(* Elliptic Integral 1. Kind *)
    F[x_, k_] := f[x, Sqrt[k]];
   
    (* Jacobi Amplitude *)
    Am[x_, k_] := π/(2 K[Sqrt[k]]) x + 2 \!\(
    \*UnderoverscriptBox[\(\[Sum]\), \(n = 1\), \(5\)]
    \*FractionBox[\(
    \*SuperscriptBox[\(q[
    \*SqrtBox[\(k\)]]\), \(n\)] Sin[2\ n\ ξ[x,
    \*SqrtBox[\(k\)]]]\), \(n \((1 +
    \*SuperscriptBox[\(q[
    \*SqrtBox[\(k\)]]\), \(2 n\)])\)\)]\);
   
    (* Jacobi Elliptic Function *)
    Dn[x_, k_] := Sqrt[1 - k Sin[Am[x, k]]^2];
   
    (* Definitions *)
    f[f_, k_] := \!\(
    \*SubsuperscriptBox[\(\[Integral]\), \(0\), \(f\)]\(
    \*FractionBox[\(1\),
    SqrtBox[\(1 -
    \*SuperscriptBox[\(k\), \(2\)]\
    \*SuperscriptBox[\(Sin[u]\), \(2\)]\)]] \[DifferentialD]u\)\);
    K[k_] := f[π/2, k];
    q[k_] := Exp[(-π K[Sqrt[1 - k^2]])/K[k]];
    ξ[x_, k_] := (π x)/(2 K[k]);