Analysis II (Math 131B), Fall 2024

MacQuarrie Hall 233, MW noon-1:15pm (Sec 01, code 49232)

Last update of this page: Tue Dec 10, 2:00pm

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, Mon Dec 16, 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 from the publisher, 35% off, i.e., for a little over $50!
• Click here to go to our Canvas page.
• Our weekly problem sessions will be held every Friday, 12:30-2:30pm, by Zoom.
• Review references for:
  • Math 131A:
    • Summary of main theoretical points of 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)
  • 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; Model outline for PS01. 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	Revision target date	Problems to be turned in
PS00	n/a	n/a	Mon Aug 26	n/a	See handout
PS01	2.1, 2.2, 2.3, 2.4	Wed Aug 28	Wed Sep 04	Tue Oct 01	See handout
PS02	2.5, 3.1, 3.2	Mon Sep 09	Wed Sep 11	Tue Oct 29	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	Mon Sep 16	Wed Sep 18	Tue Oct 29	See handout
PS04	3.5, 4.1, 4.2, 4.3, 4.4.	Thu Sep 26	Mon Sep 30	Tue Oct 29	3.5.2, 4.1.3, 4.2.1, 4.3.5, 4.3.6, 4.4.1.
PS05	4.5, 4.6, 5.1, 5.2, 5.3.	Thu Oct 03	Mon Oct 07	Tue Dec 10	4.5.1, 4.5.3, 4.5.4, 4.5.5, 4.6.2, 5.1.2, 5.2.2.
PS06	6.1, 6.2, 6.4, 7.1.	Thu Oct 10	Mon Oct 14	Tue Dec 10	6.1.2, 6.2.5, 6.2.6, 6.4.1, 6.4.4, 7.1.3, 7.1.7.
PS07	7.2, 7.3, 7.4, 7.5.	Thu Oct 24	Mon Oct 28	Tue Dec 10	7.2.4, 7.2.5, 7.3.1, 7.3.4, 7.3.6, 7.4.3, 7.5.4.
PS08	7.6, 8.1, 8.2.	Thu Oct 31	Mon Nov 04	Tue Dec 10	7.5.10, 7.6.2, 7.6.4, 7.6.7, 8.2.2, 8.2.4, 8.2.7.
PS09	8.3, 8.4, 8.5.1, 8.5.3.	Wed Nov 13	Mon Nov 18	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, 12.1, 12.2.	Fri Nov 22	Tue Dec 03	TBA	4.7.3, 4.8.3, 12.1.1, 12.2.4, 12.2.5, 12.2.8, 12.2.10.
PS11	12.3, 12.4.	Thu Dec 05	Mon Dec 09	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, 2.1-2.5, 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, 4.8, 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 (Mon Dec 16, 9:45am-noon).
• Grading: Final course grades consist of:
  

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