2-year integral

Yes! This integral took me 2 years to figure out! Of course, I didn't just do this integral without doing anything else. I intensively spent several days 2 years ago on it and gave up. One year ago, I tried it ago and still failed. However, this integral is strongly related to my research. I borrowed its result from another research paper. If it is wrong, all of my research work should be modified, and some of them are even going to be useless. Therefore, I spent some time on it again these days. Excitingly, I finally figured it out!
Click to see this integral

film 2012 ( No any comment on the plot )

I went to the theater in the Butler plaza last Saturday. That was my first time to the theater in the US. Before that, I sometimes went to Gator night movies for free.
The theater here is nice and the ticket price is also inexpensive.

Science in the film 2012 is kind of absurd.
First, it says the the mantle of the earth is heated by the neutrinos. However, in reality, neutrinos barely interact with matter. There are tons of neutrinos filled in space. Neutrinos also penetrate our bodies quite frequently. Though, we don't feel that because those particles do not interact with us. That's also a reason that neutrinos are not easily detected even though they are believed to be everywhere. If neutrinos can heat the earth mantle as the film describes, we all will be burned to ashes by neutrinos before those earthquakes or volcano eruptions occur.
Second, geomagnetic poles can indeed reverse their orientation. However, they change gradually, ranging from tens of thousands to many millions of years. How can they change suddenly as they happen in the film? Well....probably, if the earth is heated thoroughly and loses it magnetism in a short time...

Anyway, the visual effects are good...

Uncertainty relations of angular momentum and the angular position

Recently I happened to read an article on the bbs ( bulletin board system ) about the uncertainty principle of the angular momentum and the angular position. The author was confused by some linkage between the quantum and classical limits, which is not so interesting to me. However, his confusion, at some point, brought me to reconsider the uncertainty relations between the angular momentum ( more precisely, Lz) and the angular position ( more precisely, the azimuthal angle ). First, there is an upper limit for the angle. This makes angle very different from the position in translational sense, which can span all over the space to infinity. Second, the angular momentum operator is no more Hermitian on the state \hat{phi}|n> where |n> is the eigenstate of Lz and \hat{phi} is the "physical" angle operator. There are two ways to get around it. One is to keep it as usual but carefully trace the boundary terms: Don't miss it! The other is to create an periodic angle operator which keep Lz always Hermitian. The latter approach seems more rigirous. However, the former way is very physically meaningful. I trid to find some research papers that discussed this issue. The most original seems to be written by Dr. Judge. He adopt the first approach. The report is only one page long. Excellent! I like a short article because I am not that patient. This is very interesting paper.
**** D. Judge, Phys. Lett. 5, 189 (1963)***

I read it and took notes below.
http://www.phys.ufl.edu/~chungwei/paper/uncertainty_angular.pdf