Mystery: Mathworld: Elliptic Integral Singular Value When the elliptic modulus <http://mathworld.wolfram.com/EllipticModulus.html> [image: k] has a singular value, the complete elliptic integrals may be computed in analytic form in terms of gamma functions <http://mathworld.wolfram.com/GammaFunction.html>. Abel (quoted in Whittaker and Watson 1990, p. 525) proved that whenever K(1-k)/K(k) = (a + b√n)/(c + d√n) [parameter notation], where [image: a], [image: b], [image: c], [image: d], and [image: n] are integers <http://mathworld.wolfram.com/Integer.html>, [image: K(k)] is a complete elliptic integral of the first kind <http://mathworld.wolfram.com/CompleteEllipticIntegraloftheFirstKind.html>, then the elliptic modulus <http://mathworld.wolfram.com/EllipticModulus.html> [image: k] is the root <http://mathworld.wolfram.com/Root.html> of an algebraic equation with integer <http://mathworld.wolfram.com/Integer.html> coefficients <http://mathworld.wolfram.com/Coefficient.html>. (i.e., it's an algebraic number?) I can't make this work, except for a=c=0, and n rational, where it always works. E.g., no dice for K'/K = golden ratio. Who spazzed: Abel, W&W, eww, or me? Anyway here are two carefully simplified cases. n = 25/4 from yesterday: gosper.org/K÷K=2÷5.pdf <http://gosper.org/K÷K=2÷5.pdf>and n = 3/2: gosper.org/sesquith.pdf . Does anyone know why we fuss over the nth singular value when n can apparently be any rational? —rwg