DescriptionRational Elliptic Functions (xi varied, x=(-1,1)).svg	English: A graph of the fourth-degree rational elliptic function, R4(ξ,x) on the interval [−1,1]. ξ is the selectivity factor, and is varied in this graph.
Date	29 January 2009
Source	Own work
Author	Inductiveload
Permission (Reusing this file)	Public domainPublic domainfalsefalse I, the copyright holder of this work, release this work into the public domain. This applies worldwide. In some countries this may not be legally possible; if so: I grant anyone the right to use this work for any purpose, without any conditions, unless such conditions are required by law.

xp2[xi_] :=
  Module[{g, num, den},
   g = Sqrt[4*xi^2 + (4*xi^2*(xi^2 - 1))^(2/3)];
   num = 2*xi^2*Sqrt[g];
   den = Sqrt[8*xi^2*(xi^2 + 1) + 12*g*xi^2 - g^3] - Sqrt[g^3];
   num/den
   ];
xz2[xi_] := xi^2/xp2[xi];

t[xi_] := Sqrt[1 - 1/xi^2];

r1[xi_, x_] := x;
r2[xi_, x_] := ((t[xi] + 1)*x^2 - 1)/((t[xi] - 1)*x^2 + 1);
r3[xi_, x_] := 
  x*((1 - xp2[xi])*(x^2 - xz2[xi]))/((1 - xz2[xi])*(x^2 - 
        xp2[xi]));
r4[xi_, x_] :=
  Module[{num, den},
   num = (1 + t[xi]) (1 + Sqrt[t[xi]])^2*x^4 - 
     2 (1 + t[xi]) (1 + Sqrt[t[xi]])*x^2 + 1;
   den = (1 + t[xi]) (1 - Sqrt[t[xi]])^2*x^4 - 
     2 (1 + t[xi]) (1 - Sqrt[t[xi]])*x^2 + 1;
   num/den
   ];
   
Plot[
 Table[
  r4[xi, x],
  {xi, {1.0001, 1.001, 1.01, 1.1, Infinity}}],
 {x, -1, 1},
 Axes -> False]