(c562) circumference_ellipse(a,b)/2/%pi 2 b 2 a elliptic_ec(1 - --) 2 a (d562) ----------------------- %pi is clearly a legitimate mean(a,b), satisfying an identity: (c563) (circumference_ellipse(1-x,1+x)/2/%pi = rhs(d448)) 2 (1 - x) 2 (x + 1) elliptic_ec(1 - --------) 2 (x + 1) (d563) ----------------------------------- = %pi 2 2 2 (2 elliptic_ec(x ) - (1 - x) (x + 1) elliptic_kc(x )) ------------------------------------------------------- %pi (c564) taylor(%,x,0,12) 2 4 6 8 10 12 x x x 25 x 49 x 441 x (d564)/T/ 1 + -- + -- + --- + ----- + ------ + ------- + . . . = 4 64 256 16384 65536 1048576 2 4 6 8 10 12 x x x 25 x 49 x 441 x 1 + -- + -- + --- + ----- + ------ + ------- + . . . 4 64 256 16384 65536 1048576 (i.e. 2F1(-1/2,-1/2,1,x^2)) so it is >= arithmetic >= geometric means. Both the "aem" and the "egm" converge quadratically, like the agm, but to what? --rwg PS, John Brillhart challenged me to find the discriminant of the nth Legendre polynomial discriminant. I hope he doesn't mind my ducking and throwing open the question...