Analysis II (Math 131B), Fall 2023

MacQuarrie Hall 235, MW noon-1:15pm (Sec 01, code 44629)

Last update of this page: Fri Dec 08, 11:00pm

Ongoing useful links:

• Class Gradescope page
• How to get started on Gradescope (includes several useful videos)
• The Gradescope mobile app for scanning documents into Gradescope using your smartphone
• Older tips for scanning documents into Gradescope

Announcements:

• Final exam, Thu Dec 14, 9:45am, in our usual room.
• The textbook for this class is Fourier Series, Fourier Transforms, and Function Spaces: A Second Course in Analysis, Hsu, MAA Press. On sale now 35% off from the publisher, i.e., for a little over $50!
• Click here to go to our Canvas page.
• Our weekly problem sessions will be held every Friday, 12:00-2:00pm, by Zoom.
• Review references for:
  • Math 131A: Elementary Analysis: The Theory of Calculus, by Kenneth Ross (any edition is fine); Principles of Mathematical Analysis, by Walter Rudin (any edition is fine). Click here for a bird's-eye view of 131A.
  • Methods of proof: See my Math 108 course notes or Reading, Writing, and Proving, by Daepp and Gorkin.

FUA (Frequently Used Acronyms):

Administrative handouts:

Course notes:

Proofs and strategies content: Writing proofs. How to do a problem set outline. Problem solving strategies in analysis.

Analysis content: Important definitions involving epsilon.

Topics for exams:

Exam 1, Exam 2, Exam 3, Final exam.

Sample exams:

Final from 131A.

Exam 1, Exam 2, Exam 3, Final exam.

Online lectures:

Click on the link in the "Sections covered" column for a copy of the slides for the day; click on the link in "Extra slides" for other stuff displayed that day; and click on the link(s) in the "Video" column to watch the lecture on YouTube. Or just check out our class playlist here.

Problem sets:

HW	New definitions	Outline due	Completion due	Last revision due	Problems to be turned in
PS00	n/a	n/a	Wed Aug 23	n/a	See handout
PS01	2.1, 2.2, 2.3, 2.4	Mon Aug 28	Wed Aug 30	Mon Oct 30	See handout
PS02	2.5, 3.1, 3.2	Wed Sep 06	Mon Sep 11	Wed Dec 06	2.5.4, 3.1.2, 3.1.4, 3.1.6, 3.1.13, 3.2.4, 3.2.6.
PS03	3.3, 3.4	Wed Sep 13	Mon Sep 18	TBA	3.3.2, 3.3.3, 3.3.4, 3.4.3, 3.4.4, 3.4.6.
PS04	3.5, 4.1, 4.2, 4.3, 4.4.	Wed Sep 28	Mon Oct 02	Wed Dec 06	3.5.2, 4.1.4(a,b), 4.2.1, 4.3.5, 4.3.15 (no longer 4.3.6), 4.4.1.
PS05	4.5, 4.6, 5.1, 5.2, 5.3.	Wed Oct 04	Mon Oct 09	Wed Dec 06	4.5.1, 4.5.2, 4.5.4, 4.5.6, 4.6.2, 5.1.1, 5.2.2.
PS06	6.1, 6.2, 6.4, 7.1.	Wed Oct 11	Mon Oct 16	TBA	6.1.2, 6.2.4, 6.2.6, 6.4.1, 6.4.4, 7.1.3, 7.1.6.
PS07	7.2, 7.3, 7.4, 7.5.	Wed Oct 25	Mon Oct 30	Wed Dec 06	7.2.4, 7.2.5, 7.3.1, 7.3.4, 7.3.5, 7.4.3, 7.5.5.
PS08	7.6, 8.1, 8.2.	Thu Nov 02	Wed Nov 06	Wed Dec 06	7.5.10, 7.6.2, 7.6.4, 7.6.7, 8.2.2, 8.2.5, 8.2.7.
PS09	8.3, 8.4, 8.5.1, 8.5.3.	Wed Nov 08	Tue Nov 14	TBA	8.3.1, 8.3.2, 8.3.3, 8.4.1, 8.4.3, 8.4.6, 8.5.5.
PS10	4.7, 4.8, 12.1, 12.2.	Tue Nov 28	Fri Dec 01	TBA	4.7.3, 4.8.3, 12.1.1, 12.2.3, 12.2.5, 12.2.8, 12.2.10.
PS11	12.3, 12.4.	n/a	Thu Dec 07	TBA	12.3.1, 12.3.3, 12.3.5, 12.3.7, 12.4.2.

Reading assigned in course:

From textbook: 1.1-1.2, 2.1, 2.2, 2.3, 2.4, 2.5, 3.1-3.5, 4.1-4.6, 5.1-5.3, 6.1-6.2, 6.4, 7.1-7.6, 8.1-8.4, 8.5.1, 8.5.3, 4.7, 12.1, 12.2, 12.3, 12.4.

Proof notes: Sec 1-7, Sec 23 (epsilonics).

Further topics:

1. More theory
   • The Fourier transform: See Chs. 12-13 of our textbook.
   • The Discrete Fourier Transform (DFT) and the Fast Fourier Transform (FFT).
   • Lebesgue integration: See Appendix B of Spectral Theory.
   • Distributions (generalized functions).

2. Practical applications
   • The Black-Scholes model for options pricing.
   • Can you hear the shape of a drum? (No: Gordon/Webb/Wolpert, Buser/Conway/Doyle/Semmler.)
   • Wavelets and jpegs.
   • Musical acoustics and Fourier analysis: The sampling theorem.
   • Spherical harmonics: See Applied Analysis, 7.7-7.8. Application to medical imaging.
   • Auto-tune: This is probably outside the scope of this class, but for a start, try here, here, and here. (If you don't know what Auto-Tune is, check out some famous examples.)

3. Less practical applications
   • Theta functions and the heat equation.
   • Maass forms and number theory
   • Quantum computing.
   • Way beyond the scope of our course, but the solution of the regularity problem for Navier-Stokes is worth one meeellion dollars.

Other useful links:

• If you're interested in typesetting math, the system I use to make my handouts is called LaTeX. If you have a Windows PC, you can download LaTeX for free at http://www.miktex.org; you will also need a good text editor, like the shareware program WinEdt. PCTeX also makes a good commercial version; student prices are $150 for the "Publisher" version, which I recommend if you have the money, $50 for the "Writer" version, which I recommend if you don't. If you own a Mac, you can download a complete system called TeXShop for free. If you own a Linux computer, you already own a copy of this program, as it's been a standard Unix program for about 20 years.

  LaTeX is well worth learning, if you have some free time; it's used for manuals and technical reports throughout nerd-dom. For self-study, I highly recommend Learning LaTeX (Griffiths and Higham) for beginners, and the LaTeX book by Leslie Lamport as a reference.

• Instructor: Tim Hsu.
• Textbook: Fourier Series, Fourier Transforms, and Function Spaces: A Second Course in Analysis, Hsu, MAA Press.
• Reference texts: Elementary Analysis: The Theory of Calculus, by Kenneth Ross (any edition is fine; Principles of Mathematical Analysis, by Walter Rudin (any edition is fine).
• Homework: Outline and final version due once each week, except in exam weeks. For details, see the handout on homework in Math 131B.
• Exams: Three in-class 75-minute exams, plus 2 hour, 15 min final exam (Thu Dec 14, 9:45am-noon).
• Grading: Final course grades consist of:
  

  Homework:	20%
  Exam 1:	14%
  Exams 2 & 3:	18% each
  Final:	30%