(Some of my eavesdroppers' ISPs seem to be blacklisting osots.com, so I'm sending from Yahoo, and setting Rich another chore.) Does anyone know how to widen Yahoo's composition window to a usable width? Meanwhile, I'll try an attachment. If it linebreaks that too, I'll switch to gmail. (Sorry, Rich!) --rwg ____________________________________________________________________________________ Pinpoint customers who are looking for what you sell. http://searchmarketing.yahoo.com/ Rich's old Hakmem (http://www.inwap.com/pdp10/hbaker/hakmem/geometry.html) ####### ITEM 1 (Schroeppel): (1/3)! and (2/3)! are interexpressible. (1/4)! and (3/4)! are interexpressible. Thus these two pairs are of dimensionality one. (1/10)! and (2/10)! are sufficient to express (N/10)! for all N. (1/12)! and (2/12)! are sufficient to express (N/12)! for all N. (1/3)! and (1/4)! are sufficient to express (N/12)! for all N. Thus the three cases above are of dimensionality two. PROBLEM: Find some order to this dimensionality business. The reflection and multiplication formulas: pi Z Z! (-Z)! = --------- sin(pi Z) (N-1)/2 -NZ-1/2 (2 pi) N (NZ)! = Z! (Z-1/N)! (Z-2/N)! ... (Z-(N-1)/N)! ######## can now be rephrased as an applied math problem: Write a simplfier that recognizes when a monomial of fractional factorials is elementary. E.g., 1 2 7/8 (sqrt(6) + sqrt(2)) %pi (--)! (-)! 4 3 sqrt(-----------------------) 12 3 2 ---------- = ------------------------------------. 1 27 (-)! 4 I know of no CAS that can do this. (But haven't tried Mma 6.0). Presumably from its lack of a D.E., we "know" Gamma has no nice series. But the above implies 1/8 %pi 1 1 3 sqrt(-----------------) Beta(-, -) sqrt(6) + sqrt(2) 4 3 sqrt(---------------------------------------) 1 2 (-)! = ---------------------------------------------, 4 2 further evidence that <rational>! = algebraic(hypergeometrics(rationals)). Lastly, I have a 105 page Table of the Gamma Function for Complex Arguments, National Bureau of Standards Applied Mathematics Series : 34 (Issued Agust 6, 1954).

From the preface:

The table of the complex gamma function presented in this volume is of fundamental importance in both pure and applied mathematics. The tabulation was prompted by urgent and specific needs in the fields of atomic and nuclear research. [...] Does anyone know of a physics formula with a naked Gamma? I'm guessing that they really needed only Betas and similar Gamma quotients that are merely hypergeometric sums, but involving multiple arguments, making them impractical to tabulate. "Scientist don't need computers-- all we need are a few computers to make them tables." Refutation: oo [ 2 2 2 ] /===\ [ (k + a) (k + b) (2 k + a) (2 k + b) - k ] | | [ ------------------------------------------ ------------------------ ] | | [ 2 b + a b + a + 1 b + a ] | | [ 4 (k + 1) (k + ----- + 1) (k + ---------) ] k = 0 [ 2 2 ] [ ] [ 0 1 ] [ 1 ] [ 0 ---------- ] = [ Beta(a, b) ] [ ] [ 0 1 ] (To hell with hypergeometric notation, but this is essentially a 4F3[1/4] that improves on some F[1/8]/F[1/8]s in Gosper, Strip mining in the abandoned orefields of Nineteenth Century mathematics, in Computers in Mathematics (Lecture Notes in Pure and Applied Mathematics, vol. 125), D. Chudnovsky & R. Jenks, eds. pp 261-283, m. dekker, N. Y.) I.e., a decently convergent series, but of two arguments, so you can't tabulate it for complex a,b. --rwg Video games are a feminist plot to render boys academically noncompetitive.