MAT 4010/5160 — §101    Complex Variables,   Spring '15

"They that can give up essential liberty to obtain a little temporary safety
deserve neither liberty nor safety"   — Benjamin Franklin. (1706-1790)

Homework List

✈ Jump down to '♞ This Week.'

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Week 1

• Monday, Jan 12 — First day of class
  
  1. Carefully read the Class Information slides.
  2. Watch the Khan Academy videos:
     1. Basic Complex Analysis
     2. Absolute Value of a Complex Number
  3. Read §1.1.
          (The text lists for $155 but there are new copies at Amazon.com for under $60.)
  4. §1.1, pg 7. #1, 2, 9, 15, 25, 31, 33, 59.
  
  
  
• Wednesday, Jan 14
  §1.2, pg 13. #1, 3, 12, 15, 17→25 odd, 27, 33b, 34b, 38, 42, 45, 50.
  
  
• Friday, Jan 16
  §1.3, pg 20. #1, 3, 9, 11, 13, 15, 19, 27, 33, 37, 38, 45.
  §1.3, pg 20. #49: Prove Lagrange's Trigonometric Identity \[ \sum_{k=0}^n \cos(k\theta) = \frac12 + \frac{\sin\big((n+\frac12)\theta\big)}{2\sin\big(\frac12 \theta\big)}, \qquad \theta\in(0,2\pi) \] which is very important in Fourier Analysis.
  1. Recall the geometric sum formula \[ 1+z+z^2+\cdots + z^n = \frac{1-z^{n+1}}{1-z} \]
  2. Substitute \(z=e^{i\theta}\)
  3. Simplify and consider the real part of the formula
  
  

Week 2

• Monday, Jan 19 — No class: Dr Martin Luther King, Jr, Day
  
  
• Wednesday, Jan 21
  §1.4, pg 25. #1, 7, 13, 17, 19, 21, 23, 28, 31, 37, 39.
  
  
• Friday, Jan 23
  §1.5, pg. #1, 3, 5, 10, 11, 19, 23, 25 (for 19 & 23), 27, 32, 34, 40, 46.
  
  

Week 3

• Monday, Jan 26
  ◊ Look at the complex functions on the "Animated Function Families" web page.
  §2.1, pg 52. #1, 3, 6, 11, 20, 26, 29, 36, 39.
  §2.2, pg 60. #1, 3, 9, 10, 15, 17, 28.
  
  
• Wednesday, Jan 28
  §2.2, pg 60. #27, 29, 31, 35.
  §2.3, pg 69. #1, 3, 6, 13, 23, 24, 33. For algebraists: #40.
  ¤ Fig 2.2.7, pg 59. Maple version (copy/paste into Maple):
  1. f := z -> z^2 + I*z - Re(z);
  2. C := t -> 2*exp(I*t);
  3. plot([Re(f(C(t))),Im(f(C(t))), t=0..2*Pi]);
  
  
• Friday, Jan 30
  §3.1.1, pg 118. #1, 3, 8, 9, 11, 15, 18, 20, 21, 25.
  
  

Week 4

• Monday, Feb 2
  §3.1.2, pg 118. #27, 29, 31, 36, 38, 53, 57, 61.
  
  
• Wednesday, Feb 4
  ◊ Möbius Transformations Group Project: Due Wednesday, Feb 11.
  §3.2, pg 128. #1, 5, 9, 18, 22, 23, 33.
  
  
• Friday, Feb 6
  §3.3, pg 135. #1, 3, 7, 9, 17, 21, 25, LA 35, 36. (LA = Look at)
  
  

Week 5

• Monday, Feb 9
  §3.4, pg 140. #1, 5, 6, 11, 18, 27.
  
  
• Wednesday, Feb 11
  §4.1, pg 164. #5, 7, 9, 11, 15, 17, 18, 20, 23, 25, 29, 31, 35, LA 50.
  
  
• Friday, Feb 13 — Paraskevidekatriaphobia!
  §4.1, pg 164. #3 (cf. \(\sinh(i t)\)), 4, 13, 37, 57→ 65 odd.
  ¤ Maple:
  • Principal branch of complex log: ln(a+b*I)    (capital 'I')
  • numerical aaproximation: evalf(L)    (or use decimals in the numbers)
  • Eg: \(\ln(2.0+3.0\cdot I) = 1.282474679 + 0.9827937232\,I\)
  
  

Week 6

• Monday, Feb 16
  §4.3, pg 182. #1, 5, 9, 15, 21, 33.
  §5.1, pg 207. #8, 9, 11, 13, 23, 27, 31, 33.
  
  
• Wednesday, Feb 18 — It's "Elm Farm Ollie Day!"
  §5.2, pg 216. #1→15 odd.
  
  
• Friday, Feb 20
  ◊ Test I is next Friday
  §5.2, pg 216. #2→14 even.
  
  

Week 7

• Monday, Feb 23
  §5.3, pg 224. #1, 3, 5, 9, 13, 16a.
  
  
• Wednesday, Feb 25
  §5.3, pg 224. #11, 15, 18, 19, 23, 25, 31.
  ¤ An old test (with a different textbook) and a test from a far away class live in "Quiz Test" in the Class Notes directory.
  
  
• Friday, Feb 27
  ¡Test I is today Monday (c.f. Weather.com)! — A chance to demonstrate your excellence!
  ◊ Review Quiz:
  • Chapter 1: pg 43.
  • Chapter 2: pg 100.
  • Chapter 3: pg 148.
  • Chapter 4: pg 198.
  • Chapter 5: pg 254.
  
  

Week 8

• Monday, Mar 2
  ¡Test I is today! — A chance to demonstrate your excellence!
  ¤ Test 1 Topics list
  
  
• Wednesday, Mar 4
  §5.4, pg 231. #1→19 odd.
  
  
• Friday, Mar 6
  §5.4, pg 231. #27, 28, 29, LA 31.
  ◊ How would you prove the result:
     Theorem. Let \(f\) be continuous on the domain \(D\). Then \(f\) has an antiderivative on \(D\) iff \(f\) is independent of path on \(D\).
  

Spring Break Week: Monday, Mar 9 → Friday, Mar 13. Let's be careful out there!
☝ Saturday, March 14, is \(\large\pi\)-Day of the Century! 3.14.15 @ 9:26:53.


Week 9

• Monday, Mar 16
  §5.5, pg 241. #1, 3, 5, 7, 15, 28.
  
  
• Wednesday, Mar 18
  §5.5, pg 241. #29, 30.
  
  
• Friday, Mar 20
  §5.5, pg 241. #11, 13, 17, 19.
  
  

Week 10

• Monday, Mar 23
  ◊ Chapter 5 Review Quiz, pg 254. #1→ 20.
  §5.6, pg 252. #1, 5. These are for physics majors.
  
  Happy Birthday to Joseph Louiville (24 Mar, 1809)
  
  
• Wednesday, Mar 25
  
  
• Friday, Mar 27
  ◊ Project day. Newton Basins. Due Wed, April 1.
  
  

Week 11

• Monday, Mar 30
  ◊ Project day. Newton Basins
  
  
• Wednesday, Apr 1 — Hilaria!
  §6.2, pg 275. #1, 3, 5, 11, 13, 25, 31, 36.
  
  
• Friday, Apr 3
  §5.5, pg 241. #32, 34.
  ¤ Prove the real variables trigonometic identity \(~\sin(2x) = 2\sin(x)\cos(x)~\) holds for all \(z\in\mathbb{Z}\) using the Identity Theorem.
  
  

Week 12

• Monday, Apr 6
  ¤ No classes — Easter Holiday.
  
  
• Wednesday, Apr 8
  §6.3, pg 286. #1, 3, 7, 11, 17, 27.
  
  
• Friday, Apr 10
  §6.3, pg 286. #29, 31, 33, 35→38 [Maple syntax: series(fcn, var=center, order)].
  
  

Week 13

• Monday, Apr 13
  §6.3, pg 286. #32.
  ¤ Set \(f(z)=\dfrac{1}{(z-1)(z-3)^2}\).
  1. Find a Laurent expansion around the point \(z=3\).
  2. What is the annulus for this expansion?
  
  
• Wednesday, Apr 15 — Tax Day.
  §6.4, pg 292. #1, 3, 5, 7, 12, 15, 18, 26, LA 33.
  
  
• Friday, Apr 17
  ¡Test II available!
  ◊ Cauchy's Residue Thm
  
  

Week 14

• Monday, Apr 20
  ◊ Real Integrals via Residues
  
  
• Wednesday, Apr 22
  ◊ Summations via Residues
  
  
• Friday, Apr 24
  ¡Test II solutions are due!
  ¤ Integral and Summation Exercises:
  1. \(\displaystyle \int_0^\infty \frac{2x^2-1}{x^4+5x^2+4}\,dx \qquad\qquad\qquad \text{Answer: } \pi/4\)
     
     
  2. \(\displaystyle \int_{-\infty}^\infty \frac{x}{(x^2+1)(x^2+2x+2)}\,dx \qquad\qquad \text{Answer: } -\pi/5\)
     
     
  3. \(\displaystyle \int_0^\infty \frac{1}{x^4+1}\,dx \qquad\qquad\qquad\qquad \text{Answer: } \pi\sqrt{2}\,/4\)
     
     
  4. \(\displaystyle \int_{-\infty}^\infty \frac{\sin(x)}{x^2+4x+5}\,dx \qquad\qquad\qquad \text{Answer: } -\pi/(e\cdot\sin(2))\)
     
     
  5. \(\displaystyle \int_{-\infty}^\infty \frac{x\sin(\pi x)}{x^2+2x+5}\,dx \qquad\qquad\qquad \text{Answer: } -\pi\,e^{-2\pi}\)
     
     
  6. \(\displaystyle \int_{-\pi}^\pi \frac{1}{1+\sin^2(x)}\,dx \qquad\qquad\qquad \text{Answer: } \sqrt{2}\,\pi\)
     
     
  7. \(\displaystyle \int_{0}^{2\pi} \frac{\cos^2(3x)}{5-4\cos(2x)}\,dx \qquad\qquad\qquad \text{Answer: } 3\pi/8\)
     
     
  8. \(\displaystyle \sum_{n=1}^\infty \frac{1}{n^4} \qquad\qquad\qquad\qquad\qquad\qquad \text{Answer: } \pi^4/90\)
  
  

Week 15

• Monday, Apr 27
  ¤ Wilkinson's polynomial*, \[ \begin{split} w(z) =\ &z^{20} - 210\, z^{19} + 20615\, z^{18} - 1256850\, z^{17} + 53327946\, z^{16} - 1672280820\, z^{15} \\ &+ 40171771630\, z^{14} - 756111184500\, z^{13} + 11310276995381\, z^{12} - 135585182899530\, z^{11} \\ &+ 1307535010540395\, z^{10} - 10142299865511450\, z^9 + 63030812099294896\, z^8 - 311333643161390640\, z^7 \\ &+ 1206647803780373360\, z^6 - 3599979517947607200\, z^5 + 8037811822645051776\, z^4 - 12870931245150988800\, z^3 \\ &+ 13803759753640704000\, z^2 - 8752948036761600000\, z + 2432902008176640000 \end{split} \] (w := z^20 - 210 z^19 + 20615 z^18 - 1256850 z^17 + 53327946 z^16 - 1672280820 z^15 + 40171771630 z^14 - 756111184500 z^13 + 11310276995381 z^12 - 135585182899530 z^11 + 1307535010540395 z^10 - 10142299865511450 z^9 + 63030812099294896 z^8 - 311333643161390640 z^7 + 1206647803780373360 z^6 - 3599979517947607200 z^5 + 8037811822645051776 z^4 - 12870931245150988800 z^3 + 13803759753640704000 z^2 - 8752948036761600000 z + 2432902008176640000);
  
  is important in numerical analysis. Use Rouche's theorem to find a circle containing all of \(w(z)\)'s roots.
  * J H Wilkinson (1984). "The perfidious polynomial." Studies in Numerical Analysis, ed by G H Golub, pp 1-28. (Studies in Mathematics, vol. 24). MAA.
  
  
• Wednesday, Apr 29
  ¤ Check what happens to the roots of \(w(x)\) when adding \(10^{-10}\) to the \(x^{19}\) term.
  
  
• Friday, May 1
  ◊ The Final Exam has been released.
  
  

Week 16 — Final Exam Week    ♞  (This Week)

• Tuesday, May 5
  ¤ Final Exam: 9:00 to 12:30
  

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Last modified: Wednesday, 01-Feb-2023 08:32:12 EST

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