(I just knew someone would ask this someday...)
I had talked (ranted?) about this issue at some length here, so I'd like you to read that first. I'll just give an executive summary here: Mathematica uses the parameter convention, while the formula you found on Wikipedia is based on the modulus convention (quickly betrayed by the comma separating the two arguments).
In fact, even the convention in the first argument is different, so EllipticF[]
is not exactly the Mathematica function you want. What you need here, in Mathematica's convention, is denoted as InverseJacobiSN[]
:
D[InverseJacobiSN[Sqrt[x + 1], 1/2], x] /. x -> 1 // FullSimplify
ComplexInfinity
After looking through my Schwarz-Christoffel notes, I'm not sure where the Wikipedia formula was obtained. In any case, allow me to present a short demonstration of the conventional conformal mapping between the upper half plane and a rectangle (though I cheat here and use the inverse map, JacobiSN[]
, in the following code):
a = 1/2; b = 1; (* 2 a × b rectangle *)
m = ModularLambda[I b/a];
{ParametricPlot[With[{w = JacobiSN[EllipticK[m] (u + I v)/a, m]},
{Re[w], Im[w]}], {u, -a, a}, {v, 0, 99 b/100},
BoundaryStyle -> AbsoluteThickness[2], Mesh -> 15,
MeshShading -> {{Transparent}},
MeshStyle -> {Directive[AbsoluteThickness[2], ColorData[97, 1]],
Directive[AbsoluteThickness[2], ColorData[97, 2]]},
PlotRange -> {{-5, 5}, {0, 10}}],
ParametricPlot[{u, v}, {u, -a, a}, {v, 0, 99 b/100},
BoundaryStyle -> AbsoluteThickness[2], Mesh -> 15,
MeshShading -> {{Transparent}},
MeshStyle -> {Directive[AbsoluteThickness[2], ColorData[97, 1]],
Directive[AbsoluteThickness[2], ColorData[97, 2]]}]}
// GraphicsRow

As a check of correctness, try restricting the range of v
to {0, b/2}
. The mapping on the left should yield a semicircular region with radius m^(-1/4)
.