Bill Gosper: Then I bashed on it so hard that it got unnervingly ungrotesque: (*) EllipticK[1/4 (2 + (-2 + t)/Sqrt[1 - t])] == (1 - t)^(1/4) EllipticK[t] How could anything this simple be new?
WDS: Let s=Sqrt[1-t]. Then (*) is rewritten as EllipticK[ 1/2 - (1+s^2)/(4s) ] == EllipticK[ -(1-s)^2/(4s) ] == s^(1/2) EllipticK[1-s^2] Now let s=1-m. Then this is EllipticK[ -m^2/(4(1-m)) ] == (1-m)^(1/2) EllipticK[(2-m)m]. E.Salamin: Complains WDS formula is wrong.
--WDS: arg. Well, I just started with Gosper's "ungrotesque" formula taking its correctness on faith, then worked by hand to produce mine. Doing same thing over again now: Gosper: EllipticK[1/4 (2 + (-2 + t)/Sqrt[1 - t])] == (1 - t)^(1/4) EllipticK[t] Let t=1-s^2: EllipticK[1/4 (2 - (1+s^2)/s)] == (s)^(1/2) EllipticK[1-s^2] EllipticK[(2s-1-s^2)/(4s)] == (s)^(1/2) EllipticK[1-s^2] EllipticK[-(1-s)^2/(4s)] == (s)^(1/2) EllipticK[1-s^2] which is same as I got before. Now let s=1-m: EllipticK[-m^2/(4(1-m))] == (1-m)^(1/2) EllipticK[1-(1-m)^2] EllipticK[-m^2/(4(1-m))] == (1-m)^(1/2) EllipticK[2m-m^2] EllipticK[-m^2/(4(1-m))] == (1-m)^(1/2) EllipticK[(2-m)m] which is same as I got before, OK? So either I made same error over again because my brain is truly decaying, or Gosper was wrong, or Salamin was wrong, you figure out which. I was just trying to produce a nicer equivalent version. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)