User castal - MathOverflow

Answer by castal for Closed form of divergent infinite product?

2010-06-21T21:21:22Z

Take a look at the first dozen pages of Andrews and Askey, which you can read online - http://books.google.com/books?id=nMm13WXpLt8C&lpg=PP1&dq=andrews%20askey&pg=PA1#v=onepage&q&f=false

Already on page 3, they give the product representation of 1/Gamma, which is essentially your function, modified to make it convergent.

On page 10, they treat the reflection formula, which shows that 1/Gamma is "half of the sine function", i.e it contributes the zeros on the negative x axis.

Answer by castal for Reference request for category theory works which quickly prove the theorem which generalises the 1st isomorphism theorem for groups/rings/...

2010-05-24T15:33:38Z

Try "Introduction to the Theory of Categories and Functors" by Bucur and Deleanu. They have a treatment of the isomorphism theorems in abelian categories beginning on page 101. They start by noting that the first isomorphism theorem follows from the definition of abelian category. Then they go on to prove the second and third theorems. Their discussion of abelian categories only begins on page 87, and they finish the third isomorphism theorem on page 111. If you already know the preliminaries, you might be able to get right to the heart of the matter without too much trouble.

Edit: I just stumbled onto a paper by Barros and Pombo, "A direct proof of Noether’s second isomorphism theorem for abelian categories". It's only 8 pages long. They reference Bucur & Deleanu, Freyd, and Grothendieck. You can find it at http://www.scielo.org.co/pdf/rcm/v43n1/v43n1a04.pdf.

Answer by castal for Why is the Gamma function shifted from the factorial by 1?

2010-04-11T02:22:04Z

This drives me crazy, too. However, Andrews, Askey and Roy give their reasons for preferring the Legendre definition in their book on special functions. Click on the link to page 6, http://books.google.com/books?id=kGshpCa3eYwC&lpg=PP1&dq=andrews%20askey%20roy&pg=PA6#v=onepage&q=legendre&f=false

I'm not sure exactly what they are referring to in their section 1.10, but this may have something to do with it. Click on the link to page 39, http://books.google.com/books?id=kGshpCa3eYwC&lpg=PP1&dq=andrews%20askey%20roy&pg=PA39#v=onepage&q=jacobi&f=false

Answer by castal for Heuristic argument for the prime number theorem?

2010-02-26T16:34:37Z

You might look at Courant and Robbins in the section at the end, "The Prime Number Theorem Obtained by Statistical Methods".

Also, there is an article by Montgomery and Wagon, in which they mention the "Mertens Paradox" and extend the argument of Courant and Robbins.

Answer by castal for How can I conclude that I live in a solar system?

2010-02-07T00:03:50Z

Take a look at Terence Tao's pdf slides (4.3 Mb), http://terrytao.files.wordpress.com/2009/09/cosmic-distance-ladder2.pdf. Kepler makes an appearance and there's much more besides.

Comment by castal

2010-05-07T01:49:44Z

By the way, I wonder if you might not also be interested in "From a Geometrical Point of View: A Study of the History and Philosophy of Category Theory" By Jean-Pierre Marquis. Here is a link to it on Google books: <books.google.com/…;

Comment by castal

2010-05-06T17:51:57Z

The other day, I happened to be wondering just how mathematics came to be the study of "sets with structure" and "structure-preserving maps". Dominguez is a really great source for this information. I had thought that this was a 20th century phenomenon, but a quick perusal seems to indicate that Dedekind played a very prominent role in getting it started. Anyway, +1 for a great reference.

Comment by castal

2010-04-24T19:23:59Z

I have to agree with Harry. Euclid's "Elements" was not even known to western european scholars until 1120 A.D. But it came to be regarded as a paragon of the systematic development of a body of ideas, and was studied by every educated european until the twentieth century. Even the "self-evident truths" of the Jefferson's Declaration of Independence traces back to Euclid. The Wikipedia page (broken link, not sure how to fix:) (en.wikipedia.org/wiki/Euclid's_Elements) goes into more detail on the extent of Euclid's influence.

Comment by castal

2010-02-26T16:52:51Z

By the way, you might be interested to know that this is how Euler cracked the [Basel problem](en.wikipedia.org/wiki/…)