Chapter 10
Hitting the Ends

§10.5 Solving it Analytically: Conformal Mapping

§10.5.1 Solving the transcendental equation

p[ρ_, prec_] := 
      Block[{$MinPrecision = prec}, 
        p /. FindRoot[
            ArcCot[ρ] == ((p π)/2) + 
                Arg[EllipticK[(E^(2 I p π))]], {p, (10^(-8)), (10^(-3))}, AccuracyGoal -> prec, 
            WorkingPrecision -> prec]]

\(\({Precision == #, \(p\_*\) == 
              p[10, #]} &\) /@ {MachinePrecision, 17} // TableForm\)

\(\(p\_*\) == \((p /. 
          FindRoot[
            ArcCot[10] == \(p\ π\)\/2 - 
                Arg[ArithmeticGeometricMean[
                    1, \((1 - I)\) \(\@Sin[
                            p\ π]\) E\^\(I\ p\ π/2\)]], {p, 10\^\(-8\), 10\^\(-5\)}, 
            AccuracyGoal -> MachinePrecision])\)\)

§10.6 The Joy of Elliptic Functions

N[(2/π) ArcSin[(Sqrt[ModularLambda])[10  I]], 
      100]