|Week 12 (July 23 - July 29)
||[Jul. 23rd, 2006|12:47 am]
Math Reading Group
Section 2.4 - The Barnes Integral for the Hypergeometric Function|
- The Mellin transform (but why does it work?)
- Proof of Barnes's integral using residues and contour-integral estimates.
- Some sideways integrals of Γ(s)-products, and some hypergeometric identities.
I think these identities are proved in this way because nice proofs appear, not because contour integration is the natural way to deal with these particular identities.
Prof. Doron Zeilberger discovered a wonderful algorithm to produce proofs of most of these identities (it is in the book too) but the proofs are so unintuitive and boring to check that there is still a big market for "elegant" memorable proofs of things. Besides, I haven't even memorized his algorithm yet! not for lack of trying.
If this interests you you can read Petkovšek, Wilf, and Zeilberger, "A=B", which is on the Internet. You can also read Concrete Mathematics, which has a lucid explanation. Check out Zeilberger's Maple code, on his site.