Computations - WR

Determination of auxiliary functions:

Solve[t == z/(1 - t), t]

t = 1/2*(1 - (Sqrt[1 - 4 z]));

Solve[d == z*d/(1 - t)^2 + z*D[t, z] - t, d] // 
      FullSimplify

d = ((2 (z^2))/((1 + (Sqrt[1 - 4 z])) (1 - 4 z)));

Solve[
        w == 
          d + z*w/(1 - t)^2 + 
            2  z^2*D[t, z]*(d + z*D[t, z])/(1 - t)^3, w] // 
      FullSimplify

w = ((z^2)/(((1 - 4 z))^2));

Solve[r1 == z/(1 - r1 - t), r1] // FullSimplify

r1 = 
        (1/4) (1 - (Sqrt[2 + 2 (Sqrt[1 - 4 z]) - 20 z]) + (Sqrt[1 - 4 z]));

Solve[r2 == z/(1 - t)^2*(r1 + r2), r2] // 
      FullSimplify

r2 = (((1 - (Sqrt[2]) (Sqrt[1 + (Sqrt[1 - 4 z]) - 10 z]) + (Sqrt[1 - 4 z])) z)/(2 (1 + (Sqrt[1 - 4 z]) - 4 z)));

{t, d, w, D[t, z], r1, r2, D[r1, z], D[r2, z]}

Clear[t, d, w, r1, r2]

We will write q1 for (Sqrt[1 - 4 z]) and q2 for (Sqrt[2 + 2 (Sqrt[1 - 4 z]) - 20 z])as abbreviations.

Solving a linear system in terms of wr1 and wr2

Solve[{dr1 == r1^2*(dr1 + d)/z + z*r1' - r1, 
            dr2 == 2  d*t*r2/z + t^2*(dr1 + dr2)/z + z*r2' - r2, 
            wr1 == 
              dr1 + r1^2*(wr1 + w)/z + 
                2  r1^3*(r1' + t')*(dr1 + d + z*r1' + z*t')/z, 
            wr2 == 
              dr2 + 2  t^3*w*(r1 + r2)/z^2 + t^2*(wr1 + wr2)/z + 
                2  t^3*(dr1 + dr2 + z*r1' + z*r2')*t'/z + 
                2  t^3*(r1' + r2')*(d + z*t')/z + 
                6  t^4*t'*(d + z*t')*(r1 + r2)/z^2}, {dr1, dr2, wr1, 
            wr2}] // Simplify;

wr1 + wr2 /. %[[1]] // Simplify;

Writing wr in terms of q1 and q2

% /. {t -> (1/2) (1 - q1), 
            d -> ((2 (z^2))/((1 + q1) q1^2)), w -> ((z^2)/q1)^4, 
            t' -> (1/q1), r1 -> (1/4) (1 - q2 + q1), 
            r2 -> (((1 - q2 + q1) z)/(2 q1 (1 + q1))), 
            r1' -> (1/4) (-((-20 - (4/q1))/(2 q2)) - 
                    (2/q1)), 
            r2' -> ((1 - q2 + q1)/(2 q1 (1 + q1))) - (((-4 - (2/q1)) (1 - q2 + q1) z)/(2 q1^2 (((1 + q1))^2))) + (((-((-10 - (2/q1))/q2) - (2/q1)) z)/(2 q1 (1 + q1)))} // Simplify;

% /. {z -> 1/4*(1 - q1^2)} // Simplify

% /. {q2 -> Sqrt[5  q1^2 + 2  q1 - 3]} // Simplify

Solve[{(8 (q1^6) (-3 + 
                  5 q1) (((3 - 5 q1 + (Sqrt[-3 + 2 q1 + 5 (q1^2)])))^2)) == 0, q1 == Sqrt[1 - 4  z]}, {q1, z}]

Expanding wr around the singularity

%% /. {q1 -> Sqrt[1 - 4  z]} // FullSimplify

Series[%, {z, 4/25, -1}]

The dominating part is thus given by (16/(9375 (((z - (4/25)))^2)))=(1/(15 (((1 - ((25  z)/4)))^2))), which gives the asymptotics we are searching for.

(1/(15*Gamma[2]))*4*Sqrt[Pi]

1/(4*4/25)