Just another nag/brag about TLINREL, the unreasonably effective, nine line Taylor-series analog of an integer relation-finder. Here it "discovers" Legendre's relation given all possible linear and quadratic combinations of K, E, K', and E', despite the messy logs and digammas introduced by shifting to 0 the point of expansion of K' and E'. (c72) (makelist((%pi*hyper_2f1(a/2,1/2,1,x)/2,hypersimp(%%)=%%),a,[1,-1]), append(%%,multthru(hyplin1mz(subst(1-x,x,%%))))) 1 1 %pi hyper_2f1(-, -, 1, x) 2 2 (d72) [elliptic_kc(x) = -------------------------, 2 1 1 %pi hyper_2f1(- -, -, 1, x) 2 2 elliptic_ec(x) = ---------------------------, 2 inf ==== 2 \ 1 n elliptic_kc(1 - x) = - ( > (-) x / 2 n ==== n = 0 1 2 (log(x) - 2 psi (n + 1) + 2 psi (n + -))/n! )/2, 0 0 2 inf ==== \ 1 3 n elliptic_ec(1 - x) = 1 - x ( > (-) (-) x / 2 n 2 n ==== n = 0 3 1 (log(x) - psi (n + 2) + psi (n + -) - psi (n + 1) + psi (n + -)) 0 0 2 0 0 2 /(n! (n + 1)!))/4] (c73) factor(tlinrel(makelist(lhs(e)=taylor(rhs(e),x,0,11),e, append(%,%^2,%*part(%,[2,3,4,1]),part(%,[1,2])*part(%,[3,4]))))) (d73) 98304 (2 elliptic_kc(1 - x) elliptic_kc(x) - 2 elliptic_ec(1 - x) elliptic_kc(x) - 2 elliptic_kc(1 - x) elliptic_ec(x) + %pi) 4 3 3 2 2 (54 log (x) - 864 log(2) log (x) + 242 log (x) + 5184 log (2) log (x) . . . 4 3 2 + 13824 log (2) - 15488 log (2) + 11168 log (2) - 3908 log(2) + 617) 4 4 3 3 /(%pi x (3432 log (x) - 54912 log(2) log (x) + 8362 log (x) . . . 3 2 - 535168 log (2) - 832096 log (2) + 161444 log(2) + 227688)) = 0 Expansion to 11th order was probably excessive, but higher order -> more confidence in the finding. Although the probability seems slim of TLINREL uncovering a functional relation that traditional analysts have somehow overlooked, the payoff would be large compared to the analogous detection of merely numerical relations. --rwg