MA2104 Differensiallikninger og kompleks funksjonsteori 2006

Contents: Messages · Examinations · Syllabus · Exercises · Lectures

Lecturer:

Harald Hanche-Olsen

Messages (inverse chronological order)

• 2006-12-03: I have finally put together the formula sheet(s) for the exam. You will find it under the Examinations heading below. Please let me know if you find any misprints or errors in it.

  I have set aside office hours on Tuesday 5^th and Monday 11^th December, during the hours 13:00–16:00 both days. I'll be in my office and ready to answer questions.

• Old messages.

Examinations:

A collection of earlier exam problems, including midterm tests, is available.

Midterm exam: During usual lecture hours: Thursday 26^th October 12:15–14:00 in S6. Results.

Final exam: Tuesday 12^th December, 09:00–13:00.
You may bring your HP 30S calculator. A ^newformula sheet (two sheets, actually) will be appended to the exam.

Syllabus:

HH-O: The Cauchy integral theorem. (PDF: A5/A4.) Basically, the note gives a more fundamental definition of the path integral, and a different proof of the Cauchy integral theorem. It gives some proofs in much more detail than I expect you to absorb.

Paul R. Chernoff: Pointwise Convergence of Fourier Series. The American Mathematical Monthly, 87 (1980), 399–400. (The link requires a subscription. NTNU subscribes, so you can get the paper so long as you are within the NTNU computer network. Use the DOWNLOAD link at the top of the page, and select high quality output to get something that looks good when you print it.)

HH-O: The Riemann–Lebesgue lemma. (PDF: A5/A4.) A much simpler proof than the one in the book (on p.~504). We need it for the proof in Chernoff's paper.

HH-O: The classification of isolated singularities. (PDF: A5/A4.) Section 4.6 of the book, in very short form.

Nakhlé H. Asmar: Applied Complex Analysis with Partial Differential Equations (Prentice Hall; ISBN: 0130892394).
Some changes may occur.

Final syllabus:

Chap	Read	Comments
1	All
2	1–4
3	1–7, 9	Skip proofs in 3.5 and 3.9
4	1–6
5	1–4	except pages 338–342 and 348–351
7	1–4
8	1–4, 6	except pages 558–560

Exercises

There are no mandatory exercises. If you wish feedback on some of your work, you can hand in your solutions to exercises marked with an asterisk (*), preferably no later than Monday a week and a half after the exercises have been posted. I have arranged for a box on the third floor of «Nordre lavblokk», see separate map. Solutions to the first two exercises will be posted here on Monday, and then one further solution will be posted each Monday henceforth. (Both for problems with and without the asterisk.)

(Week numbers referer to the week when problems are posted.)

• Older exercises
• Week 42: solution; see also animations for problems 7.2.5, 7.2.9, 7.2.13 and note how the latter overshoots at the discontinuity: This is an example of the Gibbs phenomenon.
  7.1: 8, 12*, 15
  7.2: 5, 9*, 13, 17
  7.5: 11.
• Week 43: solution; the integrals in this problem are known as the Fresnel integrals.
  Consider the function exp(−x²). Integrate it over the triangular path [0,M,M+iM,0] and let M→∞. Note that then the integral along the x axis converges to a known limit. Prove that the integral along the vertical bit goes to zero, and by considering the integral along the hypothenuse, find the integrals of cos x² and sin x² from 0 to ∞.
• Week 44: solution
  4.4: 22* (hint: partial fraction decomposition), 23
  4.5: 1, 13* (check out example 4 first), 25*
  7.5: 15*, 17 (look at Example 3 first)
  8.2: 5(a)*, 9(a).
• Week 45: ^newsolution
  5.1: 15, 20*, 25
  5.2: 7*, 11, 13* (Hint: the “double angle trick” may be useful)
  8.2: 8*
  Exam of 2003-12-15: 1*, 4.
• Week 46: ^newsolution
  From old exam problems:
  MNFMA213 1998-12-10: 5
  MNFMA214 2002-05-16: 2
  MNFMA214 2003-05-19: 4, 5
  MA2104 2004-12-13: 3, 5.

Lectures

• Week 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46

• Week 47: These final lectures were used mainly for summarizing.

  2006-11-22: Summarized Fourier series and separation of variables.
  I had not covered the convolution product before, so I did now, with the important consequence Parseval's identity.

  2006-11-23: Summarized complex function theory.
  I spent a little extra time on the exponential function and the logarithm, since these important functions are a bit easier to understand with all the theory we now have available.

Harald Hanche-Olsen Oppdatert: 2006–12–03 14:07