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];

(*Use particular values for low-order functions*)
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]));

(*Use nesting property for higher-degree functions*)
r4[xi_, x_] := r2[r2[xi, xi], r2[xi, x]];

ellgain[xi_, w_, w0_, ep_] := 1/Sqrt[1 + ep^2*r4[xi, w/w0]^2];

Plot[
 ellgain[1.05, w, 1, 0.5],
 {w, 0, 3}]

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