(This was too easy for the kids, but it feels good when you find the words an Authority missed.) See http://dlmf.nist.gov/22.3.F2.mag and http://dlmf.nist.gov/22.3.F3.mag "For cn(x,k) the curve for k=1/sqrt2=0.70710… is a boundary between the curves that have an inflection point in the interval 0≤x≤2K(k), and its translates, and those that do not;..." a) why don't they just say "at x=K(k)" vs "in the interval 0≤x≤2K(k)"? b) Huh? cn(x,k) always has an inflection point there! But *something* happens at k=1/sqrt2. What should they have said? --rwg Caution: DLMF uses K(modulus) vs K(parameter), so if you plot this with Macsyma or Mma, the magic k = 1/2, not 1/sqrt2. In either case, K=4 (1/4)!^2/Sqrt[Pi] ~ 1.85407 (This K property is imperfectly analogous to the graphical definition of Halphen's constant, about 69% into http://gosper.org/thetpak.html .)