I now have six more math books in my room than I did yesterday. I’m so bad at resisting temptation when it comes to books. (I only had to buy two of the books, and of those two, one will be a textbook for next quarter, so I’m not as obsessive about buying books as I might have seemed had you glossed over the stuff in parentheses.)

Labutin got to use 2*2=4 as a lemma in a proof today. He said “Now using this lemma that 2*2=4, we can prove the theorem.” He failed to prove this theorem on Wednesday since he had made the assumption that 2*2=2.

For some reason, there’s a lot of furniture in the hallway of South Hall 6th floor. Maybe something is being done to some of the offices.

In music theory, we were subjected to a dreadful recording of the fifth Brandenburg Concerto. At least there was a harpsichord rather than a piano, but it was still dreadful.

We’re playing Beethoven’s Second Symphony, the first movement of Schumann’s Piano Concerto, the first movement of Mozart’s Violin Concerto in D, and Maher’s Kindertotenlieder. It seems like a very ambitious program given that the concert is on March 10th, but it will have to work somehow, and I’m sure it will.

There’s a concert tonight, and Beethoven’s Sonata for Cello and Piano in A Major will be performed. Yay!

Finally, here’s an interesting problem I thought of. Ryavec said it will be really hard. For which sets of points S in C with |z|=1 (z in S) is it possible to find a function f such that f is analytic on S and is not analytic everywhere else on |z|=1? I was thinking of functions centered at 0, but I don’t know how well that’s defined if f isn’t defined as a Laurent series.

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## About Simon

Hi. I'm Simon Rubinstein-Salzedo. I'm a mathematics postdoc at Dartmouth College. I'm also a musician; I play piano and cello, and I also sometimes compose music and study musicology. I also like to play chess and write calligraphy. This blog is a catalogue of some of my thoughts. I write them down so that I understand them better. But sometimes other people find them interesting as well, so I happily share them with my small corner of the world.

You should be able to do this one:

Exercise 1. Given an analytic function f, prove that the maximal set on which it extends is open in C.

Exercise 2. Given an open set S in C, show that there is a function which is analytic on S, but does not extend to any larger set (Hint: construct a sequence of analytic functions that converges uniformly on every compact subset of S to a function that has dense set of singularities on the boundary of S).

Exercise 3. Combine 1 and 2 to obtain the full answer to your question.

Ok I agree with all that, but I wasn’t talking about analytically continuing f. I guess it doesn’t make much sense otherwise, unfortunately, since I can just restrict it to any region I want. But I think I meant defining f in terms of a Laurent series or a Dirichlet series (that wouldn’t work, but perhaps there could be something like that that does) or something natural like that and considering the region of analyticity of f without doing any analytic continuation. Thanks for the hints though!

I am the same guy who wrote the first comment.

It seems that you want to your function “live on” S^1, so it would be as meaningless to extend it as extending a general function of one real-variable to a function of two variables. If that is what you mean, then you are looking for a concept of a manifold, and in this case complex analytic manifold. However, S^1 still would not work because one needs to have at least one dimension, and S^1 has no subsets homeomorphic to C.

Boris

I don’t want it to live on S^1. I want to live on D union S^1 (or is it just D?). I guess I didn’t make that very clear. Sorry about that.