MA598C_2014_postmortem_mjk
I struggled more than I expected I would for this qualifying exam. I have a few comments about the exam:
- I was kind of surprised that there were no problems which asked to compute a real integral by taking a detour through the complex plane. There was also no problem which made use of the Schwarz Lemma. These were standard features of past qualifying exams written by Professor Bell. I guess you can't rely on patterns to always hold true. At the end of the day, you need to have an understanding of the subject that allows you to think on your feet to solve problems.
- There was a problem on the exam which asked one to compute (without using a memorized formula) the residue of a quotient of two analytic functions at a point. I tried using the Residue Theorem but found myself unable to compute the integral. I ended up computing the first few terms of the series expansion of the denominator function and then writing out the first few terms of the formal inverse of this power series. I wrote the numerator as a power series, multiplied, and picked out the residue. In short: it paid to know how to write the series expansion for $ 1/\sin(z) $ from knowledge of the expansion of $ \sin(z) $. Serge Lang's Complex Analysis has a nice section dealing with formal power series.
- I didn't work through as many past qualifying exams as I probably should have. While I spent a lot of time reviewing class notes and really digesting proofs, I think I would have done better had I worked more problems. If you work a lot of problems, you'll probably recognize one on the exam. In fact, the last question on the exam we had seen in the MA598 class!
Overall, I didn't feel too confident leaving the exam room. While I had ideas for all of the problems, I didn't solve them all. Despite expecting to have to take the exam again (or at least take another qualifying exam), I was quite pleased to find that I passed.
