Week 12 (July 23  July 29) 
[Jul. 23rd, 200612:47 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 contourintegral 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. 

