Section B.7 Problem Sheet 7
Due 27 March at 4pm, either in class or in the pigeon hole in 4W.
- Exercise 4.4.1 (PS7) Directly checking \(\abs x\) is subharmonic (probably the easiest problem to start)
- Exercise 4.3.1 (PS7) Reflecting harmonic functions (chronologically first)
Definition 4.12 is a relatively complicated, and so it seems worth taking the time to verify it directly in Exercise 4.4.1 for a simple example before launching into more theoretical investigations of its consequences. Since this is probably the easiest problem on the sheet, I’ve decided to list it first. Exercise 4.3.1 is an absolutely classic exercise, related to the Schwarz reflection principle in complex analysis, and almost every student of harmonic functions does a version of it at some point or another. It at least starts to explain how Theorem 4.11 can be useful. As we will use Lemma 4.16 heavily in Section 4.5, it is important that we finish its proof in Exercise 4.4.2 and Exercise 4.4.3. It’s also good to get some more practice with subharmonic functions. While Exercise 4.4.2 is relatively straightforward once you unwind the definitions, Exercise 4.4.3 is a bit more subtle. I highly recommend that you draw yourself a little Venn diagram when thinking about it.
Please feel free to email me, or drop by office hours (Wednesdays 1:15–2:05 in 4W 1.12). I am also more than happy to meet one-on-one or in a small group.
