Current reading and homework assignments

Due Monday, 2 May
There will be a final exam on this date

Reading

• S-S, Chapter 8, Appendix A, 1-3.
• Lecture notes from 21 April
• Lecture notes from 26 April
• Lecture notes from 28 April

• Find all the conformal maps from the upper half plane to itself.
• S-S, section 8.5, #3,10,12
• Show that if the Taylor series Σ_k=0 a_k z^k has radius of convergence r > 0,
  then on the disc |z| ρ < r,
  Σ_k=n a_k z^k = O(zⁿ).
• Use Cauchy's integral formula to prove:
  Let f(z) be holomorphic in a closed annular sector S and f(z) = O(z^p) as z tends to infinity in S. Then in any proper annular subsector C with the same vertex as S, f^(m) = O(z^p-m). (By definition, an annular sector is the intersection of a sector with the exterior of a disc centered at its vertex.)
• Find asymptotic expressions for the integrals from 0 to infinity of
  1. exp(-t - s t^a) t^b-1
  2. t^s exp(-t) log t
  and for the integral from 0 to π²/4 of exp(x cos t^1/2).
• Stay tuned for more suggestions!

Past homework assignments

• Prof. Cain's elementary class notes on complex numbers
• Prof. Loss's class notes on complex numbers
• Prof. Harrell's elementary lectures on complex numbers (requires Maple)

Reading

• Stein-Shakarchi, chapters 1-2

• 18 January

Exercises

Hand in at least 5 of the following, and think about all of them:
• S-S, section 1.4, #3,7,12, 16 (d) and (e), 19. 25.
• S-S, section 2.6, #6,10.

Reading

• Stein-Shakarchi, chapter 2

• 20 January
• 25 January

Exercises

• S-S, section 2.6, #2,9, 12.

Nothing - continued from previous week.

Due Thursday, 10 February

There is a scheduled test on this date.

Reading

• Stein-Shakarchi, chapter 3

• 1 February
• 3 February
• 8 February

Exercises

For test preparation, work the problems in chapters 2 and 3 (at least sections 1-4) of S-S, in particular:
• S-S, section 2.6, #7,8, 15, which last will be on the test verbatim
• S-S, section 2.7, #3
• S-S, section 3.8, #2,3,13,15
• S-S, section 3.9, #3
• Doing all the rest of them would prepare you even better!

Due Thursday, 17 February

Reading

• Stein-Shakarchi, chapter 3

• 15 February (Thanks, Mitch!)

Do all problems on the first test that you did not hand in at the time.

Due Thursday, 24 February

Reading

• Stein-Shakarchi, chapter 3-4

• 15 February
• 17 February
• 22 February (includes scan of exercises).
  Note: The final argument, about the analyticity and bound of the Fourier transform of f, of moderate decrease, was incorrect. It will be corrected on Thursday.

Exercises
Work out at least three and hand them in. We can begin Thursday with a discussion of the problems.

• Consider the transcendental equation
  tan z = a z for a > 0.
  Use Rouché's theorem to determine the number of solutions of this question. Distinguish among real roots and complex roots, and between the cases a < 1 and 1.
  Hint: Consider a large square bounded by N ( ± 1 ± i).
  For the solution, click here.
• S-S, section 3.8, #9, 12, 14, 20, 21
  For a solution to Problem 9, click here.
• S-S, section 3.9 #3 (if I fail to do this in class on Tuesday),

Due Thursday, 3 March

Reading

• Stein-Shakarchi, 4

• 24 February (includes scan of exercises).
• 29 February

Exercises

• S-S, section 4.4, #1, 2.

Due Thursday, 10 March

Reading

• Stein-Shakarchi, 4, Appendix B

• 29 February
• 3 March
  This contains only the part about Fourier and Hermite. The rest of the lecture consisted in working through S-S section 4.3, back to front.

Exercises

• S-S, section 4.4, #8.
• Do the exercises in the lecture notes

Due Thursday, 17 March

There is a scheduled test on this date.

Reading

• S-S, Appendix B
• 8 March
• 10 March

Exercises

For test preparation, work the problems in chapters 3 and 4 of S-S, in particular:
• S-S, section 4.4, #6-10, at least one of which will be on the test verbatim. (How about number 9?)
• Doing all the rest of them would prepare you even better!
• A Möbius problem: Suppose that a Möbius transformation T carries one pair of concentric circles into a pair of concentric circles. Prove that the ratios of their radii are the same.
• Another one. The circles of Apollonius are the loci of points satisfying

  
               |(z-a)/(z-b)| = r/|k|            (1)
  
  

  for fixed a,b . There is one more parameter than necessary because I want the formula to reflect the fact that they are the images under a Möbius transformation of the form

  
               w = k (z-a)/(z-b)
  
  

  of concentric circles around the origin. Denote by C₁ the set of all circles through two points a,b, and by C₂ the circles of Apollonius with the points a,b in (1). Sketch these and prove as many cool properties as possible, including
  1. The families C₁ and C₂ meet at right angles.
  2. Reflection in a C₁ transforms every C₂ into a C₂ and vice versa.
  What are the Möbius transformations that preserve the points a,b, and how do they affect the families C₁ and C₂? (Reference: Ahlfors, Complex Analysis.)

Due Thursday, 1 April

Reading

• S-S, Chapter 5
• Lecture notes from 15 March, by Prof. Croot (corrected).
• Lecture notes from 26 March

Exercises None due this week, and that is no April Fool's joke!

Due Thursday, 8 April

Reading

• S-S, Chapter 5
• Lecture notes from 26 March
• Lecture notes from 5 April

• S-S, section 5.6, #2,3

Due Thursday, 15 April

Reading

• S-S, Chapter 5-6
• Lecture notes from 7 April
• Lecture notes from 12 April

• S-S, section 5.6, #5,6, 10(a).

Due Thursday, 21 April

Reading

• S-S, Chapter 6, 8
• Lecture notes from 14 April
• Lecture notes from 19 April

Homework will no longer be collected, only suggested. Feel free to discuss it in class.
• Prove the formula for Γ(2 z) stated in class on Tuesday, 19 April. (Basically same as S-S section 6.3, #3.)
• S-S, section 6.3, # 5,13,14
• S-S, section 8.5, # 10,13,14,

• The 6321 home page
• Evans Harrell's home page

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