible' for that)
  2. Prove Cauchy's theorem. (Yep, that too)
  3. Calculate ∳ (z-a)ⁿ dz where n=0, ± 1, ± 2,... is an integer, a is an arbitrary complex number, over a curve C (1) that does not include a, (2) that consists of a circle of radius R centered at a, (3) any simple closed curve that includes a but is not necessarily a nice circle. Does the answer depend on n?
  4. Exercises: 3.1. 1, 2, 3, 4, 5, 6, 7, 8, 9.
     #7, 8 were essentially done in class. #9 is pretty easy, but worthwhile to do. Stop and smell the roses once in a while.

• Homework 35, Fri. 4/22/2011 : Study Sections 3, 3.1, 3.2 of the `complex' notes.
  • We considered the curve integrals of these two `simple' complex functions:
    
    Plotting software is having a tough time with them because all the action is dominated by z ≈ 0!

  1. Make a sketch of f(z)=1/z (i) along the x-axis, (ii) along the y-axis, (iii) along an arbitrary radial z=r exp(i θ₀)
  2. Make a sketch of f(z)=1/z² (i) along the x-axis, (ii) along the y-axis, (iii) along an arbitrary radial z=r exp(i θ₀)
  3. What are the two real vector fields F(x,y) and G(x,y) that underlie the complex integrals ∫ (z-a)^-1 dz and ∫ (z-a)^-2 dz where a is an arbitrary complex number?
  4. How would Cauchy have figured out ∳ cosh(z) (z-i &pi)^-1 dz and what would be his result?
  5. Exercises: 3.1. 1, 2, 3, 4, 5, 6, 7, 8, 9.
  6. Exercises: 3.2. 3, 6, 7.
  7. Calculate ∳ z^1/3 dz over a circle of radius R centered at z=0. What would Cauchy have to say? How would you parametrize the curve? Do the integral directly using (i) θ=0 → 2π, (ii) θ=-π → π and (iii) θ=2π → 4π (why not?) Look back at HWK 31, #9 and try to understand what is going on.

• Homework 36, Mon. 4/25/2011 : Study Sections 3, 3.1, 3.2 of the `complex' notes.
  1. What are all the possible values of ∳ (z-a)ⁿ dz where n is an integer and a is an arbitrary complex number? (Done in class, in the notes, in HWK 34, in your shower, in your sleep!)
  2. What are all the possible values of (1) ∳ (cos z) z^-3 dz, (2) ∳ (sin z) z^-3 dz? (Done in class)
  3. What is Cauchy's theorem? What are the Cauchy-Riemann equations? What is Cauchy's formula? What is Cauchy's generalized formula? Are these all the same since they all say ''Cauchy''? Write down each of them on a blank sheet of paper.
  4. Justify Cauchy's formula.
  5. Justify Cauchy's generalized formula. What does Cauchy's formula have to do with problem #1?
  6. What are all the possible values of (1) ∳ (cos z^-1) z³ dz and (2) ∳ (sin z^-1) z³ dz? (Done in class) How is blindly applying Cauchy's formula working out for you?
  7. Complex jargon: What is a pole? What is a pole of order n? What is an essential singularity? What is a branch point? Give examples of each. Why do we care?
  8. Are Taylor series useful? Give one or more examples.
  9. What is ∳ dz = ? Give an elementary explanation, don't start showing off by using all sorts of unnecessary fancy theorems!
  10. A challenge: What is ∫ 1/(z²-1) dz = ? over the curve z = r exp(i θ) with (1) r=2 |cos θ|, (2) r=2 cos 2θ, (3) a lemniscate of Bernoulli: z=x + i y with y= x sin(t), and x=2 cos(t)/(1+(sin(t))²), all with θ and t =0 → 2 π. (The challenge is mostly Math 222 level, parametric curves). What the heck is a Lemniscate ? you may say.

• Homework 37, Wed. 4/27/2011 : Study Section 4 of the `complex' notes. Examples 1 and 2

• Homework 38, Fri. 4/29/2011 : Study Section 4 of the `complex' notes. Examples 1, 2, 3, 4.
  1. Calculate the real integral ∫_-∞^∞ dx/(1+x⁴). Show and justify your steps.
  2. If z=R e^{i θ} with θ =0 → π prove that |1+z²| ≥ R²-1 ≥ (R-1)². For what θ does = hold? (if any).
  3. Prove that ∳ dz/ (1+zⁿ) over the curve z=R e^{i θ} with θ =0 → π goes to zero as R → ∞ for n=2, 3, 4, 5, ...
  4. Prove that ∳ dz/ (3-z³) over the circle of radius R goes to zero as R → ∞ AND as R → 0 as well. What is ∫ |dz| for that curve (for finite R)?
  5. Prove that ∫ dz/ (4+z²) over the straight line from z=R to z=i R goes to zero as R → ∞ . What is ∫ |dz| for that curve (for finite R)?
  6. Calculate integrals (86) and (87) in the `complex' notes
  7. Exercises: 4. 1,2,3. #4 if you can. #4 is a a very important result in Fourier transforms, and underneath it is the mathematical version of Heisenberg's uncertainty principle

• Homework 39, Mon. 5/2/2011 : Study Section 4 of the `complex' notes. Example 4 (done in class today).
  1. Math 221 check: Make a sketch of sin(x)/x for x real. For what x is sin(x)/x=0? for what x is sin(x)/x=± 1/x?
  2. Show that ∫_-∞^∞ sin(π x) / (π x) dx = (1/π) ∫_-∞^∞ sin(x) / (x) dx
  3. What is the integral of sin(x)/x from -∞ to +∞ ? Compute using complex integration methods. Justify/explain carefully.
  4. What is the integral of sin(x)/x from 0 to +∞ ? Think before you crunch.
  5. What is the Sinc function ? What is the integral of the Sinc function from -∞ to +∞ ? (also known as the Dirichlet integral )
  6. Explain equations (91), (92), (93) and (94) in the complex notes.
  7. Math 221 check: For what θ is 2 θ/π ≤ sin &theta? Why does 2 θ/π ≤ sin &theta imply that exp(-R sin &theta) ≤ exp(-2R &theta/π) for any R > 0?
  8. What is Jordan's lemma? Can you explain it? What is the point of Jordan's lemma?
  9. Exercises: 4. 1,2,3,4,5

• Homework 40, Wed. 5/4/2011 : Study Section 4 of the `complex' notes.
  1. Exercises: 4. 4 and 5, Fourier transform of a Gaussian and Fresnel integrals. Two classics, discussed and sketched in class today. Redo them completely on your own `dotting i's and crossing t's'.
  2. Also discussed Vector Calc exercise 1.7.5, also a classic. Related to what we've been doing with complex variables (that is transforming a real integral over x into a complex integral over z=x+i y) but not identical. In 1.7.5 we turn a 1D integral over x into a real 2D integral over x, y, but not a complex integral.
  3. Calculate the area under the function e^-x2, also e^-x2/a2.
  4. Read the last section on `branch cuts'.