On Tue, Sep 8, 2015 at 2:28 PM, Bill Gosper <billgosper@gmail.com> wrote:
Just a couple of days ago I wrote in a draft
"Here are some other K transformations that are immediate specializations of hypergeometric transformations: EllipticK[z]==EllipticK[1/2 - (1/4)*Sqrt[-((z - 2)^2/(z - 1))]]/(1 - z)^(1/4) "
Two misfortunes: This isn't quite right, and I lost its faulty derivation to a Mma front-end crash.
The rather fishy plot of the error difference,
Plot3D[Abs[Subtract @@ % /. z -> u + I*v], {u, 3/4, 9/4}, {v, -1, 1}, AxesLabel -> Automatic, PlotRange -> All]
is gosper.org/marlin.png .
I don't recall a prior case of two "analytic" functions agreeing everywhere but on a line segment. (In this case all the discrepancies have arg π/4.) --rwg
Maybe they're commonplace. Here's a nice little shark's fin: Plot3D[Abs[-Sqrt[(z/(-1 + z))] EllipticK[1/(1 - z)] + EllipticK[1/z]] /. z -> x + I*y, {x, -1, 2}, {y, -1, 1}, PlotRange -> All, AxesLabel -> Automatic] Remove the PlotRange -> All to see the effects of the cruel Chinese fishermen. --rwg